1
Abstract
Let k be a number field, and let A ⊂ P1 (k) be a finite set of rational points. Deligne
and Goncharov have defined the motivic fundamental group π1mot (X, x) of X := P1 \ A
with base-point x being either a k-point of X or a tangential base-point. We extend
the construction of the motivic fundamental group to the setting of a smooth S-scheme
p : X → S with section x : S → X, in case S is itself smooth over a field, X satisfies
the Beilinson-Soulé vanishing conjectures and the motive of X in DM (S)Q is a mixed
Tate motive. Finally, letting Gal(MT(X)) be the Tannaka group of the Tannakian
category of mixed Tate motives over X, we identify π1mot (X, x) with the kernel of the
map p∗ : Gal(MT(X)) → Gal(MT(S)).
1
Tate motives and the fundamental group
Marc Levine
Dept. of Math.
Northeastern University
Boston, MA 02115
U.S.A.
[email protected]
October 13, 2008
Contents
1 Differential graded algebras
1.1 Adams graded cdgas . . . . . . . . . .
1.2 The bar construction . . . . . . . . . .
1.3 The category of cell modules . . . . . .
1.4 The derived category . . . . . . . . . .
1.5 Weight filtration . . . . . . . . . . . .
1.6 Bounded below modules . . . . . . . .
1.7 Tor and Ext . . . . . . . . . . . . . . .
1.8 Change of ring . . . . . . . . . . . . .
1.9 Finiteness conditions . . . . . . . . . .
1.10 Model structure . . . . . . . . . . . . .
1.11 Minimal models . . . . . . . . . . . . .
1.12 t-structure . . . . . . . . . . . . . . . .
1.13 Connection matrices . . . . . . . . . .
1.14 The homotopy category of connections
1.15 Summary . . . . . . . . . . . . . . . .
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9
9
10
12
13
14
18
20
21
22
23
24
25
31
32
37
2 Relative theory of cdgas
2.1 Definitions and model structure . . . . .
2.2 Path objects and the homotopy relation
2.3 Indecomposables . . . . . . . . . . . . .
2.4 Relative minimal models . . . . . . . . .
2.5 Relative bar construction . . . . . . . . .
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38
38
43
43
44
51
∗
The author gratefully acknowledges the support of the Humboldt Foundation and support of the NSF
via grant DMS-0457195
2
2.6
2.7
2.8
Base-change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connection matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Semi-direct products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Motives over a base
3.1 Effective motives over a base . . . . . .
3.2 T tr -spectra and the category of motives
3.3 Tensor product in SptS
T tr (S) . . . . . .
3.4 Motives with Q-coefficients . . . . . . .
3.5 Geometric motives . . . . . . . . . . .
3.6 Tate motives . . . . . . . . . . . . . . .
4 Cycle algebras
4.1 Cubical complexes . . . . . . . . . . .
4.2 The cycle cdga in DM eff (S) . . . . . .
4.3 Products and internal Hom in Shtr
Nis (S)
4.4 Equi-dimensional cycles . . . . . . . .
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53
54
55
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56
57
59
61
62
64
65
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68
69
71
73
77
78
78
84
85
87
90
93
5 N (S)-modules and motives
5.1 The contravariant motive . . . . . . .
5.2 The dual motive and cycle complexes
5.3 Cell modules and Tate motives . . .
5.4 Motives and NS -modules . . . . . . .
5.5 From cycle algebras to motives . . .
5.6 The cell algebra of an S-scheme . . .
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6 Motivic π1
6.1 Cosimplicial constructions . . . . .
6.2 The motive of a cosimplicial scheme
6.3 Motivic π1 . . . . . . . . . . . . . .
6.4 Simplicial constructions . . . . . .
6.5 The comparison theorem . . . . . .
6.6 The fundamental exact sequence . .
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94
. 94
. 96
. 98
. 99
. 99
. 101
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Introduction
In [11], P. Deligne defined the motivic fundamental group of X = P1 \{0, 1, ∞} over a number
field k as an object in the category of systems of realizations. This is a Tannakian category
over Q, which he constructed as tuples (Betti, de Rham, `-adic, crystalline), with compatibilities between them, a definition close to the one given by U. Jannsen [24]. The Betti-de
Rham component is the mixed Hodge structure, defined by J. Morgan [32], on the nilpotent completion limN Q[π top (X, x)]/I N of the topological fundamental group π1top (X(C), x),
←−
for all complex embeddings k ⊂ C, where the base-point x is either a point in X(k) or a
non-trivial tangent vector at x̄ ∈ (P1 \ X)(k).
3
A. Beilinson [12, proposition 3.4] showed that for any smooth complex variety X, and
for base-point x ∈ X(C), the ind-system
lim HomQ (Q[π top (X, x)]/I N , Q),
−→
N
which is a Hopf algebra over Q, arises from the cohomology of a cosimplicial scheme Px (X).
As pointed out by Z. Wojtkoviak [39], the Hopf algebra structure on limN (Q[π top (X, x)]/I N )∨
−→
similarly arises from operations on Px (X). These key results have many consequences. For
instance, one can use Px (X) to define the mixed Hodge structure on limN Q[π top (X, x)]/I N ,
←−
cf. [18]. Even more, the cosimplicial scheme Px (X), regardless of the geometry of X, defines
an ind-Hopf algebra object Hgm (Px (X)) in Voevodsky’s triangulated category of motives
DMgm (k)Q [15, chapter V]; here
Hgm : Sm/k op → DMgm (k)
is the “cohomological motive” functor, dual to the canonical functor Mgm : Sm/k →
DMgm (k). If in addition X is the complement in P1k of a finite set of k-rational points, then
Hgm (Px (X))Q is actually an ind-Hopf algebra in the full triangulated subcategory DMTgm (k)
of DMgm (k)Q spanned by the Tate objects Q(n).
As explained in [31], if a field k satisfies the Beilinson-Soulé vanishing conjecture, that
is, if the motivic cohomology H p (k, Q(q)) vanishes for p ≤ 0, q > 0, there is a t-structure
defined on DMTgm (k), the heart of which is the abelian category MT(k) of mixed Tate
motives over k. MT(k) is a Q-linear, abelian rigid tensor category with the structure of a
functorial exact weight filtration W∗ . Taking the associated graded object with respect to
W∗ defines a neutral fiber functor grW
∗ , endowing MT(k) with the structure of a Tannakian
category over Q.
By the work of Borel [6], we know that if k is a number field, then k does satisfy the
Beilinson-Soulé conjecture. Thus Beilinson’s construction allows one to define the ind-Hopf
algebra object H 0 (Hgm (Px (X)) in MT(k), if k is a number field. In [12, théorème 4.4] P.
Deligne and A. Goncharov show that the dual π1mot (X, x) of H 0 (Hgm (Px (X))Q ), which is
a pro-group scheme object in MT(k), yields Deligne’s original motivic fundamental group
upon applying the appropriate realization functors, in case x ∈ X(k) and X ⊂ P1k is the
complement of a finite set of k-points of P1 . In addition, they show that, even for a tangential base-point x, there is a pro-group scheme object π1mot (X, x) in MT(k) which maps to
Deligne’s motivic fundamental group under realization, without, however, making an explicit
construction of π1mot (X, x) in this case. Using this construction as starting point, they go on
to construct a motivic fundamental group for any unirational variety over the number field
k, as a pro-group scheme over the larger Tannakian category of Artin-Tate motives MAT(k)
(see [12] for details).
Using recent work of Cisinski-Déglise [10], one now has available a reasonable candidate
for the category of motives over a base X, at least if X is a smooth variety over a perfect
field k. The resulting triangulated category DM (X) has Tate objects ZX (n) which properly
compute the motivic cohomology of X (defined using Voevodsky’s category DMgm (k)). In
addition, if X ⊂ P1k is an open defined over a number field k, then the observation made
in [31] carries over to the full triangulated subcategory DMT(X) of the category DM (X)Q
generated by the Tate objects QX (n). Thus, assuming k is a number field, there is a heart
4
MT(X) ⊂ DMT(X) which is a Q-linear abelian rigid tensor category, and which receives
MT(k) by pull back via the structure morphism p : X → Spec k.
By Tannaka duality, we therefore have the Tannaka group schemes G(MT(X), grW
∗ ) and
∗
∗
)
over
Q,
and
the
functors
p
:
MT(k)
→
MT(X),
x
:
MT(X)
→
MT(k)
give
G(MT(k), grW
∗
a canonical split short exact sequence
/
/K
1
G(MT(X), grW
∗ )o
p∗
x∗
/
G(MT(k), grW
∗ )
/ 1,
where K is defined as the kernel of p∗ . The splitting x∗ also defines an action of the Tannaka
group G(MT(k), grW
∗ ) on K, which lifts the Q group-scheme K to a group-scheme object Kx
in MT(k).
In [12, section 4.19], Deligne and Goncharov use the group-scheme π1mot (X, x) over MT(k)
to define MT(X) as the category of MT(k) representations of π1mot (X, x). In [12, section 4.22]
they ask about the relationship between MT(k(P1 )), defined as above as a subcategory of
Voevodsky’s category DM (k(P1 ))Q , and limX⊂P1 MT(X) (this is the formulation for k = Q̄,
−→
in general, one needs to use the Artin-Tate motives MAT). The purpose of this article is
to give an answer to this question in the following form: the intrinsic definition of MT(X)
mentioned above is equivalent to the category of Kx -representations in MT(k), assuming
P1 \ X consists of k-rational points.
We now describe our main result for X as above.
Theorem 1 Let k be a number field, A ⊂ X(k) a finite (possibly empty) set of k-points
of P1 , let X := P1 \ A and take x ∈ X(k). Then the pro-group scheme objects Kx and
π1mot (X, x) are isomorphic as pro-group-schemes in MT(k).
The equivalence of MT(X) with the category of Kx -representations in MT(k) follows directly
from this.
In fact, we have a more general result. Let S be a smooth k-scheme, and let X → S
be a smooth S-scheme with a section x : S → X. One can easily extend the construction
of π1mot (X, x) to this setting, if we assume that X satisfies the Beilinson-Soulé vanishing
conjectures, and, in addition, that the motive of X in DM (S)Q is in the Tate subcategory
DMT(S) (see definition 6.3.1 for details). Note that S also satisfies the Beilinson-Soulé vanishing conjectures, as the section x identifies H ∗ (S, Q(∗)) with a summand of H ∗ (X, Q(∗)).
Replacing MT(k) with MT(S), we have the split exact sequence as above
1
/
K
/
G(MT(X), grW
∗ )o
p∗
x∗
/
G(MT(S), grW
∗ )
/
1.
defining the pro-group scheme object Kx of MT(S). Our main result in this more general
setting is
Theorem 2 Suppose X satisfies the Beilinson-Soulé vanishing conjectures, and suppose that
the motive of X in DM (S)Q is in the Tate subcategory DMT(S) of DM (S)Q . Then the progroup scheme objects Kx and π1mot (X, x) are isomorphic as pro-group-schemes in MT(S).
We now explain the ideas that go into the proof. In [2] S. Bloch and I. Kriz construct a
group-scheme GBK (k) over Q, by applying the bar construction to the cycle algebra Nk :=
5
Q ⊕ ⊕r≥1 Nk (r). The rth component Nk (r) of Nk is a shifted, alternating version of Bloch
cycle complex,
Nkm (r) = z r (k, 2r − m)Alt ;
the alternation makes the product on Nk strictly graded-commutative. The additional grading r is the Adams grading. The reduced bar construction gives us the Adams graded Hopf
algebra H 0 (B̄(Nk )) and GBK (k) is the pro group scheme Spec H 0 (B̄(Nk )). Bloch-Kriz define
the category of “Bloch-Kriz” mixed Tate motives over k, MTBK (k), as the finite dimensional
graded representations of GBK (k) in Q-vector spaces.
In [26], I. Kriz and P. May consider, for an Adams graded commutative differential graded
f
algebra (cdga) A = Q · id ⊕ ⊕r≥1 A(r) over Q, the “bounded” derived category DA
of Adams
f
graded dg A modules. DA
admits a functorial exact weight filtration, arising from the Adams
f
grading; in case A is cohomologically connected, DA
has a t-structure, defined by pulling
f
b
back the usual t-structure on DQf ∼
⊕
D
(Q)
via
the
functor
M 7→ M ⊗LA Q from DA
to DQf .
= n
f
In particular, they define the heart HA
. Next, assuming A cohomologically connected, they
construct an exact functor
f
ρ : Db co-repfQ (H 0 (B̄(A))) → DA
where co-repfQ (H 0 (B̄(A))) is the category of graded co-representations of H 0 (B̄(A)) in finitef
dimension Q-vector spaces. Furthermore, they show that ρ identifies the categories HA
and
f
0
co-repQ (H (B̄(A))) (although ρ is not in general an equivalence). For those who prefer
group-schemes to Hopf algebras, let GA := Spec H 0 (B̄(A)). Then GA is a pro-affine group
scheme over Q with Gm action, and co-repfQ (H 0 (B̄(A))) is equivalent to the category of
graded representions of GA in finite dimensional Q-vector spaces.
Taking A = Nk , and noting that the Beilinson-Soulé vanishing conjectures for k are
equivalent to the cohomological connectedness of A, this gives an equivalence of the heart
f
HN
with the Bloch-Kriz mixed Tate motives MTBK (k).
k
M. Spitzweck [37] (see [29, section 5] for a detailed account) defines an equivalence
f
θk : DN
→ DMT(k) ⊂ DMgm (k)Q
k
for k an arbitrary field. In addition, under the assumption that k satisfies the BeilinsonSoulé conjectures, or equivalently, that Nk is cohomologically connected, θk restricts to an
equivalence
f
θk : HN
→ MT(k).
k
From the discussion above, this gives an equivalence of co-repfQ (H 0 (B̄(Nk ))) with MT(k),
and in fact identifies GBK (k) n Gm as the Tannaka group of (MT(k), gr∗W ).
Our first task is to extend this picture from k to X. To this aim, one defines the cycle
algebra N (X) by replacing k with X in the definition of Nk and modifying the construction
further by using complexes of cycles which are equi-dimensional over X. This yields an
Adams graded cdga over Q together with a map of Adams graded cdgas p∗ : N (k) → N (X)
arising from the structure morphism p : X → Spec k. Essentially the same construction as
for k gives an equivalence
f
θX : DN
(X) → DMT(X) ⊂ DM (X)Q
6
(∗)
and if X satisfies the Beilinson-Soulé vanishing conjectures, θX restricts to an equivalence
f
HN
(X) ∼ MT(X). Defining the Q pro-group scheme GBK (X) as above,
GBK (X) := GN (X) = Spec (H 0 (B̄(NX ))),
we also have the equivalence of MT(X) with the graded representations of GBK (X) in finite
dimensional Q-vector spaces, giving the identification of GBK (X) n Gm with the Tannaka
W
W
group of (MT(X), grW
∗ ), and identifying p∗ : G(MT(X), gr∗ ) → G(MT(k), gr∗ ) with the
map
p̃ × id : GBK (X) n Gm → GBK (k) n Gm
induced from p∗ : N (k) → N (X).
A k-point x of X gives an augmentation x : N (X) → N (k). We discuss the general
0
theory of augmented cdgas in section 2, leading to the relative bar construction HN
(B̄N (A, ))
f
of a cdg N algebra A with augmentation : A → N , as an ind-Hopf algebra in HN
. Let
0
GA/N () = Spec HN (B̄N (A, )) and let GA/N ()Q be the pro-group scheme over Q gotten
f
from GA/N () by applying the fiber functor grW
∗ : HN → VecQ . Note that Tannaka duality
gives a canonical action of GN on GA/N ()Q .
Of course, in order to make a reasonable relative bar construction, one needs to use a
good model for A as an N -algebra. This is provided by using the relative minimal model
A{∞}N of A over N , for which the derived tensor product is just the usual tensor product.
In section 2.8, especially theorem 2.8.3, we show that
1. GA/N ()Q = Spec H 0 (B̄(A{∞}N ⊗N Q)).
2. There is an exact sequence of pro-group schemes over Q:
p∗
→ GN → 1
1 → GA/N ()Q → GA −
The splitting ∗ to p∗ defines a splitting ∗ : GN → GA to p∗ .
3. The conjugation action of GN on GA/N ()Q given by the splitting ∗ is the same as the
canonical action.
To do this, we use an alternate description of dg modules over an Adams graded cdga
+
N
,
L that of flat dg connections. Kriz and May describe dg modules M over N as N :=
r>0 N (r)-valued connections over M ⊗N Q (for the canonical augmentation N → Q). Writ+
ing A{∞}+
N as N ⊕ I, with this decomposition coming from the augmentation A{∞}N →
0
0
N , the absolute (i.e. A{∞}+
N -valued) connection on H (B̄(A)) = H (B̄(A{∞}N )) induces
0
a N + -valued connection on H 0 (B̄(A{∞}N ⊗N Q)). Similarly, the structure of HN
(B̄N (A, ))
f
+
as an ind-Hopf algebra in HN gives an N -valued connection on
0
HN
(B̄N (A, )) ⊗N Q = H 0 (B̄(A{∞}N ⊗N Q)).
Using this description, it is easy to make the identifications necessary for proving (1)-(3)
above. H. Esnault has interpreted this argument as saying that GA/N () is the Gauß-Manin
connection of GA associated to A/N .
Applying this theory to the splitting x : N (X) → N (k), the Q pro-group scheme K, and
the lifting Kx to a MT(k) pro-group scheme, gives us the isomorphism of pro-group schemes
K∼
= Spec H 0 (B̄(N (X){∞}N (k) ⊗N (k) Q))
7
f
and the isomorphism of pro-group scheme objects in HN
(k)
0
Kx ∼
(B̄N (k) (N (X), x )).
= Spec HN
(∗∗)
0
One can make the dg N (k)-module HN
(B̄N (k) (N (X), x )) explicit as an object in MT(k)
via Spitzweck’s theorem. This relies on a crucial property of the transformation from dg
N (k) modules to motives (see theorem 5.6.2 for a more general statement):
Take X ∈ Sm/k. If the motive of X in DM (k)Q is in DMT(k) and X satisfies the BeilinsonSoulé vanishing conjectures, then the motive of NX {∞}N (k) is canonically isomorphic to
Hgm (X)Q .
The explicit decription of the Beilinson simplicial scheme underlying the Deligne-Goncharov
construction, together with this essential fact, allows one to conclude that Kx with its MT(k)
structure induced by the Gauß-Manin connection is precisely π1mot (X, x), when x comes from
a rational point x ∈ X(k) (see sections 6.5 and 6.6). In other words, we have the isomorphism
of pro-group schemes over MT(k):
0
π1mot (X, x) ∼
(B̄Nk (NX , x )).
= Spec HN
Combining this with our identification (∗∗) proves theorem 1. Replacing k with a more
general base-scheme S ∈ Sm/k, the program outlined above proves theorem 2.
In this article, we do not consider the case of the base-point x being a non-trivial tangent
vector at some point x̄ ∈ P1 \ X. As mentioned above, Deligne-Goncharov [12] show in this
case as well that the motivic π1 , defined by Deligne [11] as a system of realizations, comes
from MT(k). This defines π1mot (X, x) as an object in MT(k), but does not give a direct
construction in MT(k). However, the results of [28] give a section x to p∗ : N (k) → N (X)
(in the homotopy category of cdgas) for tangential base-points x as well as for k-points, so
we do have a relative bar construction available even for tangential base-points. In order
to extend our main theorem 6.6.1 to this case, one should define realization functors on the
categories of Tate motives, described as dg modules over the cycle algebra, and check that
0
the realization of Spec HN
(B̄Nk (NX , x )) agrees with Deligne’s motivic π1 .
Outline: The paper is organized as follows: We begin in section 1 with a review of the
theory of dg modules over an Adams-graded cdga, following for the most part the discussion of Kriz-May [26], but adding some new material dealing with the category of “weightbounded” modules. In section 2 we describe an extension of the classical model structure
on cdgas over a field of characteristic zero (cf. [7]) to the category of cdgas over a cdga.
This enables us to extend the theory of minimal models and the bar construction to the
relative case. We conclude this section with our main result on the relative bar construction,
theorem 2.8.3. In fact, the reader who is moderately familiar with the Kriz-May theory of
dg modules over a cdga could simply skim the first two sections to absorb our notation, and
accept theorem 2.8.3 on faith for the first reading.
We then proceed to a review of the recently available theory of motives over a basescheme, due to Cisinski-Déglise [9, 10], in section 3. Next, in section 4, we take a look
at generalizations of the Bloch-Kriz cycle algebra to a functorial construction for smooth
8
schemes over k, modifying a construction of Joshua [25]. In section 5, we describe the “cohomological motive” functor to the Cisinski-Déglise category and show how a Q-version of this
functor can be described using the cycle algebra. This section is the technical heart of the
paper. In it, we prove our main results relating motives and cycle algebras: our generalization of Spitzweck’s representation theorem, theorem 5.3.2, identifying the derived category
of dg modules over the cycle algebra N (S) to the triangulated category of Tate motives
over S, and our two main results relating the cycle complex of a smooth S-scheme X to the
geometric motive of X, theorem 5.5.3 and theorem 5.6.2. We put everything together in section 6, giving our generalization of the Deligne-Goncharov motivic π1 and proving our main
results, theorem 1 and theorem 2 (these are corollary 6.6.2 and theorem 6.6.1, respectively).
Acknowledgements: Together with H. Esnault, we gave a seminar in the winter 2006-7 at
the university of Duisburg-Essen on [12], to try to understand the constructions and results
of Deligne-Goncharov, as well as the various constructions of mixed Tate motives and the
relationships between them, as developed in the works of Bloch, Bloch-Kriz, Kriz-May and
Spitzweck, and summarized in [29]; this paper is to a large extent a product of that seminar.
We thank all the seminar participants for their willingness to give talks. In particular we
thank Phùng Hô Hai for various discussions on Tannakian categories.
Most importantly, this paper is a revision of a joint work with Hélène Esnault [14]. This
joint work also contained a proof of theorem 1, with proof along the same lines as the one
given here. It was Esnault who had originally suggested relating the Deligne-Goncharov
motivic π1 to the Bloch-Kriz cycle Hopf algebras as a way of answering the question of
Deligne and Goncharov on the relation of MT(k(t)) to π1 (X, x)-representations in MT(k).
This paper would never have existed had it not been for the many fruitful discussions and
numerous insights Esnault has shared with us; we take this opportunity to thank her for her
crucial contribution to this work. Finally, we would like to thank the referee for making a
number of useful suggestions.
1
Differential graded algebras
We fix notation and recall some basic facts on commutative differential graded algebras
(cdgas) over Q. This material is taken mainly from [26], with some refinements and additions.
In what follows a cdga will always mean a cdga over Q.
1.1
Adams graded cdgas
Definition 1.1.1 (1) A cdga (A∗ , d, ·) (over Q) consists of a unital, graded-commutative Qalgebra (A∗ := ⊕n∈Z An , ·) together with a graded homomorphism d = ⊕n dn , dn : An → An+1 ,
such that
1. dn+1 ◦ dn = 0.
2. dn+m (a · b) = dn a · b + (−1)n a · dm b; a ∈ An , b ∈ Am .
A∗ is called connected if An = 0 for n < 0 and A0 = Q · 1, cohomologically connected if
H n (A∗ ) = 0 for n < 0 and H 0 (A∗ ) = Q · 1.
9
(2) An Adams graded cdga is a cdga A together with a direct sum decomposition into
subcomplexes A∗ := ⊕r≥0 A∗ (r) such that A∗ (r) · A∗ (s) ⊂ A∗ (r + s). In addition, we require
that A∗ (0) = Q · id. An Adams graded cdga is said to be (cohomologically) connected if the
underlying cdga is (cohomologically) connected.
For x ∈ An (r), we call n the cohomological degree of x, n := deg x, and r the Adams
degree of x, r := |x|.
Note that an Adams graded cdga A has a canonical augmentation A → Q with augmentation
ideal A+ := ⊕r>0 A∗ (r).
1.2
The bar construction
We let Ord denote the category with objects the sets [n] := {0, . . . , n}, n = 0, 1, . . ., and
morphisms the non-decreasing maps of sets. The morphisms in Ord are generated by the
coface maps δin : [n] → [n + 1] and the codegeneracy maps σin : [n] → [n − 1], where δin is
the strictly increasing map omitting i from its image and σin is the non-decreasing surjective
map sending i and i + 1 to i. For a category C, we have the categories of cosimplicial objects
in C and simplicial objects in C, namely, the categories of functors Ord → C and Ordop → C,
respectively. For a cosimplicial object X : Ord → C, we often write δin and σin for the coface
maps X(δin ) and X(σin ), and for a simplicial object S : Ordop → C, we often write dni and
sni for the face and degeneracy maps S(δin ) and S(σin ).
Let A be a cdga. We begin by defining the simplicial cdga B• (A) as follows: Tensor
product (over Q) is the coproduct in the category of cdgas, so for a finite set S, we have
A⊗S , giving the functor A⊗? from finite sets to cdgas. Thus, if we have a simplicial set S
such that S[n] is a finite set for all n, we may form the simplicial cdga A⊗S , n 7→ A⊗S[n] . We
have the representable simplicial sets ∆[n] := HomOrd (−, [n]); setting [0, 1] := ∆[1] gives us
the simplicial cdga
B• (A) := A⊗[0,1] .
The two inclusions [0] → [1] define the maps i0 , i1 : ∆[0] → ∆[1]. Letting {0, 1} denote
the constant simplicial set with two elements, the maps i0 , i1 give rise to the map of simplicial
sets i0 q i1 : {0, 1} → [0, 1], which makes B• (A) into a simplicial A ⊗ A = A⊗{0,1} algebra.
Suppose we have augmentations 1 , 2 : A → Q. Define B̄• (A, 1 , 2 ) by
B̄• (A, 1 , 2 ) := B• (A) ⊗A⊗A Q
using 1 ⊗ 2 : A ⊗ A → Q as structure map. Since B̄n (A, 1 , 2 ) is a complex for each
n, we can form a double complex by using the usual alternating sum of the face maps
dni : B̄n+1 (A, 1 , 2 ) → B̄n (A, 1 , 2 ) as the second differential, and let B̄(A, 1 , 2 ) denote
the total complex of this double complex. We use cohomological grading throughout, so
B̄n (A, 1 , 2 )m has total degree m − n. For 1 = 2 = , we write B̄(A, ) or simply B̄(A); this
is the reduced bar construction for (A, ). As is usual, we denote a decomposable element
x1 ⊗ . . . ⊗ xn of B̄(A) by [x1 |, . . . |xn ]. Note that
X
deg([x1 | . . . |xm ]) = −m +
deg(xi ).
i
10
The bar construction B̄ := B̄(A) has the following structures: a differential d : B̄ → B̄
of degree +1 coming from the differential in A, a product
∪ : B̄ ⊗ B̄ → B̄
[x1 | . . . |xp ] ∪ [xp+1 | . . . |xp+q ] =
X
sgn(σ)[xσ(1) | . . . |xσ(p+q) ]
σ
where the sum is over all (p, q) shuffles σ ∈ Sp+q (and the sign is the graded sign of σ, taking
into account the degrees of the xi ) , a co-product
δ : B̄ → B̄ ⊗ B̄
δ([x1 | . . . |xn ]) :=
n
X
(−1)i deg([xi+1 |...|xn ]) [x1 | . . . xi ] ⊗ [xi+1 | . . . |xn ]
i=0
and an involution
ι : B̄ → B̄,
ι([x1 |x2 | . . . |xn−1 |xn ]) := (−1)m [xn |xn−1 | . . . |x2 |x1 ]; m =
X
deg(xi ) · deg(xj ),
1≤i<j≤n
making (B̄(A), d, ∪, δ, ι) a differential graded Hopf algebra over Q, which is graded-commutative with respect to the product ∪. The cohomology H ∗ (B̄(A)) is thus a graded Hopf
algebra over Q, in particular, H 0 (B̄(A)) is a commutative Hopf algebra over Q.
Let I(A) be the kernel of the augmentation H 0 (B̄(A)) → Q induced by . The coproduct
δ on H 0 (B̄(A)) induces the structure of a co-Lie algebra on γA := I(A)/I(A)2 .
From the formula for the coproduct, we see that, modulo tensors of degree < m, we have
δ([x1 | . . . |xm ]) = 1 ⊗ [x1 | . . . |xm ] + [x1 | . . . |xm ] ⊗ 1
This implies that the pro-affine Q-algebraic group G := Spec H 0 (B̄(A)) is pro-unipotent. In
addition, in case A is cohomologically connected, H 0 (B̄(A)) is, as a Q-algebra, a polynomial
algebra with indecomposables γA (see, e.g., [2, theorem 2.30, corollary 2.31]).
Suppose A = ⊕r≥0 A∗ (r) is an Adams graded cdga, with canonical augmentation :
A → Q. The Adams grading on A induces an Adams grading on B• (A) and thus on B̄(A);
explicitly B̄(A) has the Adams grading B̄(A) = ⊕r≥0 B̄(A)(r) where the Adams degree of
[x1 | . . . |xm ] is
X
|[x1 | . . . |xm ]| :=
|xj |.
j
0
0
Thus H (B̄(A)) = ⊕r≥0 H (B̄(A)(r)) becomes an Adams graded Hopf algebra over Q, commutative as a Q-algebra. We also have the Adams graded co-Lie algebra γA = ⊕r>0 γA (r).
Remark 1.2.1 Let A be a cohomologically connected Adams graded cdga. The Adams
grading equips the pro-unipotent affine Q group scheme G := Spec H 0 (B̄(A)) with a grading,
or, equivalently, with a Gm -action. Thus γA is a positively graded nilpotent co-Lie algebra,
and there is an equivalence of categories between the continuous graded co-representations
of H 0 (B̄(A)) in finite dimensional graded Q-vector spaces, co-repfQ (H 0 (B̄(A))), and the
continuous graded co-representations of γA in finite dimensional graded Q-vector spaces,
co-repfQ (γA ).
11
1.3
The category of cell modules
Kriz and May [26] define a triangulated category directly from an Adams graded cdga A
without passing to the bar construction or using a co-Lie algebra. We recall some of their
work here, with some extensions.
Let A∗ be a graded algebra over Q. We let A[n] be the left A∗ -module which is Am+n
in degree m, with the A∗ -action given by left multiplication. If A∗ (∗) = ⊕n≥0,r≥0 An (r) is
a bi-graded Q-algebra, we let A<r>[n] be the left A∗ (∗)-module which is Am+n (r + s) in
bi-degree (m, s), with action given by left multiplication.
Definition 1.3.1 Let A be a cdga.
(1) A dg A-module (M ∗ , d) consists of a complex M ∗ = ⊕n M n of Q-vector spaces with
differential d, together with a graded, degree zero map A∗ ⊗Q M ∗ → M ∗ , a ⊗ m 7→ a · m,
which makes M ∗ into a graded A∗ -module, and satisfies the Leibniz rule
d(a · m) = da · m + (−1)deg a a · dm; a ∈ A∗ , m ∈ M ∗ .
(2) If A = ⊕r≥0 A∗ (r) is an Adams graded cdga, an Adams graded dg A-module is a dg
A-module M ∗ together with a decomposition into subcomplexes M ∗ = ⊕s M ∗ (s) such that
A∗ (r) · M ∗ (s) ⊂ M ∗ (r + s). We say x ∈ M ∗ has Adams degree s if x ∈ M ∗ (s), and write this
as |x| = s.
(3) An Adams graded dg A-module M is a cell module if
(a) M is free as a bi-graded A-module, where we forget the differential structure. That
is, there is a set J and elements bj ∈ M nj (rj ), j ∈ J, such that the maps a 7→ a · bj
induces an isomorphism of bi-graded A-modules
⊕j∈J A<−rj >[−nj ] → M.
(b) There is a filtration on the index set J:
J−1 = ∅ ⊂ J0 ⊂ J1 ⊂ . . . Jn ⊂ . . . ⊂ J
such that J = ∪∞
n=0 Jn and for j ∈ Jn ,
X
dbj =
aij bi .
i∈Jn−1
A finite cell module is a cell module with finite index set J.
We denote the category of dg A-modules by MA , the A-cell modules by CMA and the
finite cell modules by CMfA .
12
1.4
The derived category
Let A be an Adams graded cdga and let M and N be Adams graded dg A-modules. Let
HomA (M, N ) be the Adams graded dg A-module with HomA (M, N )n (r) the A-module
consisting of maps f : M → N with f (M a (s)) ⊂ N a+n (s + r), f (am) = (−1)np af (m)
for a ∈ Ap and m ∈ M , and with differential d defined by df (m) = d(f (m))(−1)n f (dm)
for f ∈ Hom(M, N )n (r). Similarly, let M ⊗A N be the Adams graded dg A-module with
underlying module M ⊗A N and with differential d(m ⊗ n) = dm ⊗ n + (−1)deg m m ⊗ dn.
For f : M → N a morphism of Adams graded dg A-modules, we let Cone(f ) be the
Adams graded dg A-module with
Cone(f )n (r) := N n (r) ⊕ M n+1 (r)
and differential d(n, m) = (dn + f (m), −dm). Let M [1] be the Adams graded dg A-module
with M [1]n (r) := M n+1 (r) and differential −d, where d is the differential of M . A sequence
of the form
f
j
i
M→
− N→
− Cone(f ) →
− M [1]
where i and j are the evident inclusion and projection, is called a cone sequence.
Definition 1.4.1 Let A be an Adams graded cdga over Q. We let MA denote the category
of Adams graded dg A-modules, KA the homotopy category, i.e. the objects of KA are the
objects of MA and
HomKA (M, N ) = H 0 (HomA (M, N )(0)).
The category KA is a triangulated category, with distinguished triangles those triangles
which are isomorphic in KA to a cone sequence.
Definition 1.4.2 The derived category DA of dg A-modules is the localization of KA with
respect to morphisms M → N which are quasi-isomorphisms on the underlying complexes
of Q-vector spaces. For M in DA , we denote the nth cohomology of M , as a complex of
Q-vector spaces, by H n (M ).
We define the homotopy category of A-cell modules, resp. finite cell modules, as the full
subcategory of KA with objects in CMA , resp. in CMfA ,
KCMfA ⊂ KCMA ⊂ KA .
Note that for A = Q, MQ is just the category of complexes of graded Q-vector spaces,
and DQ is the unbounded derived category of graded Q-vector spaces.
Proposition 1.4.3 ([26, construction 2.7]) Let A be an Adams graded cdga. Then the
evident functor
KCMA → DA
is an equivalence of triangulated categories. Explicitly, let f : M 0 → M be a quasi-isomorphism in MA , N ∈ CMA . Then the induced map
f : HomKA (N, M 0 ) → HomKA (N, M )
is an isomorphism.
13
f
We let DA
⊂ DA be the full subcategory with objects those M isomorphic in DA to a
finite cell module. As an immediate consequence of proposition 1.4.3, we have
f
Proposition 1.4.4 KCMfA → DA
is an equivalence of triangulated categories.
Example 1.4.5 (Tate objects) For n ∈ Z, let Q(n) be the object of CMfA which is the
free rank one A-module with generator bn having Adams degree −n, cohomological degree
0 and dbn = 0, i.e., Q(n) = A<n>. We sometimes write QA (n) for Q(n); Q(n) is called a
Tate object.
1.5
Weight filtration
Let M be an Adams graded dg A-module which is free as a bi-graded A-module. Choose a
basis B := {bj | j ∈ J}, M = ⊕j A · bj . Write
X
dbj =
aij bi ; aij ∈ A.
i
Since |aij | ≥ 0 and d has Adams degree 0, it follows that
|bi | ≤ |bj | if aij 6= 0.
Thus, we have the subcomplex
WnB M = ⊕{j,
|bj |≤n} A
· bj
of M .
The subcomplex WnB M is independent
of the choice of basis: if B 0 = {b0j } is another basis
P
0
0
and if |bj | = n, then as bj = i eij bi and |eij | ≥ 0, it follows that b0j ∈ WnB M and hence
0
0
WnB M ⊂ WnB M . By symmetry, WnB M ⊂ WnB M . We may thus write Wn M for WnB M .
This gives us the increasing filtration as an Adams graded dg A-module
W∗ M : . . . ⊂ Wn M ⊂ Wn+1 M ⊂ . . . ⊂ M
with M = ∪n Wn M .
Similarly, for n ≥ n0 , define Wn/n0 M as the cokernel of the inclusion Wn0 M → Wn M ,
i.e., Wn/n0 M is the Adams graded dg A-module with basis the bj having n0 < |bj | ≤ n and
>n
with differential induced by the differential in Wn M . We write grW
for
n for Wn/n−1 and W
W∞/n .
It is not hard to see that Wn M is functorial in M . In particular, if f : M → M 0 is a
homotopy equivalence of cell modules with homotopy inverse g : M 0 → M , then f and g
restricted to Wn M and Wn M 0 give inverse homotopy equivalences Wn f : Wn M → Wn M 0 ,
Wn g : Wn M 0 → Wn M . Thus the W filtration in CMA defines a functorial tower of endofunctors on KCMA :
. . . → Wn → Wn+1 → . . . → id
(1.5.1)
Lemma 1.5.1 1. The endo-functor Wn is exact for each n.
2. For n0 ≤ n ≤ ∞, the sequence of endo-functors Wn0 → Wn → Wn/n0 canonically extends to a distinguished triangle of endo-functors.
14
Proof For (1), it follows directly from the definition that Wn transforms a cone sequence
into a cone sequence. For (2), take M ∈ CMA . The sequence
0 → Wn0 M → Wn M → Wn/n0 M → 0
is split exact as a sequence of bi-graded A-modules. Thus (2) follows from the general fact
that a sequence in CMA
p
i
0 → N0 →
− N→
− N 00 → 0
that is split exact as a sequence of bi-graded A-modules extends canonically to a distinguished
triangle in KCMA . To see this, choose a splitting s to p (as bi-graded A-modules), and define
t : N 00 → N 0 [1] by i ◦ t = s ◦ dN 00 − dN ◦ s. It is then easy to check that t is a map of complexes
and (s, t) : N 00 → N ⊕ N 0 [1] defines the map of complexes
(s, t) : N 0 → Cone(i)
making the diagram
N0
N0
/
i
i
/
N
N
p
/
/
N 00
/
N 0 [1]
/
N 0 [1]
t
(s,t)
Cone(i)
commute. In particular, (s, t) is an isomorphism in KCMA . One sees similarly that another
choice s0 of splitting leads to a homotopic map (s0 , t0 ).
Note that it is not necessary for M to be a cell module to define Wn M ; being free as
a bi-graded A-module suffices. However, it is not clear that Wn M is a quasi-isomorphism
invariant in general. To side-step this issue, we use instead
Definition 1.5.2 Define the tower of exact endo-functors on DA
. . . → Wn → Wn+1 → . . . → id
using (1.5.1) and the equivalence of categories in proposition 1.4.3. We define Wn/n0 , grW
n
and W >n on DA similarly.
Remark 1.5.3 Since KCMA → DA is an equivalence of triangulated categories, the natural
distinguished triangles
Wn0 → Wn → Wn/n0 → Wn0 [1]
in KCMA give us natural distinguished triangles
Wn0 → Wn → Wn/n0 → Wn0 [1]
in DA .
One uses the weight filtration for inductive arguments, for example:
Lemma 1.5.4 Let M be a finite A-cell module. Suppose N is a summand of M in DA .
Then there is a finite A-cell module M 0 with N ∼
= M 0 in DA .
15
Proof By proposition 1.4.3 there is an isomorphism N 0 ∼
= N in DA with with N 0 an object in
CMA . Thus we may assume that N is a cell module. Since KCMA → DA is an equivalence,
N is a summand of M in KCMA . Write M = N ⊕ N 0 in KCMA and let p : M → M be the
projection M → N followed by the inclusion N → M .
Since M is finite, there is a minimal n with Wn M 6= 0. Thus Wn−1 N is homotopy
equivalent to zero and N ∼
= W∞/n−1 N in KCMA . Hence, we may assume that Wn−1 N = 0
in CMA . Similarly, we may assume that M = Wn+r M and N = Wn+r N in CMA for some
r ≥ 0. We proceed by induction on r.
As A∗ (0) = Q·id, it follows that Wn M = A⊗Q M0 for a finite complex of finite dimensional
graded Q-vector spaces M0 . Indeed, choose a finite bi-graded A-basis {bj } for Wn M and
let M0 be the finite dimensional Q-vector space
Pspanned by the bj . Since Wn−1 M = 0, all
the bj have Adams degree n. Writing dbj = i aij bi and noting that the differential has
Adams degree 0, it follows that |aij | = 0 for all i, j, i.e., aij ∈ Q · id. Consequently M0 is a
subcomplex of M and Wn M = A ⊗Q M0 as an Adams graded dg module.
But such an M0 is homotopy equivalent to the direct sum of its cohomologies; replacing
M0 with ⊕n H n (M0 )[−n] and changing notation, we may assume that dM0 = 0. Thus Wn M =
A ⊗Q M0 for M0 a finite dimensional bi-graded Q-vector space; using again the fact that
A(r) = 0 for r < 0 and A(0) = Q · id, we see that Wn p = id ⊗ q with q : M0 → M0 an
idempotent endomorphism of the bi-graded Q-vector space M0 . Thus Wn N ∼
= A ⊗ im(q),
hence Wn N is homotopy equivalent to a finite A-cell module. This also takes care of the
case r = 0.
Using the distinguished triangle
Wn N → N → Wn+r/n N → Wn N [1]
we may replace N with the shifted cone of the map Wn+r/n N → A⊗im(q)[1]. Since Wn+r/n N
is a summand of Wn+r/n M , it follows by induction on r that Wn+r/n N is homotopy equivalent
to a finite cell module, hence the cone of Wn+r/n N → A ⊗ im(q) is homotopy equivalent to
a finite cell module as well.
+w
Definition 1.5.5 Let DA
⊂ DA be the full subcategory of DA with objects M such that
∼
Wn M = 0 for some n. Similarly, let CM+w
A ⊂ CMA be the full subcategory with objects M
+w
such that Wn M = 0 for some n and let KCM+w
A be the homotopy category of CMA .
+w
Lemma 1.5.6 1. The natural map KCM+w
A → KCMA is an equivalence of KCMA with
the full subcategory of KCMA with objects the M such that Wn M ∼
= 0 in KCMA for n << 0.
+w
2. The equivalence KCMA → DA induces an equivalence KCM+w
A → DA .
+w
Proof Since KCM+w
A is the homotopy category of the full subcategory CMA of CMA ,
∼
the functor KCM+w
A → KCMA is a full embedding. Suppose that Wn M = 0 in KCMA . We
>n
have the cell module W M and the distinguished triangle
Wn M → M → W >n M → Wn M [1]
in KCMA . Thus the map M → W >n M is an isomorphism in KCMA ; since W >n M is in
+w
CM+w
A , the essential image of KCMA in KCMA is as described.
16
For (2), following definition 1.5.2, Wn M is defined by choosing an isomorphism P → M
in DA with P ∈ CMA and taking Wn M := Wn P . Since Wn P = Wn M ∼
= 0 in DA , it follows
+w
+w
∼
that Wn P = 0 in KCMA , so P is isomorphic to an object in KCMA . Thus DA
is the
essential image of KCM+w
in
D
.
Since
KCM
→
D
is
an
equivalence,
this
proves
(2).
A
A
A
A
+w
Remark 1.5.7 Take M ∈ DA
. Then there is an n0 such that Wn M ∼
= 0 for all n ≤ n0 .
>n0
M
is
an isomorphism in
Indeed, by definition, Wn0 M ∼
0
for
some
n
.
Thus
M
→
W
=
0
>n0
∼
M = 0 is an isomorphism in DA .
DA . If n < n0 , then Wn M → Wn W
Another result using induction on the weight filtration is
Lemma 1.5.8 Let M be an Adams graded dg A-module.
1. M is a finite A-cell module if and only if M is free and finitely generated as a bi-graded
A-module.
2. M is in CM+w
if and only if M is free as a bi-graded A-module and there is an inA
teger r0 such that |m| ≥ r0 for all m ∈ M .
Proof We first prove (1). Clearly a finite A-cell module is free and finitely generated as a
bi-graded A-module. Conversely, suppose M is free and finitely generated over A; choose a
basis B for M .
Clearly WnB M = 0 for n << 0; let N be the minimum integer n such that WnB M 6= 0
and let BN be the set of basis elements b of Adams degree N . Since A(0) = Q · id, it follows
that BN forms a Q basis for WN M and the differential on BN is given by
X
deα =
aαβ eβ
β
with aαβ ∈ Q and eβ ∈ BN . Changing the Q basis for WNB M , we may assume that the subset
0
BN
of BN of eα such that deα = 0 forms an Q basis for the kernel of d on the Q-span of BN .
Since d2 = 0, the two-step filtration
0
⊂ BN
BN
exhibits WN M as a finite cell module.
The result follows by induction on the length of the weight filtration: By induction
WB>N M := M/WnB M is a finite cell module with basis say {b0j | j ∈ J} for some filtration
on J. Since M = WNB M ⊕ WB>N M as an A-module, we just take the union of the two bases,
and the concatenation of the filtrations, to present M as a finite cell module.
The proof of (2) is similar. In fact, the same proof as for (1) shows that the sub-dg
B
A-module WnB M of M is in CM+w
A for all n and that we may find an A basis Bn for Wn M
and a filtration
∅ = Bnr0 −1 ⊂ Bnr0 ⊂ . . . ⊂ Bn2n−1 ⊂ Bn2n = Bn
that exhibits WnB M as a cell module. In addition, we may assume that Bi with its filtration
is just Bn2i with the induced filtration, for all i ≤ n. Thus, taking the union of the Bn gives
the desired basis for M , showing that M is in CM+w
A .
17
1.6
Bounded below modules
+
Definition 1.6.1 Let DA
⊂ DA be the full subcategory with objects the Adams graded dg
n
A-modules M having H (M ) = 0 for n << 0, as usual, we call such an M bounded below.
Lemma 1.6.2 Suppose that A is cohomologically connected, and M is an Adams graded dg
A-module with H n (M ) = 0 for n < n0 . Then there is a quasi-isomorphism P → M with P
an A-cell module having basis {eα } with deg(eα ) ≥ n0 for all α. If in addition there is an r0
such that H n (M )(r) = 0 for all (r, n) with r < r0 , we may find P → M as above satisfying
the additional condition |eα | ≥ r0 for all α.
Proof We first note the following elementary facts: Let V = ⊕n,r V n (r) be a bi-graded
Q-vector space, which we consider as a complex with zero differential. Then the complex
A ⊗Q V is a cell-module, since a bi-graded Q basis for V gives a bi-graded A basis with 0
differential. In addition, the map v 7→ 1 ⊗ v gives a map
V n := ⊕r V n (r) → H n (A ⊗ V ).
Finally, suppose there is an n0 such that V n0 6= 0 but V n = 0 for all n < n0 . Then as
H n (A) = 0 for n < 0 and H 0 (A) = Q, the map
V n → H n (A ⊗Q V )
is an isomorphism for all n ≤ n0 .
We begin the construction of P → M by taking V to be a bi-graded Q subspace of
⊕n≥n0 M n representing ⊕n H n (M ), giving the map of Adams graded dg A modules
φn0 : P0 := ⊕n≥n0 A ⊗ H n (M )[−n] → M.
From the discussion above, we see that φn0 is an isomorphism on H n for n ≤ n0 and a
surjection on H n for n > n0 . If in addition there is an r0 such that H n (M )(r) = 0 for r < r0
and all n, then P0 has a bi-graded A-basis S0 with |v| ≥ r0 for each v ∈ S0 .
Suppose by induction we have constructed a sequence of inclusions of A-cell modules
P0 → P1 → . . . → Pr−1
and maps of Adams graded dg A-modules
φn0 +i : Pi → M
with the following properties:
1. The Pi have A-bases S(i) := S0 q . . . q Si . In addition, for all i ≥ 1, all the elements
in Si are of cohomological degree n0 + i − 1, and for v ∈ Si , dv is in Pi−1 .
2. The map Pi → Pi+1 is the one induced by the inclusion S(i) ⊂ S(i + 1).
3. φn0 +i : Pi → M induces an isomorphism on H n for n ≤ n0 + i and a surjection for all
n.
18
4. If H n (M )(r) = 0 for r < r0 and all n, then v ∈ S(i) has Adams degree |v| ≥ r0 .
nr
be a biWe now show how to continue the induction. For this, let nr = n0 +r and let V ⊂ Pr−1
graded Q-subspace of representatives for the kernel of the surjection H nr (Pr−1 ) → H nr (M ).
Let
Pr := Pr−1 ⊕ A ⊗Q V
as bi-graded A-module, where the differential is given by using the differential on Pr−1 ,
setting
nr
d((0, 1 ⊗ v)) = (v, 0) ∈ Pr−1
nr
, there is an
for v ∈ V and extending by the Leibniz rule. Note that, for v ∈ V ⊂ Pr−1
nr −1
mv ∈ M
with dmv = φr−1 (v); chosing a bi-graded Q-basis Sr for V and extending the
assignment s 7→ ms from Sr to all of V by Q-linearity, we have a Q-linear map
f : V → M nr −1
with d(f (v)) = φr−1 (v) for all v ∈ V . Thus, we may define the map of dg A-modules
φr : Pr → M
by using φr−1 on Pr , f on 1 ⊗ V and extending by A-linearity. Clearly Pr is an A-cell module
with A-basis S(r) := S(r − 1) q Sr .
In case H n (M )(r) = 0 for r < r0 and all n, clearly all bi-homogeneous elements of V
have Adams degree ≥ r0 , so |v| ≥ r0 for all v ∈ Sr .
We can compute the cohomology of Pr by using the sequence of A-cell modules
0 → Pr−1 → Pr → A ⊗Q V → 0,
where we consider V as a complex with zero differential, which is split exact as a sequence of
bi-graded A-modules. The resulting long exact cohomology sequence shows that Pr−1 → Pr
induces an isomorphism in cohomology H n for n < nr − 1 and we have the exact sequence
∂
0 → H nr −1 (Pr−1 ) → H nr −1 (Pr ) → V →
− H nr (Pr−1 ) → H nr (Pr ) → 0.
In addition, one can compute the coboundary ∂ by lifting the element 1 ⊗ v ∈ (A ⊗Q V )nr −1
to the element (0, 1 ⊗ v) ∈ Prnr −1 and applying the differential dPr . From this, we see that
the sequence
∂
0→V →
− H nr (Pr−1 ) → H nr (Pr ) → 0
is exact, hence H nr −1 (Pr−1 ) → H nr −1 (Pr ) is an isomorphism. This also shows that φr : Pr →
M induces an isomorphism on H n for n ≤ nr and the induction continues.
If we now take P to be the direct limit of the Pr , it follows that P is an A-cell module
with basis elements all in cohomological degree ≥ n0 , and that the map φ : P → M induced
from the φr is a quasi-isomorphism. If there is an r0 such that H ∗ (M )(r) = 0 for r < r0 ,
then by (4) above, the basis S := ∪r S(r) clearly has |e| ≥ r0 for all e ∈ S. This completes
the proof.
19
1.7
Tor and Ext
The Hom functor HomA (M, N ) and tensor product functor M ⊗A N define bi-exact bifunctors
HomA : KCMop
A ⊗ KCMA → DA
⊗A : KCMA ⊗ KCMA → KCMA .
Via proposition 1.4.3, these give well-defined derived functors of HomA and ⊗A :
op
RHomA : DA
⊗ DA → DA
⊗LA : DA ⊗ DA → DA .
Restricting to KCMfA , we have the derived functors for the finite categories
f op
f
f
RHomA : DA
⊗ DA
→ DA
f
f
f
⊗LA : DA
⊗ DA
→ DA
.
In both settings, these bi-functors are adjoint:
RHomA (M ⊗L N, K) ∼
= RHomA (M, RHomA (N, K)).
+w
We have as well the restriction of ⊗L to DA
:
+w
+w
+w
⊗LA : DA
⊗ DA
→ DA
.
The derived tensor product makes DA into a triangulated tensor category with unit
f
+w
+
1 := A and DA
, DA
and DA
are triangulated tensor subcategories. By lemma 1.5.4, Df is
+w
closed under taking summands in DA ; this property is obvious for DA
.
∨
Define M := RHomA (M, A) and call M strongly dualizable if the canonical map M →
M ∨∨ is an isomorphism in DA . Note that M is strongly dualizable if M is rigid, i.e., there
exists an N ∈ DA and morphisms δ : A → M ⊗LA N and : N ⊗LA M → A such that
(idM ⊗ ) ◦ (δ ⊗ idM ) = idM
(idN ⊗ δ) ◦ ( ⊗ idN ) = idN
We have
f
Proposition 1.7.1 ([26, theorem 5.7]) M ∈ DA is rigid if and only if M is in DA
, i.e.,
∼
M = N in DA for some finite A-cell module N .
The precise statement found in [26, theorem 5.7] is that M is rigid if and only if M is a summand in DA of some finite cell module, so the proposition follows from this and lemma 1.5.4;
Kriz and May are working in a more general setting in which lemma 1.5.4 does not hold.
Example 1.7.2 For n ≥ 0, Q(±n) ∼
= (Q(±1))⊗n and for all n, Q(n)∨ ∼
= Q(−n).
20
1.8
Change of ring
If φ : A → A0 is a homomorphism of Adams graded cdgas, we have the functor
− ⊗A A0 : MA → MA0
which induces a functor on cell modules and the homotopy category
φ∗ : KCMA → KCMA0 .
Via proposition 1.4.3, we have the change of rings functor
φ∗ : DA → DA0
on the derived category. By proposition 1.4.3 and lemma 1.5.6, the respective restrictions of
φ∗ define exact tensor functors
+w
+w
φ∗ : DA
→ DA
0
f
f
φ∗ : DA
→ DA
0.
From [26] we have
Theorem 1.8.1 ([26, proposition 4.2]) If φ is a quasi-isomorphism, then
φ∗ : DA → DA0
is an equivalence of triangulated tensor categories.
Noting the φ∗ is compatible with the weight filtrations, the theorem immediately yields
Corollary 1.8.2 If φ is a quasi-isomorphism, then
+w
+w
φ∗ : DA
→ DA
0
is an equivalence of triangulated tensor categories.
In addition, we have
Corollary 1.8.3 If φ is a quasi-isomorphism, then
f
f
φ∗ : DA
→ DA
0
is an equivalence of triangulated tensor categories.
Proof Since an equivalence of tensor triangulated categories induces an equivalence on the
subcategories of rigid objects, the result follows from theorem 1.8.1 and proposition 1.7.1.
+w
+w
Proposition 1.8.4 Let φ : A → B be a map of cdgas. Then φ∗ : DA
→ DB
is conserva∼
∼
tive, i.e., φ∗ (M ) = 0 implies M = 0, or equivalently, if φ∗ (f ) is an isomorphism then f is
an isomorphism.
21
Proof Take M ∈ D+w , and let
S := {n | M ∼
= W >n M }.
Then S 6= ∅; we claim that either M ∼
= 0 or S has a maximal element. Indeed, if S has no
∼
maximum then Wn M = 0 for all n. But since
lim Wn M → M
−→
n
is an isomorphism, this implies that M is acyclic, hence M ∼
= 0 in DA .
Thus, we may find a cell module P and quasi-isomorphism P → M such that Wn−1 P = 0,
but Wn P is not acyclic. In particular P has a basis {eα } with |eα | ≥ n for all α. If |eα | = n
then since there are no basis elements with Adams grading < n, we have
X
deα =
aαj ej
j
with |aαj | = 0, |ej | = n, i.e., aαj ∈ Q = A(0). Since Wn P is not acyclic, it thus follows that
(Wn P ) ⊗A Q is also not acyclic: if (Wn P ) ⊗A Q were acyclic, this complex would be zero in
the homotopy category KCMQ , which would make Wn P 0 in KCMA . As Wn (P ⊗A B) =
(Wn P ) ⊗A B and
(Wn P ) ⊗A Q = (Wn P ⊗A B) ⊗B Q
it follows that P ⊗A B is not isomorphic to zero in KCMB , and thus φ∗ (M ) is non-zero in
+w
DB
.
Example 1.8.5 Each Adams graded cdga A has a canonical augmentation : A → Q, given
by projection on A0 (0) = Q · id.
In particular, we have the functor
q := ∗ : CMA → MQ , qM := M ⊗A Q
and the exact tensor functors
q : DA → DQ ,
+w
q +w : DA
→ DQ+w ,
f
q f : DA
→ DQf .
Explicitly, q is given on the derived level by qM := M ⊗LA Q.
1.9
Finiteness conditions
MQ is just the category of graded Q-vector spaces, so DQ is equivalent to the product of the
unbounded derived categories
Y
DQ ∼
D(Q).
=
n∈Z
Similarly
DQf ∼
= ⊕n∈Z Db (Q),
22
where Db (Q) is the bounded derived category of finite dimensional Q-vector spaces. Finally,
[Y
Y
D(Q) ⊂
DQ+w ∼
D(Q).
=
N n≥N
n∈Z
Remark 1.9.1 The inclusion Q → A splits , identifying DQ , DQ+w , etc., with full subcate+w
gories of DA , DA
, etc. Under this identification, and the decomposition of DQ intoQ
its Adams
W
graded pieces described above, the functor q is identified with the functor gr∗ := n∈Z grW
n .
Indeed, if P is an A-cell module with basis {eα }, then as A(r) = 0 for r < 0 and A(0) = Q·id,
the differential d decomposes as d = d0 + d+ with
X
X
d0 e α =
a0αβ eβ , d+ eα =
a+
αβ eβ
β
β
+
where |a0αβ | = 0, |a+
αβ | > 0. Since d has Adams degree 0, it follows that |eβ | < |eα | if aαβ 6= 0,
and |eβ | = |eα | if a0αβ 6= 0. Thus grW
∗ P is the complex of graded Q-vector spaces with Q
0
basis {eα } and with dgrW
e
=
d
e
.
α As qP has exactly the same description, we have the
∗ P α
W
identification of gr∗ and q as described.
f
+w
Lemma 1.9.2 Let M be in DA
. Then M is in DA
if and only if
b
1. grW
n M is in D (Q) ⊂ D(Q) for all n.
∼
2. grW
n M = 0 for all but finitely many n.
f
Proof It is clear that M ∈ DA
satisfies the conditions (1) and (2). Conversely, suppose
+w
M ∈ DA satisfies (1) and (2). If M ∼
= 0, there
Q is nothing to prove, so assume M is not
isomorphic to 0. By proposition 1.8.4, qM = n grW
n M is not isomorphic to zero. Take N
W
minimal such that grN M is not isomorphic to zero. By (2), there is a maximal N 0 such that
grW
N 0 M is not isomorphic to zero.
b
∼ s
If N = N 0 , then M ∼
= grW
N M is in D (Q) by (1), hence M = ⊕i=1 A<−N >[mi ], and thus
f
M is in DA . In general, we apply remark 1.5.3, giving the distinguished triangle
>N
grW
→ grW
NM → M → M
N M [1];
f
f
>N ∼
note that grW
= 0 for n > N 0 . By induction on N 0 − N , M >N is in DA ; since DA is a
n M
f
full triangulated subcategory of DA , closed under isomorphism, it follows that M is in DA
.
1.10
Model structure
Let cdga denote the category of Adams graded commutative differential graded algebras
over Q. In the non-Adams graded case, Bousfield and Guggenheim [7] have defined a model
structure on cdgas with weak equivalences the quasi-isomorphisms. As we are interested in
possibly non-connected Adams graded cdgas, we modify their definitions slightly.
Definition 1.10.1 1. A morphism φ : A → B in cdga is a weak equivalence if φ induces an
isomorphism
φ∗ : H n (A(r)) → H n (B(r))
23
for all n, r ≥ 1.
2. A morphism φ : A → B in cdga is a fibration if φ(r) : A(r)n → B(r)n is surjective
for all n, r ≥ 1
3. A morphism φ : A → B in cdga is a cofibration if φ has the left lifting property
with respect to acyclic fibrations.
The proof that this defines a model structure on cdga is word for word the same as the proof
in [7, chapter 4]; we will give details of the proof in §2, where we discuss a more general
situation. We denote the homotopy category of cdga by H(cdga).
1.11
Minimal models
Definition 1.11.1 A cdga A is said to be generalized nilpotent if
1. As a graded Q-algebra, A = Sym∗ E for some Z-graded Q-vector space E, i.e., A =
Λ∗ Eodd ⊗ Sym∗ Eev . In addition, En = 0 for n ≤ 0.
2. For n ≥ 0, let A(n) ⊂ A be the subalgebra generated by the elements of degree ≤ n. Set
A(n+1,0) = A(n) and for q ≥ 0 define A(n+1,q+1) inductively as the subalgebra generated
by A(n) and
n+1
An+1
(n+1,q+1) := {x ∈ A(n+1) |dx ∈ A(n+1,q) }.
Then for all n ≥ 0,
A(n+1) = ∪q≥0 A(n+1,q) .
Note that a generalized nilpotent cdga is automatically connected.
Definition 1.11.2 Let A be a cdga. An n-minimal model of A is a map of cdgas
s : A{n} → A,
with A{n} generalized nilpotent and generated (as an algebra) in degrees ≤ n, such that s
induces an isomorphism on H m for 1 ≤ m ≤ n and an injection on H n+1 .
Remark 1.11.3 Let s : A{n} → A be an n-minimal model of A. Then A{n}(n−1) ⊂ A{n} is
clearly generalized nilpotent and the inclusion in A{n} is an isomorphism in degrees ≤ n − 1.
Thus H p (A{n}(n−1) ) → H p (A{n}) is an isomorphism for p ≤ n − 1 and injective for p = n,
and hence s : A{n}(n−1) → A is an n − 1-minimal model.
Define the above notions for Adams graded cdgas by giving everything an Adams grading.
By lemma 2.4.4, a generalized nilpotent cdga is cofibrant, so the minimal model s :
A{∞} → A is a cofibrant replacement, that is, s is a weak equivalence and A{∞} is cofibrant.
Theorem 1.11.4 Let A be an Adams graded cdga.
1. For each n = 1, 2, . . . , ∞, there is an n-minimal model of A: A{n} → A.
2. If ψ : A → B is a quasi-isomorphism of Adams graded cdgas, and s : A{n} → A,
t : B{n} → B are n-minimal models, then there is an isomorphism of Adams graded cdgas,
φ : A{n} → B{n} such that ψ ◦ s is homotopic to t ◦ φ.
24
See [7, chapter 4] for a proof in the non-Adams graded case; the Adams graded case is exactly
the same, where one proceeds by a double induction, first with respect to the Adams degree
and then with respect to the cohomological degree. For details, theorem 1.11.4(1) is a special
case of proposition 2.4.9 and theorem 1.11.4(2) is a special case of proposition 2.4.14.
Corollary 1.11.5 If A is cohomologically connected, there is a quasi-isomorphism of Adams
graded cdgas A0 → A with A0 connected. Similarly, if φ : A → B is a map of cohomologically
connected Adams graded cdgas, there is a diagram of Adams graded cdgas
/
A0
A
φ
B0
/B
that commutes up to homotopy, with the vertical maps being quasi-isomorphisms, such that
A0 and B 0 are connected.
Proof For the first assertion, just take A0 = A{∞}. For the second, let B 0 = B{∞}. Since
φ : A{∞} → A is a quasi-isomorphism of A-cell modules, φ is a homotopy equivalence of
A-cell modules (proposition 1.4.3), so taking the tensor product yields a quasi-isomorphism
A{∞} ⊗A B → B.
Clearly A{∞}⊗A B is a generalized nilpotent cdga, so we need only apply theorem 1.11.4(2).
This result, together with theorem 1.8.1, corollary 1.8.2 and corollary 1.8.3, allows us to
+w
replace “cohomologically connected” with “connected” in statements involving DA , DA
or
f
DA .
1.12
t-structure
f
+w
To define a t-structure on DA
or DA
, one needs to assume that A is cohomologically
connected; by corollaries 1.8.2 or 1.8.3, we may assume that A is connected. Recall from
example 1.8.5 the functor
q := ∗ : CMA → MQ
associated to the augmentation : A → Q, and its extension to exact tensor functors on the
various derived categories.
≤0
≥0
+w
Define full subcategories DA
, DA
and HA of DA
by
≤0
+w
DA
:= {M ∈ DA
| H n (qM ) = 0 for n > 0}
≥0
+w
| H n (qM ) = 0 for n < 0}
DA
:= {M ∈ DA
+w
HA := {M ∈ DA
| H n (qM ) = 0 for n 6= 0}.
≤0
≥0
+w
with
The arguments of Kriz-May [26] show that this defines a t-structure (DA
, DA
) on DA
+
+w
heart HA . Since Kriz-May use DA instead of DA , we give a sketch of the argument here,
with the necessary modifications.
25
Q
Remark 1.12.1 As we have identified the functor q with n grW
n (remark 1.9.1) we can
≤0
+w
m
W
describe the category DA as the M ∈ DA such that H (grn M ) = 0 for all m > 0 and all
≥0
n. We have a similar description of DA
and HA .
Recall that an essentially full subcategory B of a category A is a full subcategory such
that, if b → a is an isomorphism in A with b in B, then a is in B.
Definition 1.12.2 We recall from [5] that a t-structure on a triangulated category D consists
of essentially full subcategories (D≤0 , D≥0 ) of D such that
1. D≤0 [1] ⊂ D≤0 , D≥0 [−1] ⊂ D≥0
2. HomD (M, N [−1]) = 0 for M in D≤0 , N in D≥0
3. Each M ∈ D admits a distinguished triangle
M ≤0 → M → M >0 → M ≤0 [1]
with M ≤0 in D≤0 , M >0 in D≥0 [−1].
Write D≤n for D≤0 [−n] and D≥n for D≥0 [−n].
A t-structure (D≤0 , D≥0 ) is non-degenerate if A ∈ ∩n≤0 D≤n , B ∈ ∩n≥0 D≥n imply A ∼
=
0∼
B.
=
Lemma 1.12.3 Suppose that A is connected.
≤0
1. Take M in DA
. Then there is an A-cell module P ∈ CM+w
with basis {eα } such
A
that deg(eα ) ≤ 0 for all α, and a quasi-isomorphism P → M .
≥0
2. For N ∈ DA
, there is an A-cell module P ∈ CM+w
A with basis {eα } such that deg(eα ) ≥ 0
for all α, and a quasi-isomorphism P → N .
Proof For (1) choose a quasi-isomorphism Q → M with Q in CM+w
A . Let {eα } be a basis
+
0
for Q. Decompose the differential dQ as dQ = dQ + dQ as in remark 1.9.1. Making a Q-linear
change of basis if necessary, we may assume that the collection S0 of eα with deg eα = 0 and
d0Q eα = 0 forms a basis of
ker[d0 : ⊕deg eα =0 Qeα → ⊕deg eβ =1 Qeβ ].
Let τ ≤0 Q be the A submodule of Q with basis {eα | deg eα < 0} ∪ S0 . We claim that τ ≤0 Q
is a subcomplex of Q. Indeed, we have
dQ eα = d0Q eα + d+
Q eα
X
X
=
a0αβ eβ +
a+
αβ eβ
β
β
+
with |a0αβ | = 0 = deg a0αβ , |a+
αβ | > 0. Since A is connected, deg aαβ ≥ 1. As dQ has
cohomological degree +1, it follows that deg eβ ≤ deg eα if a+
αβ 6= 0. Similarly, deg eβ =
0
deg eα + 1 if aαβ 6= 0.
26
Take eα with deg eα = −1. Since d2Q = 0, it follows that (d0Q )2 = 0, from which it follows
that eβ is in S0 if a0αβ 6= 0. Now take eα ∈ S0 . Write
X
X
0
deα =
b+
f
+
b+
αβ β
αβ fβ
deg b+
αβ =1
deg b+
αβ >1
∗≥1
with the {b+
, the fβ in the Q span of the degree ≤ −1
αβ } being chosen Q independent in A
0
part of the basis {eα } and the fβ in Q span of the degree 0 part of {eα }. We have
X
0
0
0 = d2 eα =
b+
αβ d (fβ ) + . . .
deg bαβ =1
with the . . . involving only the degree ≤ 0 part of the basis (and coefficients from A). Since
0 0
0
the b+
αβ are Q independent, we have d fβ = 0 for all β in the first sum, hence the fβ are in
the Q-span of S0 . Thus τ ≤0 Q is a subcomplex of Q, as claimed.
So far we have only needed that Q is a cell module. We will now use that Q lies in
≤0
CM+w
Q → Q is a quasi-isomorphism. By proposition 1.8.4 applied to
A . We claim that τ
the augmentation A → Q, the functor
+w
q : DA
→ DQ+w
is conservative, thus it suffices to see that qτ ≤0 Q → qQ is a quasi-isomorphism. Now,
qQ represents qM ∈ DQ , and by assumption qM is in DQ≤0 , hence qQ is in DQ≤0 . But by
construction qτ ≤0 Q → qQ is an isomorphism on H n for all n ≤ 0. Since H n (qτ ≤0 Q) = 0 for
n > 0, it follows that qτ ≤0 Q → qQ is a quasi-isomorphism, as desired.
For (2), we may assume that N is an object in CM+w
A and thus Wr0 −1 N = 0 for some
r0 . The result then follows from lemma 1.6.2.
≤0
Lemma 1.12.4 Suppose that A is connected. Then HomD+w (M, N [−1]) = 0 for M in DA
,
A
≥0
N in DA
.
Proof By lemma 1.12.3 we may assume that M and N [−1] are A-cell modules with bases
{eα } for M and {fβ } for N [−1] satisfying deg eα ≤ 0 and deg fβ ≥ 1 for all α, β. By
lemma 1.5.6, we also have
HomD+w (M, N [−1]) = HomKCM+w (M, N [−1]).
A
A
But if φ : M → N [−1] is a map in KCM+w
A , then φ is given by a degree 0 map of complexes,
so
X
φ(eα ) =
aαβ fβ
β
for aαβ ∈ A with deg(aαβ ) + deg(fβ ) = deg(eα ) Since Ai = 0 for i < 0, this is impossible.
+w
Lemma 1.12.5 Suppose that A is connected. For M ∈ DA
, there is a distinguished triangle
M ≤0 → M → M >0 → M ≤0 [1]
≤0
≥1
with M ≤0 in DA
, M >0 in DA
.
27
Proof We may assume that M is in CM+w
A . We perform exactly the same construction as
in the proof of lemma 1.12.3, giving us a sub A-cell module τ ≤0 M of M such that
(a) τ ≤0 M has a basis {eα } with deg eα ≤ 0 for all α
(b) The map qτ ≤0 M → qM induced by applying q to τ ≤0 M → M gives an isomorphism
on H n for n ≤ 0.
Let M ≤0 = τ ≤0 M and let M >0 be the cone of τ ≤0 M → M . This gives us the distinguished
triangle
M ≤0 → M → M >0 → M ≤0 [1]
+w
+w
in DA
. By construction, M ≤0 is in DA
. Applying q to the distinguished triangle gives the
+w
distinguished triangle in DQ
qM ≤0 → qM → qM >0 → qM ≤0 [1];
by (b) and the fact that H 1 (qM ≤0 ) = 0, it follows that H n (qM >0 ) = 0 for n ≤ 0. Thus M >0
≥1
is in DA
, as desired.
≤0
≥0
Theorem 1.12.6 Suppose A is cohomologically connected. Then (DA
, DA
) is a non+w
degenerate t-structure on DA
.
Proof Replacing A with its minimal model, we may assume that A is connected. The
property (1) of definition 1.12.2 is obvious; properties (2) and (3) follow from lemmata 1.12.4
and 1.12.5, respectively.
≤n
For A ∈ ∩n≤0 DA
, it follows that H n (qA) = 0 for all n, i.e., qA ∼
= 0 in DQ+w . Since q is
≥n
+w
conservative, A ∼
. The case of B ∈ ∩n≥0 DA
is similar, hence the t-structure is
= 0 in DA
non-degenerate.
f,≤0
f
f,≥0
f
f
f
≤0
≥0
Definition 1.12.7 Let DA
:= DA
∩ DA
, DA
:= DA
∩ DA
, HA
:= HA ∩ DA
=
f,≤0
f,≥0
DA ∩ DA .
f,≤0
f,≥0
Corollary 1.12.8 If A is cohomologically connected, then (DA
, DA
) is a non-degenerate
f
f
t-structure on DA with heart HA .
f
+w
is a full triangulated subcategory of DA
, closed under isomorphisms in
Proof Since DA
+w
DA
, all the properties of a non-degenerate t-structure are inherited from the non-degenerate
≤0
≥0
+w
given by theorem 1.12.6, except perhaps for the condition
t-structure on (DA
, DA
) on DA
f
≤0
≥0
+w
(3) of definition 1.12.2. So, take M ∈ DA
. Since (DA
, DA
) is a t-structure on DA
, we
have a distinguished triangle
M ≤0 → M → M >0 → M ≤0 [1]
≤0
≥0
with M ≤0 in DA
, M >0 in DA
[−1]. Applying the exact functor grW
n (see remark 1.5.3) gives
the distinguished triangle
≤0
W
>0
≤0
grW
→ grW
→ grW
[1]
n M
n M → grn M
n M
28
≤0
in the derived category of Q-vector spaces D(Q), such that grW
is in D(Q)≤0 and
n M
>0
≤0
>0
grW
is in D(Q)≥1 , i.e., H n (grW
) = 0 for n > 0, H n (grW
) = 0 for n ≤ 0.
n M
n M
n M
f
W
b
However, since M is in DA , it follows that grn M is in D (Q) for all n and is isomorphic
to 0 for all but finitely many n (lemma 1.9.2). The long exact cohomology sequence for a
>0
≤0
are in Db (Q) for all
and grW
distinguished triangle in D(Q) thus shows that grW
n M
n M
n and are isomorphic to zero for all but finitely many n. Applying lemma 1.9.2 again shows
f
M ≤0 and M >0 are in DA
.
f
Lemma 1.12.9 (1) The restriction of ⊗L to HA and HA
makes these into abelian tensor
categories.
f
+w
(2) The weight filtrations on DA
and DA
restrict to define exact functorial filtrations on
f
HA and HA .
f
f
(3) HA
is the smallest abelian subcategory of HA
containing the Tate objects Q(n), n ∈ Z
f
and closed under extensions in HA .
Proof (1) is more or less obvious: for cell modules M and N , we have q(M ⊗A N ) ∼
=
≤0
≥0
qM ⊗Q qN ; the Künneth formula for H n (qM ⊗Q qN ) thus shows that DA
and DA
are
closed under ⊗LA .
For (2), note that the augmentation : A → Q is a homomorphism of Adams graded
cdgas, and that q = ∗ . Thus q is compatible with the weight filtrations on DA and DQ (and
also on the finite categories). In particular, we have
∼ W
q(grW
n M ) = grn qM.
On the other hand, for C in DQ+w we have
C∼
= ⊕m H m (C)[−m]
m
Furthermore H m (C) is isomorphic to its associated weight graded ⊕n grW
n H (C). All this
implies that
≤0
≤0
M is in DA
⇐⇒ grW
n M is in DA for all n
≥0
≤0
≥0
+w
and similarly for DA
. Thus, the t-structure (DA
, DA
) on DA
induces a t-structure
≤0
≥0
+w
+w
. The
(Wn DA , Wn DA ) on the full subcategory Wn DA with objects the Wn M , M ∈ DA
≥0
same holds for DA , from which it follows that the truncation functors τ≤0 , τ≥0 associated
≤0
≥0
with the t-structure (DA
, DA
) commute with the functors Wn . This proves (2).
f
T
For (3), we argue by induction on the weight filtration. Let HA
⊂ HA
be any full abelian
f
subcategory containing all the Q(n) and closed under extension in HA . Since A(0) = Q · id,
f
f
the full subcategory DA
(−n) of DA
consisting of M with M ∼
= grW
n M is equivalent to the
bounded derived category of (ungraded) finite dimensional Q-vector spaces, Db (Q), with
f
the equivalence sending a complex C to Q(−n) ⊗Q C. The t-structure on DA
restricts to a
f
t-structure on DA
(−n) which is equivalent to the standard t-structure on Db (Q).
f
rn
∼
Thus, if we have M ∈ HA
, then grW
for some rn ≥ 0. If N is the minimal
n M = Q(−n)
n such that Wn M 6= 0, then we have the exact sequence
>N
0 → grW
M →0
NM → M → W
29
T
T
By induction on the length of the weight filtration, W >N M is in HA
, hence M is in HA
and
f
T
thus HA = HA .
f
Lemma 1.12.10 For N, M ∈ HA
, n ≤ m ∈ Z, we have
HomHf (W >m M, Wn N ) = 0
A
Proof If M = Q(−a), N = Q(−b) with a > b, then
HomHf (M, N ) = H 0 (A(a − b)) = 0
A
since A is connected. The result in general follows by induction on the weight filtration.
f
Proposition 1.12.11 HA
is a neutral Tannakian category over Q.
Proof Since Q(n)∨ = Q(−n), it follows from lemma 1.12.9 that M 7→ M ∨ restricts from
f
f
f
f
DA
to an exact involution on HA
. Since DA
is rigid, it follows that HA
is rigid as well. Also
(
H 0 (A(a − b)) = 0 if a 6= b
HomHf (Q(−a), Q(−b)) =
A
H 0 (A(0)) = Q · id if a = b.
By induction on the weight filtration, this implies that HomHf (M, N ) is a finite dimensional
A
f
Q-vector space for all M, N in HA
. Since the identity for the tensor product is Q(0), it
f
follows that HA is Q linear.
f
f
f
is equivalent to the
. Noting that HQ
We have the rigid tensor functor q : HA
→ HQ
category of finite dimensional graded Q-vector spaces, composing q with the functor “forget
f
the grading” from HQ
to VecQ defines the rigid tensor functor
f
ω : HA
→ VecQ .
f
f
f
is faithful.
→ VecQ is faithful, so we need only see that q : HA
→ HQ
The forgetful functor HQ
Sending M ∈ VecQ to Q(−n) ⊗ M defines an equivalence of VecQ with the full subcategory
f
f
W
grW
n H of HA consisting of M which are isomorphic to grn M . Via this identification, we
can further identify q with the functor
W
M 7→ grW
∗ M := ⊕n grn M.
f
Let f : M → N be a map in HA
such that grW
n f = 0 for all n; we claim that f = 0. By
induction on the length of the weight filtration, it follows that W >n f = 0, where n is the
mininal integer such that Wn M ⊕ Wn N 6= 0. Thus f is given by a map
f˜ : W >n M → grW
n N.
But f˜ = 0 by lemma 1.12.10, hence f = 0 as desired.
Notation 1.12.12 We denote the truncation to the heart,
+w
τ≤0 τ ≥0 : DA
→ HA ,
by HA0 .
30
1.13
Connection matrices
A convenient way to define an A-cell module is by a connection matrix (called a twisting
matrix in [26]).
Let (M, dM ) be a complex of Adams graded Q-vector spaces. An A-connection for M is
a map (of bi-graded Q-vector spaces)
Γ : M → A+ ⊗Q M
of Adams degree 0 and cohomological degree 1. One says that Γ is flat if
dΓ + Γ2 = 0.
This means the following: A ⊗Q M has the standard tensor product differential, so dΓ :=
dA+ ⊗Q M ◦ Γ + Γ ◦ dM using the usual differential in the complex of maps M to A+ ⊗Q M .
Also, we extend Γ to
Γ : A+ ⊗ M → A+ ⊗ M
using the Leibniz rule, so that Γ2 is defined.
Remark 1.13.1 Given a connection Γ : M → A+ ⊗Q M , define
d0 : M → A ⊗Q M = M ⊕ A+ ⊗Q M, m 7→ dM m ⊕ Γm
and extend d0 to dΓ : A ⊗Q M → A ⊗Q M by the Leibniz rule. Then Γ is flat if and only if
dΓ endows A ⊗Q M with the structure of a dg A-module, i.e. d2Γ = 0.
If Γ : M → A+ ⊗Q M is a connection, call Γ nilpotent if M admits a filtration by bi-graded
Q subspaces
0 = M−1 ⊂ M0 ⊂ . . . ⊂ Mn ⊂ . . . ⊂ M
such that M = ∪n Mn and such that
dM (Mn ) ⊂ Mn−1 ; Γ(Mn ) ⊂ A+ ⊗ Mn−1
for every n ≥ 0.
The following result is obvious:
Lemma 1.13.2 Let Γ : M → A+ ⊗Q M be a flat nilpotent connection. Then the dg A-module
(A ⊗Q M, dΓ ) is a cell module.
Indeed, choosing a Q basis B for M such that Bn := Mn ∩ B is a Q basis for Mn for each
n gives the necessary filtered A basis for A ⊗Q M . In addition, we have
Lemma 1.13.3 Let Γ : M → A+ ⊗Q M be a flat connection. Suppose there is an integer r0
such that |m| ≥ r0 for all m ∈ M . Then Γ is nilpotent.
Proof The proof is essentially the same as that of lemma 1.5.8(2): If M is concentrated in
a single Adams degree r0 , then Γ is forced to be the zero-map. Thus, taking M0 = ker(dM ) ⊂
M and M1 = M shows that Γ is nilpotent. In general, one shows by induction on the length
of the weight filtration that the restriction of Γ to Wn M := ⊕r≤n M (r) is nilpotent for every
n, and then a limit argument completes the proof.
31
A morphism f : (M, dM , Γ) → (M 0 , dM 0 , Γ0 ) is a map of bi-graded vector spaces
f := f0 + f + : M → A ⊗ M 0 = M 0 ⊕ A+ ⊗ M 0
such that
dΓ0 f = f dΓ .
In particular, we may identify the category of complexes of Q-vector spaces with the subcategory consisting of complexes with flat connection 0 and morphisms f = f 0 + f + with
f + = 0.
Definition 1.13.4 We denote the category of flat nilpotent connections over A by ConnA .
We let Conn+w
A be the full subcategory consisting of flat nilpotent connections on M with
M (r) = 0 for r << 0, and ConnfA the full subcategory of flat nilpotent connections on M
with M finite dimensional over Q.
It follows from lemma 1.13.3 that a flat connection on M with M (r) = 0 for r << 0 (or with
M finite dimensional over Q) is automatically nilpotent.
1.14
The homotopy category of connections
Define a tensor operation on ConnA by
(M, Γ) ⊗ (M 0 , Γ0 ) := (M ⊗ M 0 , Γ ⊗ id + id ⊗ Γ0 )
with Γ ⊗ id + id ⊗ Γ0 suitably interpreted as a connection by using the necessary symmetry
isomorphisms. Complexes of Q vector spaces act on ConnA by
(M, Γ) ⊗ K := (M, Γ) ⊗ (K, 0).
Let I be the complex
δ
Q→
− Q⊕Q
with Q in degree -1, and with connection 0. We have the two inclusions i0 , i1 : Q → I. Two
maps f, g : (M, Γ) → (M 0 , Γ0 ) are said to be homotopic if there is a map h : (M, Γ) ⊗ I →
(M 0 , Γ0 ) with f = h ◦ (id ⊗ i0 ), g = h ◦ (id ⊗ i1 ).
Definition 1.14.1 Let HConnA denote the homotopy category of ConnA , i.e., the objects
are the same as ConnA and morphisms are homotopy classes of maps in ConnA . Similarly,
we have the full subcategories
H(ConnfA ) ⊂ H(Conn+w
A ) ⊂ HConnA
with objects ConnfA , resp. Conn+w
A .
If M is an A-cell module, then let M0 be the complex of Q-vector spaces M ⊗A Q.
Using the identity splitting Q → A to the augmentation A → Q, we have the canonical
isomorphism of A-modules
A ⊗Q M0 ∼
= M.
32
Using the decomposition A = Q⊕A+ , we can decompose the differential on A⊗Q M0 induced
by the above isomorphism as
d = d0 + d+
where d0 maps Q ⊗ M0 to Q ⊗ M0 and d+ maps Q ⊗ M0 to A+ ⊗ M0 .
We can thus make M0 into a complex of Adams graded Q-vector spaces by using the
differential d0 . The map
d+ : M0 → A+ ⊗ M0
gives a connection and the flatness condition follows from d2 = 0. Nilpotence follows from
the filtration condition (definition 1.3.1(3b)) for an A-basis of M .
Conversely, if (M0 , d0 ) is a complex of Adams graded Q-vector spaces, and
Γ : M0 → A+ ⊗ M0
is a flat nilpotent connection, make the free Adams graded A-module A ⊗Q M0 a cell module
by taking dΓ to be the differential (see remark 1.13.1 and lemma 1.13.2).
Lemma 1.14.2 The correspondences
(M, dM = d0 + d+ ) 7→ ((M0 , d0 )d+ ), ((M0 , d0 )d+ ) 7→ (M = A ⊗Q M0 , d0 + d+ )
define an equivalence of the category of A-cell modules with the category of flat nilpotent
A-connections. This equivalence respectives the homotopy relations and tensor products.
Proof Indeed the functor which assigns to a flat nilpotent connection (M0 , dM0 , Γ) the cell
module (A ⊗Q M0 , dΓ ) is essentially surjective by the previous discussion, and the map on
Hom groups is an isomorphism.
Define the shift operator by (M, Γ)[1] := (M [1], −Γ[1]). Given a morphism f : (M, Γ) →
(M 0 , Γ0 ) of flat nilpotent connections, decompose f : M → A ⊗ M 0 as f := f 0 + f + . Define
the cone of f as having underlying complex Cone(f 0 ), with connection (−Γ[1] ⊕ Γ0 ) + f + .
This gives us the cone sequence
(M, Γ) → (M 0 , Γ0 ) → Cone(f ) → (M, Γ)[1].
The next result is immediate:
Lemma 1.14.3 Using the cone sequences as distinguished triangles makes HConnA into
a triangulated tensor category. The equivalence of lemma 1.14.2 passes to an equivalence
of HConnA with the homotopy category KCMA of A-cell modules, as triangulated tensor
categories.
Thus, via proposition 1.4.3 we have defined an equivalence of HConnA with DA as tri+w
and
angulated tensor categories. This restricts to equivalences of H(Conn+w
A ) with DA
f
f
H(ConnA ) with DA .
The weight filtration in DA can be described in the language of flat connections: Let M
be an Adams graded complex of Q-vector spaces, which we decompose into Adams graded
pieces as M = ⊕r M (r). Set
Wn M := ⊕r≤n M (r)
33
giving us the subcomplex Wn M of M . If Γ : M → A+ ⊗ M is a flat connection, then as Γ
has Adams degree 0, it follows that Γ restricts to a flat nilpotent connection
Wn Γ : Wn M → A+ ⊗ Wn M.
It is easy to see that this filtration corresponds to the weight filtration on DA via the equivalence of lemma 1.14.3 and proposition 1.4.3.
Let HConn+w
⊂ HConnA be the full subcategory of objects M such that Wn M ∼
= 0
A
f
+w
for some n, and let HConnA ⊂ HConnA be the full subcategory of objects M such that
⊕n H n (M ) is finite dimensional. It is easy to see that the inclusions H(ConnfA ) ⊂ HConnfA
+w
and H(Conn+w
A ) ⊂ HConnA are equivalences, giving us the equivalences
f
+w
HConnfA ∼ DA
, HConn+w
A ∼ DA .
Now suppose that A is connected. It is easy to see that the standard t-structure on the
derived category D(Q) of complexes over Q induces a t-structure on the homotopy category
+w
+w
+w
HConn+w
∼ DA
, the t-structure on DA
defined in
A . Under the equivalence HConnA
≥0
section 1.12 corresponds to the pair of subcategories (HConn≤0
,
HConn
),
hence
these
A
A
+w
give the corresponding t-structure on HConnA .
Definition 1.14.4 Suppose that A is connected. Let Conn0A denote the filtered abelian
tensor category of flat A-connections on Adams graded Q-vector spaces V with V (r) = 0
0
for r << 0. Let Conn0f
A ⊂ ConnA be the full sub-category of flat connections on finite
dimensional Adams graded Q-vector spaces.
Lemma 1.14.5 Suppose that A is connected. Then the equivalence of lemma 1.14.3 defines
an equivalence of filtered abelian tensor categories
HA ∼ Conn0A .
and this restricts to an equivalence of filtered Tannakian categories
f
HA
∼ Conn0f
A .
f
Proof The first equivalence follows from the discussion above. Also, DA
is equivalent to
f
the full subcategory HConnA of HConnA with objects the flat nilpotent connections on
complexes M such that ⊕n H n (M ) is finite dimensional, compatible with the restrictions of
the respective t-structures, giving the second equivalence.
Remarks 1.14.6
1. By lemma 1.13.3, the flat connection Γ for an object (M, Γ) in Conn0A is automatically
nilpotent.
f
2. Conn0f
A may also be defined as the full subcategory of HConnA consisting of complexes
∗
0
M with H (M ) = H (M ).
We can also give an explicit description of the truncation functors for this t-structure
in the language of flat nilpotent connections. Let (M, d) is a complex of Adams graded
Q-vector spaces with a flat nilpotent connection
Γ : M → A+ ⊗ M
34
such that (M, d, Γ) is in Conn+w
A . Then we can decompose Γ as
X
Γ :=
Γ(i)
i≥1
by writing
[A+ ⊗ M ]n+1 = ⊕i≥1 Ai ⊗ M n−i+1
and letting Γ(i),n : M n → Ai ⊗ M n−i+1 be the composition
Γn
M n −→ [A+ ⊗ M ]n+1 → Ai ⊗ M n−i+1 .
The flatness condition for Γ when restricted to the component which maps M n to A1 ⊗ M n
yields the commutative diagram
Mn
dn
Γ(1),n
/
M n+1
Γ(1),n+1
/
A1 ⊗Q M n
1⊗dn+1
A1 ⊗Q M n+1 .
This implies that Γ restricts to a flat connection τ≤n Γ on the subcomplex τ≤n M :
τ≤n Γ : τ≤n M → A+ ⊗ τ≤n M ;
τ≤n Γ is nilpotent by lemma 1.13.3.
This in turn implies that Γ descends to a connection on the quotient complex τ >n M :=
M/τ≤n M :
τ >n Γ : τ >n M → A+ ⊗ τ >n M
which is in fact a flat nilpotent connection. Indeed, the only question for flatness is for the
terms in Γ2 + dΓ which factor via Γ or d through A+ ⊗ M ∗≤n , but which have non-zero image
in A+ ⊗ τ >n M . There are three such terms:
Γ(1),n ◦ Γ(i+1−n),i , (1 ⊗ dn ) ◦ Γ(i+1−n),i , (1 ⊗ dn−1 ) ◦ Γ(i+2−n),i
where we use the convention that Γ(0),i = di . For a term of the first type, the fact that Γ(1)
commutes with d implies that the composition factors through Ai+1−n ⊗ (M n / ker dn ). The
second term similarly factors through Ai+1−n ⊗ (M n / ker dn ), while the third term goes to
zero in Ai+2−n ⊗ (M n / ker dn ).
As before, the nilpotence of τ >n Γ follows from lemma 1.13.3.
Thus for each (M, d, Γ) in Conn+w
A we have the sequence of complexes with flat nilpotent
connection
0 → (τ≤n M, d, τ≤n Γ) → (M, d, Γ) → (τ >n M, d, τ >n Γ) → 0
which is exact as a sequence of bi-graded Q-vector spaces. When we take the associated
cell modules, this gives us the canonical distinguished triangle for the t-structure we have
+w
described for DA
.
In particular, the truncation functor HAn := τ ≥n τ≤n can be explicitly described in the
language of flat nilpotent connections. Namely, the restricted connection
Γ(1),n : M n → A1 ⊗ M n
35
defines a connection (not necessarily flat) on the Adams graded Q-vector space M n for each
n, and the differential d gives a map in the category of connections
dn : (M n , Γ(1),n ) → (M n+1 , Γ(1),n+1 ).
In short, (M, d, Γ(1) ) is a complex in the category of connections. Thus Γ(1) induces a
connection on H n (M ):
H n (Γ) := H n (Γ(1) ) : H n (M ) → A1 ⊗ H n (M ).
On M n , the flatness condition for Γ, when restricted to the component which maps M n to
A2 ⊗ M n , gives the identity:
(id ⊗ dn+1 ) ◦ Γ(2),n − Γ(1),n+1 ◦ Γ(1),n + Γ(2),n+1 ◦ dn = 0
and thus H n (Γ(1) ) is flat. H n (Γ(1) ) is nilpotent by lemma 1.13.3.
The canonical quasi-isomorphism of complexes
τ ≥n τ≤n (M, dM ) → H n (M, dM )
thus gives rise to a quasi-isomorphism of complexes with flat nilpotent connection
τ ≥n τ≤n (M, dM , Γ) → (H n (M, dM ), H n (Γ(1) )).
Definition 1.14.7 Let A be a cohomologically connected cdga with 1-minimal model A{1}.
We let QA := A{1}1 and let ∂ : QA → Λ2 QA denote the differential d : A{1}1 → Λ2 A{1}1 =
A{1}2 . Then (QA, ∂) is co-Lie algebra over Q. If A is an Adams graded cdga, then QA
becomes an Adams graded co-Lie algebra.
In the Adams graded case, we let co-rep(QA) denote the category of co-modules M over
QA, where M is a bi-graded Q-vector space such that the Adams degrees in M are bounded
below.
Remark 1.14.8 Let us suppose that A is a generalized nilpotent Adams graded cdga. Then
the co-Lie algebra QA is given by the restriction of d to A1 , noting that d factors as
d : A1 → Λ2 A1 ⊂ A2 .
If now M is an Adams graded Q-vector space (concentrated in cohomological degree 0) and
Γ : M → A+ ⊗ M is a flat connection, then Γ is actually a map
Γ : M → A1 ⊗ M
and the flatness condition is just saying the Γ makes M into an Adams graded co-module
for the co-Lie algebra QA. If in addition the Adams degrees occuring in M have a lower
bound, then Γ is automatically nilpotent (lemma 1.13.3).
Thus, we have an equivalence of the category Conn0A with co-rep(QA), which restricts to
f
an equivalence of Conn0f
A with the category co-rep (QA) of finite dimensional co-modules
over QA.
Putting this together with the above discussion, we have equivalences
HA ∼ Conn0A ∼ co-rep(QA)
which restrict to equivalences
f
f
HA
∼ Conn0f
A ∼ co-rep (QA).
36
1.15
Summary
In [26] the relations between the various constructions we have presented above are discussed.
We summarize the main points here.
Definition 1.15.1 1. Let H = Q · id ⊕ ⊕r≥1 H(r) be an Adams Hopf algebra over Q.
We let co-rep(H) denote the abelian tensor category of co-modules M over H, where M
is a bi-graded Q vector space such that the Adams degrees in M are bounded below. Let
co-repf (H) ⊂ co-rep(H) be the full subcategory of co-modules M such that M is finite dimensional over Q.
2. Let γ = ⊕r≥1 γ(r) be an Adams graded co-Lie algebra over Q. We let co-rep(γ) denote the abelian tensor category of co-modules M over γ, where M is a bi-graded Q vector
space such that the Adams degrees in M are bounded below. Let co-repf (γ) ⊂ co-rep(γ) be
the full subcategory of co-modules M such that M is finite dimensional over Q.
The Adams grading induces a functorial exact weight filtration on the abelian categories
co-rep(H) and co-rep(γ) by setting
Wn M := ⊕r≤n M (r).
The subcategories co-repf (H) and co-repf (γ) are Tannakian categories over Q, with neutral
fiber functor the associated graded for the weight filtration grW
∗ .
Let H+ = ⊕r≥1 H(r) ⊂ H be the augmentation ideal, γH := H+ /H+2 the co-Lie algebra
of H. For an H co-module δ : M → H ⊗ M we have the associated γH co-module M̄ with
the same underlying bi-graded Q vector space, and with co-action δ̄ : M̄ → M̄ ⊗ γH given
by the composition
δ
M→
− M ⊗ H = M ⊕ M ⊗ H+ → M ⊗ H+ → M ⊗ γH .
Then the association M 7→ M̄ induces equvalences of filtered abelian tensor categories
co-rep(H) ∼ co-rep(γH ), co-repf (H) ∼ co-repf (γH ).
For an Adams graded cdga A, we have the Adams graded Hopf algebra χA := H 0 (B̄(A))
and the Adams graded co-Lie algebra γA := γχA . We have as well the co-Lie algebra QA
defined using the 1-minimal model of A (definition 1.14.7).
Theorem 1.15.2 Let A be an Adams graded cdga. Suppose that A is cohomologically connected.
f
(1) There is a functor ρ : Db (co-repf (χA )) → DA
. ρ respects the weight filtrations and
sends Tate objects to Tate objects. ρ induces a functor on the hearts
f
H(ρ) : co-repf (χA ) → HA
which is an equivalence of filtered Tannakian categories, respecting the fiber functors grW
∗ .
37
(2) Let A{1} be the 1-minimal model of A. Then A{1} → A induces an isomorphism
of graded Hopf algebras χA{1} → χA and graded co-Lie algebras
QA ∼
= γA{1} ∼
= γA .
(3) The functor ρ is an equivalence of triangulated categories if and only if A is 1-minimal.
(4) Sending a co-module M ∈ co-rep(χA ) to the γA co-module M̄ defines equivalences of
neutral Tannakian categories
co-rep(χA ) ∼ co-rep(γA ); co-repf (χA ) ∼ co-repf (γA ).
Putting this together with our discussion on connections in section 1.13 gives
Corollary 1.15.3 Let A be a cohomologically connected Adams graded cdga. We have equivalences of filtered abelian tensor categories
co-rep(χA ) ∼ co-rep(γA ) ∼ co-rep(QA) ∼ Conn0A
and equivalences of filtered neutral Tannakian categories
co-repf (χA ) ∼ co-repf (γA ) ∼ co-repf (QA) ∼ Conn0A ∩ ConnfA .
2
Relative theory of cdgas
The theory of cdgas over Q generalizes to a large extent to cdgas over a cdga N . In this
section, we give the main constructions in this direction that we will need. As in section 1,
all cdgas are cdgas over Q.
2.1
Definitions and model structure
We fix a base cdga N . A cdga over N is a cdga A together with a homomorphism of cdgas
φ : N → A. An augmented cdga over N has in addition a splitting π : A → N to φ. The
same notions apply for an Adams graded cdga A over an Adams graded cdga N . Let cdgaN
denote the category of Adams graded augmented cdgas over N , where a map A → B is a
dg N -algebra morphism compatible with the augmentations.
Definition 2.1.1 1. A morphism φ : A → B in cdgaN is a weak equivalence if φ induces
an isomorphism
φ∗ : H n (A(r)) → H n (B(r))
for all n, r ≥ 1.
2. A morphism φ : A → B in cdgaN is a fibration if φ(r) : A(r)n → B(r)n is surjective for all n, r ≥ 1
3. A morphism φ : A → B in cdga is a cofibration if φ has the left lifting property with
respect to acyclic fibrations.
38
As usual, we call φ : A → B a quasi-isomorphism if φ induces an isomorphism on H n for all
n.
The category cdgaN has push-outs and pull-backs: the push-out in the diagram
/
C
B
A
is A ⊗C B, with the morphisms A, B → A ⊗C B given by a 7→ a ⊗ 1, b 7→ 1 ⊗ b; the
augmentation is induced from that of A and B.The pull-back in the diagram
B
A
/
g
f
C
is A ×C B, i.e., the sub-complex of A ⊕ B of elements (a, b) with f (a) = g(b). The product is
(a, b)·(a0 , b0 ) := (aa0 , bb0 ) and the augmentation is induced from that of A and B. Additionally,
small filtered colimits and limits exist in cdgaN .
We proceed to show that definition 2.1.1 makes cdgaN a model category, closely following
[7, chapter 4]. We call a map which is a (co)fibration and a weak equivalence an acyclic
(co)fibration.
The proof of the following lemma is easy and is left to the reader.
Lemma 2.1.2 The cofibrations in cdgaN satisfy
1. Every isomorphism in cdgaN is a cofibration
2. Cofibrations are closed under push-out by an arbitrary morphism
3. If A1 → A2 → . . . is a sequence of cofibrations, then A1 → limn An is a cofibration.
−→
4. If {ij : Aj → Bj }j∈J is a set of cofibrations, then
⊗j∈J Aj → ⊗j∈J Bj
is a cofibration (where ⊗ means ⊗N ).
5. Recall that a map f : A → B is a retract of a map g : C → D if there is a commutative
diagram
A
f
/
i
C
p
/
A
g
B
j
/
D
q
/
f
B
with pi = idA , qj = idB . Then any retract of a cofibration is a cofibration.
39
Let cdga denote the category of Adams graded cdgas over Q. There is a bi-functor
⊗ : cdgaN × cdga → cdgaN
defined by letting A ⊗ B be the Adams graded cdga over N with (A ⊗ B)(r) := ⊕s A(r) ⊗Q
B(r − s), product
0
(a ⊗ b)(a0 ⊗ b0 ) := (−1)deg b deg a aa0 ⊗ bb0
and augmentation n 7→ (n) ⊗ 1.
Following [7, §4.4], we have the elementary cofibrations in cdga [7, §4.4]. As preparation
for the definition, for n ≥ 0, r ≥ 1, let S(n, r) be the cdga over Q freely generated as a gradedcommutative algebra over Q by a single element e ∈ S(n, r)(r)n with de = 0. Similarly, let
T (n, r) be the cdga over Q freely generated by elements a ∈ T (n, r)(r)n , b ∈ T (n, r)(r)n+1
with b = da. Set T (−1, 0) = Q (in degree 0).
Lemma 2.1.3 1. Let i : A → B be a cofibration in cdga. Then N ⊗ A → N ⊗ B is a
cofibration in cdgaN .
2. The following maps are cofibrations in cdga (the elementary cofibrations):
a. the map θ : S(n, r) → T (n − 1, r) with θ(e) = b
b. the map σ : Q → S(n, r) with σ(1) = 1
c. the map τ : Q → T (n, r) with τ (1) = 1.
Proof We have the restriction of scalars functor U : cdgaN → cdga with respect to the
identify map Q → N ; the functor N ⊗ − : cdga → cdgaN is left adjoint to U. As U maps
weak equivalences to weak equivalences and fibrations to fibrations, N ⊗− sends cofibrations
to cofibrations, proving (1).
(2) is an exercise, left to the reader.
Write TN (n, r) := N ⊗ T (n, r), SN (n, r) := N ⊗ S(n, r). Given A ∈ cdgaN , we have
bijections of sets (for n, r ≥ 1):
HomcdgaN (TN (n, r), A) ↔ A(r)n
HomcdgaN (SN (n, r), A) ↔ Z n (A(r)) := {y ∈ A(r)n | dy = 0}
We let φx : TN (n, r) → A be the morphism corresponding to x ∈ A(r)n and φ0y : SN (n, r) →
A be the morphism corresponding to y ∈ Z n (A(r)).
It follows from lemma 2.1.3 that the maps θ, σ, τ induce cofibrations
θ : SN (n, r) → TN (n − 1, r)
σ : N → SN (n, r)
τ : N → TN (n, r)
in cdgaN .
We refer the reader to [7, definition 4.1] for the list of axioms defining a (closed) model
category. The axioms CM1, CM2, CM3 are easy to verify and are left to the reader; CM4(b)
is satisfied by the definition of a cofibration. We need to verify the axioms CM4(a) and CM5.
For CM5, we need to show that every morphism f : A → B in cdgaN can be factored as
pi, with i a cofibration and p a fibration and either
40
(a) i is a weak equivalence, or
(b) p is a weak equivalence.
To check (a), the maps φx induce the map
φB : ⊗x∈B(r)n ,r,n≥1 TN (n, r) → B;
clearly φB is a fibration. We have the cofibration τ : N → TN (n, r), giving us the cofibration
ψB : N = ⊗x∈B(r)n ,r,n≥1 N → ⊗x∈B(r)n ,r,n≥1 TN (n, r).
Since each T (n, r) is acyclic, ψB is also a weak equivalence. Taking the push-out of ψB by
the augmentation N → A gives us the acyclic cofibration
i : A → Kf := A ⊗N ⊗x∈B(r)n ,r,n≥1 TN (n, r).
The maps f : A → B and φB give the map
p : Kf → B
which is a fibration. As i is a weak equivalence and f = pi, CM5(a) is proved.
To check (b), we form a sequence of maps
β1
β2
/ Lf (2) β3
k
k
ψ1 yyy ψ2 kkkkk
k
y
k
f
k
y
k
yy kkkk
|y
ukkk
A
/
Lf (1)
/ ...
(2.1.1)
B
To define Lf (1), set
Lf (1) := A ⊗N [⊗r,n≥1,x∈B(r)n TN (n, r)] ⊗N [⊗r,n≥1,y∈Z(B(r)n ) SN (n, r)].
Define ψ1 : Lf (1) → B by φx in the factor indexed by x ∈ B(r)n to x, and φ0y in the factor
indexed by y ∈ Z(B(r)n ). Clearly ψ1 is a fibration. We have the evident map β1 : A → Lf (1);
as in the proof of CM5(a), β1 is a cofibration. Furthermore H n (ψ1 ) : H n (Lf (1)) → H n (B)
is surjective for all n ≥ 1.
Let
R(r)n := {(w, y) ∈ Lf (1)(r)n+1 × B(r)n | dw = 0 and dy = ψ1 (w)}
Define β2 : Lf (1) → Lf (2) via the push-out diagram
⊗(w,y)∈R(r)n ,r,n≥1 SN (n + 1, r)
/
⊗θw
Lf (1)
⊗(w,y)∈R(r)n ,r,n≥1 TN (n, r)
/
β2
Lf (2)
We let ψ2 : Lf (2) → B be the map induced by ψ1 : Lf (1) → B and the maps TN (n.r) →
Lf (2). As each θw is a cofibration, β2 is a cofibration and since ψ1 is a fibration, so is ψ2 .
Note that, for n ≥ 1, H n (ψ2 ) : H n (Lf (2)) → H n (B) restricts to an isomorphism on the
image of H n (β2 ) since ker H n (ψ2 ) = ker H n (β2 ) by construction.
Iterating this procedure gives the diagram (2.1.1) with the following properties:
41
(i) each map βm is a cofibration
(ii) each map ψm is a fibration
(iii) for n ≥ 1, H n (ψm ) : H n (Lf (m)) → H n (B) restricts to an isomorphism on the image
of H n (βm ).
Let Lf := limn Lf (n), i : A → Lf , p : Lf → B the maps given on the limit by the diagram
−→
(2.1.1). Then i is a cofibration, f is a fibration, and by (iii), f is also a weak equivalence.
This proves CM5(b).
To prove CM4(a), we need to show: given a commutative diagram
/X
A
i
f
/
Y
B
with i a cofibration and a weak equivalence, and f a fibration, there exist a lifting
A
i
~
~
~
/X
~?
/
B
f
Y
i
p
0
For this, factor i : A → B as we did in the proof of CM5(a): A −
→
Ki →
− B. In particular,
i0 is a cofibration and weak equivalence, and p is a fibration. Since i is a weak equivalence,
so is p. This gives us the diagram
A
i
i0
/
Ki
p
B
B
which admits a lifting B → Ki , since i is a cofibration. Thus, it suffices to prove CM4(a) for
the cofibration i0 : A → Ki . But Ki = A ⊗N [⊗j∈J TN (r, n)], with every TN (n, r) appearing
in the tensor product having n, r ≥ 1. It is clear that there is a lifting TN (n, r) → X for
every diagram
X
TN (n, r)
/
f
Y
with f a fibration and n, r ≥ 1, giving us the desired lifting Ki → X . This completes the
proof of CM4(a), giving
Proposition 2.1.4 With cofibrations, fibrations and weak equivalences defined as in definition 2.1.1, cdgaN is a closed model category.
We denote the homotopy category of cdgaN by H(cdgaN ).
42
2.2
Path objects and the homotopy relation
Let M be a model category. As usual, we call an object A of M cofibrant if the from the
initial object ∅ → A is a cofibration, and fibrant if the map A → ∗ to the final object is a
fibration.
Recall that for an object B in a model category M, a path object for B is a factorization of
the diagonal map B → B ×B as pi, with i : B → B I a weak equivalence and p : B I → B ×B
a fibration. Let pi : B I → B, i = 1, 2 be πi ◦ p, where π1 , π2 : B × B → B are the two
projections.
Two morphisms f, g : A → B in M are right homotopic if there is a path object B I , p1 , p2
and a morphism h : A → B I with f = p1 h, g = p2 h.
The main results on model categories state that right homotopy with respect to a fixed
path object defines an equivalence relation ∼ on HomM (A, B), if A is cofibrant and B is
fibrant (see [36, chap. 1, §1, lemmas 4, 5(i) and their duals]. In addition, the category with
objects the fibrant and cofibrant (bifibrant) objects of M, and with morphisms the right
homotopy classes of morphisms in M, is equivalent to the homotopy category of M [36,
chap. 1, §1, theorem 1].
Passing to cdgaN , we give a construction of a path object for each B ∈ N .
Let cdga denote the category of commutative differential graded algebras over Q (without
Adams grading and without augmentation). We have the bi-functor
⊗ : cdgaN × cdga → cdgaN
where A ⊗B has Adams degree r summand (A⊗B)(r) := A(r)⊗Q B for r ≥ 1. The product
0
is (a ⊗ b)(a0 ⊗ b0 ) := (−1)deg b deg a aa0 ⊗ bb0 , and the augmentation is induced by that of A.
Let Ω∗ be the cdga of polynomial differential forms on A1 , that is, Ω0 := Q[t], Ω1 :=
1
ΩQ[t]/Q = Q[t]dt and the differential is the usual one. We have the unit map η : Q → Ω∗ and
two restriction maps
i∗0 , i∗1 : Ω∗ → Q
with i∗ (f ) := f (), = 0, 1. Clearly η is a quasi-isomorphism and (i∗0 , i1 ∗) : Ω∗ → Q × Q is
surjective.
For B ∈ cdgaN , let B I := B ⊗ Ω∗ , iB : B → BI the map id ⊗ η and pB : B I → B ×N B
the map (id ⊗ i∗0 , id ⊗ i∗1 ). Clearly p is a fibration and i is a weak equivalence, giving us the
desired path object. For A, B in cdgaN , we write ∼Ω for the relation on HomcdgaN (A, B)
given by right homotopy with respect to the path object B ⊗Q Ω∗ .
Note that for B ∈ cdgaN , the augmentation B → N is always a fibration, hence all
objects in cdgaN are fibrant. Thus for A cofibrant, and f, g : A → B, we have f ∼ g if and
only if f ∼Ω g.
The results from the theory of model categories, as recalled above, thus gives us
Proposition 2.2.1 The category H(cdgaN ) is equivalent to the category with objects the
cofibrant objects of cdgaN and with morphisms (for A, B cofibrant) HomcdgaN (A, B)/ ∼Ω .
2.3
Indecomposables
For A ∈ cdgaN , let A+ denote the kernel of the augmentation A → N . Let
QA := A+ /(A+ · A+ ).
43
The Leibniz rule for dA implies that dA induces a differential on QA, making (QA, d) an
Adams graded N -module. Sending A to QA thus gives a functor Q : cdgaN → MN .
Lemma 2.3.1 1. Let f, g : A → B be morphisms in cdgaN . If f ∼Ω g then H ∗ (Qf ) =
H ∗ (Qg).
2. Let f : A → B be a weak equivalence between cofibrant objects of cdgaN . Then
Qf : QA → QB is a quasi-isomorphism.
Proof For (1), it clearly suffices to show that Qi : QB → Q(B⊗Q Ω∗ ) is a quasi-isomorphism.
Since Q[t] has a unit, and Ω1 · Ω1 = 0, the evident map
Q(B ⊗Q Ω∗ ) → QB ⊗Q Ω∗
is an isomorphism; via this isomorphism Qi is transformed to
− ⊗ 1 : QB → QB ⊗Q Ω∗ .
As the unit Q → Ω∗ is a quasi-isomorphism, so is − ⊗ 1, proving (1).
For (2), since A and B are cofibrant, it follows from proposition 2.2.1 that there is a
morphism g : B → A with gf ∼Ω idA and f g ∼Ω idB . By (1), H ∗ (Qf ) has inverse H ∗ (Qg),
completing the proof.
2.4
Relative minimal models
The notions of generalized nilpotent algebras and minimal models (over Q) extend without
difficulty to augmented cdgas over N . Specifically:
Definition 2.4.1 An Adams graded cdga A over N is said to be generalized nilpotent over
N if
1. As a bi-graded N -algebra, A = Sym∗ E ⊗N for some Adams graded Z-graded Q-vector
space E, i.e., A = Λ∗ E odd ⊗ Sym∗ E ev ⊗ N , where the parity refers to the cohomological
degree. In addition, E(r)n = 0 if n ≤ 0 or if r ≤ 0.
2. For n ≥ 0, let A(n) ⊂ A be the N -subalgebra generated by the subspace E ≤n of E
consisting of elements of cohomological degree ≤ n. Set A(n+1,0) = A(n) and for q ≥ 0
define A(n+1,q+1) inductively as the N -subalgebra generated by A(n) and
n+1
An+1
(n+1,q+1) := {x ∈ A(n+1) |dx ∈ A(n+1,q) .}
Then for all n ≥ 0,
A(n+1) = ∪q≥0 A(n+1,q) .
3. If A = (Sym∗ E ⊗ N , d), satisfying (1) and (2), and there is an integer n such that
deg e ≤ n for all homogeneous e ∈ E, we say that A is generated in degree ≤ n.
44
Remark 2.4.2 We can phrase the condition (2) above differently: For each n ≥ 0, E ≤n+1
has an increasing exhaustive bi-graded filtration
E ≤n = F0 E ≤n+1 ⊂ F1 E ≤n+1 ⊂ . . . ⊂ Fm E ≤n+1 ⊂ . . . ⊂ E ≤n+1
such that
d(Fm E ≤n+1 ⊗ N ) ⊂ Sym∗ (Fm−1 E ≤n+1 ) ⊗ N
Indeed, if A = Sym∗ E ⊗ N satisfies (2), define Fm E ≤n+1 by
Fm E ≤n+1 ⊗ 1 = (E ≤n+1 ⊗ 1) ∩ A∗(n+1,m) .
Conversely, it is easy to see that the existence of such a filtration F∗ E ≤n+1 for all n implies
(2).
Remark 2.4.3 Suppose that N is connected, that is, that N (r)n = 0 for r ≥ 1, n ≤ 0.
Then the subalgebra A(n) can be defined directly from A, independent of the choice of bigraded Q-vector space E with A = Sym∗ E ⊗Q N . In fact, A(n) is just the N -subalgebra of
A generated by the elements x ∈ A with deg x ≤ n. The inductive definition of A(n,q) thus
shows that these subalgebras are also independent of the choice of E.
Lemma 2.4.4 Let A be a generalized nilpotent cdga over N . Then A is cofibrant in cdgaN .
Proof We write A as a colimit of elementary cofibrations, with N as the initial source.
Indeed, let E be a generating bi-graded Q-vector space for A with filtration F ∗ E satisfying
the properties given remark 2.4.2.
In particular, for each y ∈ F 0 E, dy = 0. Choose for each r, n ≥ 1 a Q-basis yαn,r of
F 0 E(r)n . Let LA (0) = ⊗yαn,r SN (n, r) and let β0 : LA (0) → A be the tensor product of maps
φ0yαn,r . By definition, β0 identifies LA (0) with the N -subalgebra of A generated by F 0 E. Let
i0 : N → LA (0) be the coproduct of the cofibrations σ : N → SN (n, r).
1
n
Next, for each r, n ≥ 1, choose a subset {xn,r
α } of F E(r) that maps bijectively to a
1
n
n,r
basis of grF E(r) . For each x = xα , dx is in the sub-algebra β0 (LA (0)) of A, giving us the
diagram
SN (n + 1, r)
θ
φ0dx
/
LA (0)
TN (n, r)
We let LA (1) be defined as the push-out in the diagram
φ0
⊗xn,r
SN (n + 1, r)
α
⊗xn,r θ
α
n,r
dxα
⊗xn,r
TN (n, r)
α
The maps φxn,r
together with β0 give the map
α
β1 : LA (1) → A
45
/
LA (0)
identifying LA (1) with the N -subalgebra generated by F 1 E. We have as well the cofibration
i1 : LA (0) → LA (1), defined as the push-out of ⊗xn,r
θ, giving the commutative diagram
α
i0
/ LA (0) i1 / LA (1)
kk
β0 xxx β1 kkkkk
x kkkk
x
x kk
|xukxkxkkkk
N
A
Continuing in the way, we have cofibrations in : LA (n − 1) → LA (n), injections βn :
LA (n) → A identifying LA (n) with the subalgebra of A generated by F n E, giving a commutative diagram
/ LA (0) i1 / LA (1) i2 / . . . in e/ LA (n) in+1 /
e
kk
eeeeee
β0 xxx β1 kkkkk
βn eeeeeeeee
k
e
x
e
xx kkkk
eeeeee
xkxkkekekekeeeeeeee
|x
kee
rue
N
i0
...
A
As the map limn LA (n) → A is thus an isomorphism and limn LA (n) is cofibrant, the proof
−→
−→
is complete.
Lemma 2.4.5 Let A be a generalized nilpotent cdga over a cdga N . If N is cohomologically
connected, then so is A.
Proof We have just seen that A is isomorphic to limn LA (n), with each map in : LA (n −
−→
1) → LA (n) being the push-out in a diagram of the form
φ0
⊗
xn,r
α
SN (n + 1, r)
⊗xn,r θ
α
n,r
dxα
/
LA (0)
TN (n, r)
⊗xn,r
α
with n, r ≥ 1. In particular, in is injective and we have the exact sequence
i
pn
n
0 → LA (n − 1) −
→
LA (n) −→ [⊗xn,r
SN (n, r)]+ → 0
α
where the last term is kernel of the augmentation ⊗xn,r
SN (n, r) → N .
α
SN (n, r) is the free N algebra on a generator e with deg e = n ≥ 1, |e| = r ≥ 1 and
de = 0. Thus, each finite tensor product ⊗pi=1 SN (ni , ri ) is the free N algebra on generators
e1 , . . . , ep with deg ei = ni ≥ 1, |ei | = ri ≥ 1 and dei = 0. As a dg N -module, we thus have
[⊗pi=1 SN (ni , ri )]+ ∼
= ⊕α N <−rα >[−nα ]
with rα , nα ≥ 1. As N is cohomologically connected, it follows that H m ([⊗pi=1 SN (ni , ri )]+ ) =
0 for m ≤ 0.
Since ⊗xn,r
SN (n, r) is by definition the colimit of the tensor products over finite subsets
α
n,r
of {xα }, we have
H m ([⊗xn,r
SN (n, r)]+ ) = 0
α
for m ≤ 0. By induction (starting with LA (−1) := N ), it follows that each LA (n) is
cohomologically connected, and hence so is A.
46
Proposition 2.4.6 Let A and B be generalized nilpotent cdgas over N and let f : A → B
be a weak equivalence. Then f is an isomorphism.
Before we give the proof, we note the following version of Nakayama’s lemma
Lemma 2.4.7 Let N = Q·id⊕⊕r≥1 Nr be a graded Q-algebra. Let E = ⊕r≥1 Er , F = ⊕r≥1 Fr
be graded Q vector spaces, and let A = Sym∗ E ⊗Q N , B = Sym∗ F ⊗Q N , and let φ : A → B
be an N -algebra morphism, respecting the gradings induced by the grading of E, F and N .
Let φ̄ : Sym∗ E → Sym∗ F be the map φ ⊗N idQ , with respect to the augmentation N → Q,
and let Qφ̄E → F be the map on indecomposables induced by φ̄. Then
φ is an isomorphism ⇔ φ̄ is an isomorphism ⇔ Qφ̄ is an isomorphism.
Proof The implications ⇒ are obvious. Suppose that Qφ̄ is an isomorphism. Let F n Sym∗ E
be the ideal ⊕m≥n Symm E and define F n Sym∗ F similarly. Then φ̄ induces the map gr∗F φ̄ on
the associated graded, and we clearly have
gr∗F φ̄ = Sym∗ (Qφ̄).
Thus gr∗F φ̄ is an isomorphism.
Let (Sym∗ E)(r), (Sym∗ F )(r) denote the respective degree r summands, where we use the
grading induced from that of E, F . Since E and F are positively graded, the filtration on
(Sym∗ E)(r), (Sym∗ F )(r) induced by F ∗ Sym∗ E, F ∗ Sym∗ F is finite, and thus the fact that
gr∗F φ̄ is an isomorphism implies that
φ̄ : (Sym∗ E)(r) → (Sym∗ F )(r)
is an isomorphism for each r.
Now suppose that φ̄ is an isomorphism. Let F n A be the (two-sided) ideal (⊕r≥n Nr )A
of A, and define F n B similarly. As φ is an N -algebra map, φ respects the filtrations. As
gr∗F φ = φ̄ ⊗ idN , gr∗F φ is an isomorphism. Letting A(r), B(r) be the degree r summand with
respect to the grading induced from that of E, F , the fact that N is positively graded implies
that the filtrations induced by F on A(r), B(r) are finite, and thus φ is an isomorphism.
Proof (Proof of proposition 2.4.6) Write
A = Sym∗ E ⊗Q N , B = Sym∗ F ⊗Q N ,
as N -algebras, where E and F are bi-graded Q-vector spaces, with filtrations satisfying the
conditions of remark 2.4.2. In particular, we have
d(E) ⊂ [(Sym∗ E)+ · (Sym∗ E)+ ] ⊗Q N
hence dQA = id ⊗ dN ; similarly dQB = id ⊗ dN . Thus
H ∗ (QA) ∼
= E ⊗Q H ∗ (N ), H ∗ (QB) ∼
= F ⊗Q H ∗ (N )
as bi-graded H ∗ (N )-modules, and the map H ∗ (Qf ) gives an isomorphism of bi-graded
H ∗ (N )-modules
H ∗ (Qf ) : E ⊗Q H ∗ (N ) → F ⊗Q H ∗ (N ).
47
Using the augmentation H ∗ (N ) → Q, we thus have the isomorphism
H ∗ (Qf ) : E → F
of bi-graded Q-vector spaces. But clearly H ∗ (Qf ) is just the map Qf¯ induced by f by first
applying − ⊗N Q (via the augmentation N → Q) and then taking the indecomposables.
Using just the Adams grading, and ignoring the cohomological grading, we consider
E and F as positively graded Q vector spaces, and N as a positively graded Q-algebra.
By lemma 2.4.7, it follows that f : A → B is an N -algebra isomorphism and hence an
isomorphism in cdgaN .
Definition 2.4.8 Let A be an augmented Adams graded cdga over N , n ≥ 1 an integer, or
n = ∞. An n-minimal model over N of A is a map of augmented Adams graded cdgas over
N
s : A{n}N → A,
with A{n}N generalized nilpotent over N , generated in degree ≤ n, and such that s induces
an isomorphism on H m for 1 ≤ m ≤ n and an injection on H n+1 . A minimal model over N ,
A{∞}N → A, is a relative n-minimal model for all n.
If the base-cdga N is understood, we call an n-minimal model over N a relative n-minimal
model.
Proposition 2.4.9 Let N be a cohomologically connected Adams graded cdga, A an augmented Adams graded cdga over N . Then for each n = 1, 2, . . . , ∞, there is an n-minimal
model over N : A{n}N → A.
Proof This result is the relative analog of theorem 1.11.4 and the proof is essentially the
same (see [7, chapter 7] for the details in the absolute case). The construction of the nminimal model over N is essentially the same as for cdgas over Q except that we use both
the cohomological degree and the Adams degree for induction.
In detail: The augmentation gives a canonical decomposition of A as
A=N ⊕I
with I an Adams graded dg N -ideal in A. Let E10 (1) ⊂ I 1 (1) be a Q-subspace of representatives for H 1 (I(1)), where we give E10 (1) cohomological degree 1 and Adams degree 1.
Using the N -module structure of A, we have the evident mapping
E10 (1) ⊗Q N → A,
which extends to
Sym∗ E10 (1) ⊗Q N → A
using the algebra structure. Clearly this is a map of augmented cdgas over N , and induces
an isomorphism on H 1 (−)(1), because N (r) = 0 for r < 0 and N (0) = Q · id.
One then proceeds as in the case N = Q to adjoin elements in degree 1 and Adams
degree 1 to successively kill elements in the kernel of the map on H 2 (−)(1). Since N (r) = 0
for r < 0 and N (0) = Q · id, this does not affect H 1 in Adams degree ≤ 1. Thus we have
constructed a bi-graded Q-vector space E1 (1), of Adams degree 1 and cohomological degree
48
1, a generalized nilpotent cdga over N , A1,1 := Sym∗ E1 (1) ⊗ N and a map of cdgas over N ,
A1,1 → A, that induces an isomorphism on H 1 (−)(1) and an injection on H 2 (−)(1) .
This completes the Adams degree ≤ 1 part for the construction of the 1-minimal model.
So far, we have not used the cohomological connectivity of N , this comes in now: Use the
canonical augmentation of A1,1 to write A1,1 = N ⊕ I1,1 .
Claim 2.4.10 H p (I1,1 (r)) = 0 for r > 1, p ≤ 1.
To prove the claim, we use the same filtration that we used in the proof of lemma 2.4.5.
The same induction argument as in lemma 2.4.5, using of course the cohomological connnecti
edness of N , shows that the lowest degree cohomology of I1,1 (r) comes from ⊕r−1
i=1 Sym E1 (1)⊗
r
1
0
H (N (r−i)) plus Sym E1 (1)⊗H (N (0)). Since all the elements of E1 (1) have cohomological
degree 1, this proves the claim.
To construct the Adams degree ≤ 1 part of the n-minimal model in case n > 1, we
continue the construction, first adjoining elements of Adams degree 1 and cohomological
degree 2 to generate all of H 2 (A)(1), and then adjoining elements of Adams degree 1 and
cohomological degree 2 to kill the kernel on H 3 (−)(1). Continuing in this manner gives the
generalized nilpotent cdga over N ,
A1,n := Sym∗ En (1) ⊗ N ,
with En (1) in Adams degree 1 and cohomological degree 1, . . . , n, together with a map over
N , A1,n → A, that induces an isomorphism on H i (−)(1) for 1 ≤ i ≤ n and an injection for
i = n + 1. If we are in the case n = ∞, we just take the colimit of the A1,n . In addition,
writing A1,n = N ⊕ I1,n , we prove as above
H p (I1,n (r)) = 0 for r > 1, p ≤ 1.
Now suppose we have constructed bi-graded Q-vector spaces
En (1) ⊂ En (2) ⊂ . . . ⊂ En (m)
(for fixed n with 1 ≤ n ≤ ∞) with En (j) having Adams degrees 1, . . . , j and cohomological
degrees 1, . . . , n, a differential on Am,n := Sym∗ En (m) ⊗ N making Am,n a generalized
nilpotent cdga over N , and a map Am,n → A of cdgas over N that is an isomorphism on
H i (−)(j) for 1 ≤ i ≤ n, j ≤ m, and an injection for i = n + 1, j ≤ m. In addition, writing
An,m = N ⊕ In,m , we have
H p (Im,n (r)) = 0 for r > m, p ≤ 1.
(2.4.1)
We extend En (m) to En (m + 1) by simply repeating the construction for En (1) described
above, but working in Adams degree m + 1 rather than 1; using (2.4.1) allows us to start
the construction by adjoining generators for H 1 (I(m + 1)), just as in the case of Adams
weight 1. Again, as N (r) = 0 for r < 0 and N (0) = Q · id, the inclusion Am,n → Am+1,n is
an isomorphism in Adams degree ≤ m. In addition, the argument used to prove the claim
shows that (2.4.1) extends from m to m + 1 and the induction goes through.
Taking En := ∪m En (m), we thus have a differential on A{n}N := Sym∗ En ⊗ N making
A{n}N a generalized nilpotent cdga over N , and a map A{n}N → A of cdgas over N that
is an isomorphism on H i (−) for 1 ≤ i ≤ n and an injection for i = n + 1, completing the
proof.
49
Remark 2.4.11 Suppose that both N and A are cohomologically connected. Then
A{n}N → A
induces an isomorphism on H i for all i ≤ n. In particular, the map A{∞}N → A is a
quasi-isomorphism.
Proposition 2.4.12 Suppose that N is cohomologically connected, A ∈ cdgaN . Let s :
AN → A, s0 : A0N → A be relative minimal models. Then there is an isomorphism φ :
AN → A0N in cdgaN such that s0 ◦ φ ∼Ω s.
Proof By definition, the maps s, s0 are weak equivalences in cdgaN , and thus we have the
isomorphism in H(cdgaN )
s0−1 s : AN → A0N .
Since AN and AN are both generalized nilpotent N -algebras, AN and AN are both cofibrant
(see lemma 2.4.4), and thus there is a morphism φ : AN → A0N in cdgaN representing
the isomorphism s0−1 s in H(cdgaN ). Thus φ is a weak equivalence and s0 ◦ φ ∼Ω s. By
proposition 2.4.6, φ is an isomorphism in cdgaN .
Thus, the relative minimal model is unique up to (non-canonical) isomorphism in cdgaN .
In fact, in case N is connected, the same holds for the relative n-minimal models. For this,
we first note the following simple extension of proposition 2.4.9.
Lemma 2.4.13 Suppose that N is cohomologically connected. Let sn : A{n}N → A be an
n-minimal model for some n, 1 ≤ n < ∞. Then there is a monomorphism of generalized
nilpotent cdgas over N , i : A{n}N → AN , such that
1. The morphism sn : A{n}N → A extends to a morphism s : AN → A in cdgaN .
2. s : AN → A is a relative minimal model of A.
If in addition N is connected, then A{n}N is equal to the N -subalgebra AN (n) of AN generated by elements x ∈ AN with deg x ≤ n.
Proof Write A{n}N = Sym∗ En ⊗ N as an N -algebra, where En is a bi-graded Q-vector
space with filtration satisfying the conditions of remark 2.4.2 and such that each e ∈ En
has deg e ≤ n. We now just apply the inductive construction of the relative minimal model
of A as given in the proof of proposition 2.4.9, starting with the generating vector space
En , to construct a relative minimal model AN → A as an augmented N -algebra containing
A{n}N . This proves (1).
Suppose N is connected. Let E ⊂ En be the bi-graded Q vector space of N -algebra
generators for AN constructed by the inductive procedure of proposition 2.4.9. Then E ≤n =
En≤n ; as N is connected, this immediately implies A{n}N = AN (n) .
Proposition 2.4.14 Suppose that N is connected and take A ∈ cdgaN . Suppose we have
relative n-minimal models
sn : A{n}N → A; s0n : A{n}0N → A.
Then there is an isomorphism φn : A{n}N → A{n}0N in cdgaN such that s0n ◦ φn ∼Ω sn .
50
Proof By lemma 2.4.13, we may extend sn and s0n to relative minimal models s : AN → A,
s0 : A0N → A, giving us commutative diagrams
0
i
/ A0
N
II
II
II
s0
s0n III $
i
/ AN
HH
HH
H
s
sn HHH
H$ A{n}0N
A{n}N
A;
A
in cdgaN , such that i and i0 are monomorphisms, giving identifications
A{n}N = AN (n) ; A{n}0N = A0N (n) .
By proposition 2.4.12 there is an isomorphism φ : AN → A0N in cdgaN with s0 ◦ φ ∼Ω s.
Restricting φ to AN (n) gives the isomorphism
φn : A{n}N = AN (n) → A{n}0N = A0N (n)
and a choice of a right homotopy h : AN → A ⊗Q Ω between s0 ◦ φ and s restricts to a right
homotopy between s0n ◦ φ and sn .
Remark 2.4.15 A generalized nilpotent cdga over N is automatically a cell-module over
N . Indeed, for A = Sym∗ E ⊗ N satisfying the conditions of definition 2.4.1, one has the
filtration on E ≤n given by remark 2.4.2. Combining this filtration with the filtration by
degree on Sym∗ E gives a filtration on Sym∗ E which exhibits A as an N -cell module.
2.5
Relative bar construction
One forms the bar construction for a cdga A over N just as for cdgas over Q, replacing
⊗Q with ⊗N . However, for this construction to have good cohomological properties, one
should replace A with a quasi-isomorphic cdga A0 which is a cell module over N , so that
⊗N = ⊗LN . This is accomplished by using the minimal model A{∞}. In any case, we give
the “pre-derived” definition for an arbitrary cdga A over N .
Definition 2.5.1 Let A be an augmented Adams graded cdga over N . Define the simplicial
cdga B•pd (A/N ) by
B•pd (A/N ) := A⊗N [0,1]
The inclusion {0, 1} → [0, 1] makes B•pd (A/N ) a simplicial cdga over A ⊗ A. Given two
(possibly equal) augmentations 1 , 2 : A → N , set
B•pd (A/N , 1 , 2 ) := B•pd (A/N ) ⊗A⊗A N .
pd
(A, 1 , 2 ) be the total complex associated to B•pd (A/N , 1 , 2 ).
and let B̄N
Remark 2.5.2 Let A be a generalized nilpotent algebra over N , and write A, as a bi-graded
N -algebra, as
A = Sym∗ E ⊗Q N ,
51
where E is a bi-graded Q-vector space satisfying the conditions of definition 1.11.1. Since
A(r) = 0 for r < 0 and A(0) = Q · id, each bi-homogeneous element e ∈ E has Adams degree
|e| ≥ 1. Thus, W−1 A = 0. Since
A⊗N n ∼
= Sym∗ E ⊗Q n ⊗Q N
the same holds for A⊗N n . In particular, A⊗N n is in CM+w
N for each n ≥ 0.
pd
As B̄N
(A, 1 , 2 ) is the total complex of a double complex built out of the A⊗N n , we see
pd
that the dg N -module B̄N
(A, 1 , 2 ) is in CM+w
N . Finally, if 1 = 2 = , then, using the
pd
+w
same formulas as in §1.2, B̄N (A, ) has the natural structure of a dg Hopf algebra in CMN
,
+w
and thus a Hopf algebra in DN
.
Definition 2.5.3 Let A be an augmented Adams graded cdga over N with augmentation
. Suppose that N is cohomologically connnected and let A{∞}N → A be the relative
minimal model of A over N . Define
pd
B• (A/N ) := B•pd (A{∞}N /N ), B̄N (A, ) := B̄N
(A{∞}N , {∞}).
Remarks 2.5.4
1. Still supposing N to be cohomologically connected, we may apply the truncation functor
+w
0
HN
: DN
→ HN
+w
0
to the dg Hopf algebra B̄N (A, ) in DN
, giving us the Hopf algebra HN
(B̄N (A, )) in HN .
0
We may also form the co-Lie algebra object γA/N in HN = ConnN :
0
0
γA/N := HN
(B̄N (A, ))+ /HN
(B̄N (A, ))2+
0
0
(B̄N (A, )) the augmentation ideal.
(B̄N (A, ))+ ⊂ HN
with HN
We let B̄•≤m (A/N , ) denote the restriction of the simplicial object B̄• (A/N , ) to the full
≤m
subcategory {[0], . . . , [m]} of Ord, and B̄N
(A, ) ⊂ B̄N (A, ) the associated total complex
of B̄•≤m (A/N , ).
f
f
≤m
0
If we suppose that A is in DN
, then HN
(B̄N
(A, )) is in HN
for each m, hence
f
0
HN (B̄N (A, )) has the structure of an ind-Hopf algebra in HN with
≤m
0
0
(B̄N (A, )) = lim HN
(B̄N
(A, ))
HN
−→
m→∞
in HN .
2. In our definition of B• (A/N ), we made a choice of a relative minimal model of A; by
proposition 2.4.12, this choice is unique up to (non-unique) isomorphism, and thus the same
is true for B• (A/N ). Furthermore, two different relative minimal models are canonically
0
isomorphic in H(cdgaN ), and therefore the Hopf algebra object HN
(B̄N (A, )) is indepenf
dent of the choice of relative minimal model, up to unique isomorphism. In case A is in DN
,
f
0
the same holds for HN (B̄N (A, )) as an ind-Hopf algebra in HN .
52
2.6
Base-change
We consider a quasi-isomorphism φ : N 0 → N of cohomologically connected cdgas. Given
an augmented cdga A over N with augmentation : A → N , we have A = I ⊕ N , with I
the kernel of . In particular, I is a (non-unital) N -algebra. Via φ, we make I a (non-unital)
N 0 -algebra, and thus give A0 := I ⊕ N 0 the structure of a cdga over N 0 , with augmentation
0 : A0 → N 0 the projection on N 0 with kernel I.
This construction yields the commutative diagram of cdgas
φ0
0
A
O
p0
p
0
N0
/A
O
φ
/
(2.6.1)
N
with φ and φ0 quasi-isomorphisms.
Now let f 0 : A0 {n}N 0 → A0 be a relative n-minimal model over A0 over N 0 . Since the
composition φ0 f 0 : A0 {n}N 0 → A is an N 0 -module map, φ0 f 0 factors through a unique map
f : A0 {n}N 0 ⊗N 0 N → A
of cdgas over N . Similarly, the N 0 -augmentation of A0 {n}N 0 induces an N -augmentation of
A0 {n}N 0 ⊗N 0 N , making f a map of augmented cdgas over N .
Lemma 2.6.1 f : A0 {n}N 0 ⊗N 0 N → A is a relative n-minimal model of A over N .
Proof As A0 {n}N 0 is a generalized nilpotent algebra over N 0 , with generators in degree
≤ n, the same follows for A0 {n}N 0 ⊗N 0 N as an algebra over N . φ is a quasi-isomorphism, so
φ∗ : DN 0 → DN is an equivalence of triangulated categories. We can compute cohomology
of a dg module via maps in the derived category; as A0 {n}N 0 is an N 0 -cell module, we have
φ∗ (A0 {n}N 0 ) = A0 {n}N 0 ⊗N 0 N , hence the canonical map
A0 {n}N 0 → A0 {n}N 0 ⊗N 0 N
is a quasi-isomorphism of cdgas. Since φ0 : A0 → A is a quasi-isomorphism, and A0 {n}N 0 →
A0 is a relative n-minimal model, the map on H i induced by f is an isomorphism for 1 ≤ i ≤ n
and an injection for i = n + 1, i.e., f : A0 {n}N 0 ⊗N 0 N → A is a relative n-minimal model.
Remark 2.6.2 Still assuming N and N 0 cohomologically connected, write A{n}N for the
n-minimal model A0 {n}N 0 ⊗N 0 N . We have the change of rings isomorphism
⊗
m
Nm
A0 {n}NN0 0 ⊗N 0 N → A{n}⊗
N
and the quasi-isomorphism
⊗
A0 {n}NN0 0
m
⊗
m
→ A0 {n}NN0 0 ⊗N 0 N
Thus on the bar construction
β
α
pd
pd
pd
0
0
0
0
B̄N
→ B̄N
− B̄N
(A{n}N , )
0 (A {n}N 0 , ) −
0 (A {n}N 0 , ) ⊗N 0 N →
53
the map α is a quasi-isomorphism and the map β is an isomorphism.
In particular, taking n = ∞, we have the canonical isomorphism
0
0 0
∼ 0
φ∗ (HN
0 (B̄N 0 (A , ))) = HN (B̄N (A, ))
of Hopf algebra objects in HN . Since φ∗ : HN 0 → HN is an equivalence, we are thus free to
0
replace N with a quasi-isomorphic N 0 in a study of HN
(B̄N (A, )). For instance, we may
use the minimal model N {∞} → N as a replacement for N .
2.7
Connection matrices
Generalized nilpotent algebras over N fit well into the connection matrix point of view
described in section 1.13. Indeed, suppose that A = Sym∗ E ⊗ N is generalized nilpotent
over N , with augmentation : A → N induced by writing Sym∗ E = Q ⊕ Sym∗≥1 E.
Using the augmentation of N , we write N = Q · id ⊕ N + , which writes A as
A = Sym∗ E ⊗ id ⊕ Sym∗ E ⊗ N + .
Thus, the differential on A is completely determined by its restriction to Sym∗ E ⊗ id, giving
the decomposition
d = d0 + Γ
with d0 a differential on Sym∗ E and Γ : Sym∗ E → Sym∗ E ⊗ N + a flat connection. In
addition, (Sym∗ E, d0 ) an Adams graded cdga over Q with augmentation 0 induced by the
projection to Sym0 E = Q. Finally, the connection Γ is nilpotent since Sym∗ E has all Adams
degrees ≥ 0 (lemma 1.13.3).
Using the tensor structure in the category of flat nilpotent connections, the flat nilpotent
connection Γ : Sym∗ E → Sym∗ E ⊗N + gives rise to a flat nilpotent connection on (Sym∗ E)⊗n
for all n. These fit together to give a flat nilpotent connection on the bar construction:
B̄(Γ) : B̄((Sym∗ E, d0 ), 0 ) → B̄((Sym∗ E, d0 ), 0 ) ⊗ N + .
This defines a Hopf algebra object in ConnN .
Proposition 2.7.1 Let N be cohomologically connected. The N -cell module corresponding
pd
to B̄((Sym∗ E, d0 ), 0 ) with flat nilpotent connection B̄(Γ) is isomorphic to B̄N
(A, ), as dg
+w
Hopf algebra objects in CMN .
Proof We check instead the equivalent statement that the dg Hopf algebra in ConnN
pd
corresponding to B̄N
(A, ) is (B̄((Sym∗ E, d0 ), 0 ), B̄(Γ)).
We note that we have canonical isomorphisms
A⊗N n ∼
= (Sym∗ E)⊗Q n ⊗Q N = (Sym∗ E)⊗Q n ⊗ id ⊕ (Sym∗ E)⊗Q n ⊗Q N +
respecting differentials and multiplications. Tracing this isomorphism through the definition
we have given of the flat nilpotent connection on B̄((Sym∗ E, d0 ), 0 ) completes the proof.
54
2.8
Semi-direct products
Let : A → N be an augmented Adams graded cdga over N . We suppose that N is generalized nilpotent and that A is generalized nilpotent over N . We let GA := Spec H 0 (B̄(A)),
GN := Spec H 0 (B̄(N )) be the Q-algebraic group schemes defined with respect to the canonical augmentations A → Q, N → Q. The N -algebra structure π ∗ : N → A induces the map
of algebraic groups π : GA → GN ; the augmentation gives a splitting s : GN → GA to π.
Lemma 2.8.1 The map π is flat.
Proof Following our remarks in §1.2, H 0 (B̄(A)) and H 0 (B̄(N )) are polynomial algebras
over Q on A1 , N 1 respectively, and the map
H 0 (B̄(π ∗ )) : H 0 (B̄(N )) → H 0 (B̄(A))
is just the polynomial extension of the linear injection
π ∗ : N 1 → A1 .
That is, H 0 (B̄(π ∗ )) identifies H 0 (B̄(A)) with a polynomial extension of H 0 (B̄(N )).
Lemma 2.8.2 Let e denote the identity in GN . The fiber π −1 (e) is canonically isomorphic
to Spec H 0 (B̄(A ⊗N Q)) as group schemes over Q.
Proof We have the natural map of Hopf algebras
H 0 (B̄(A)) ⊗H 0 (B̄(N )) Q → H 0 (B̄(A ⊗N Q)).
Writing A = Sym∗ E ⊗ N as an N -algebra, H 0 (B̄(A ⊗N Q)) is a polynomial algebra
on (Sym∗ E)1 , while H 0 (B̄(A)) is the polynomial algebra on A1 = (Sym∗ E)1 ⊕ N 1 , and
H 0 (B̄(N )) is the polynomial algebra on N 1 . This shows that the above map is an algebra
isomorphism.
Set K := Spec H 0 (B̄(A ⊗N Q)) = Spec H 0 (B̄(Sym∗ E)). The splitting s gives an action
of GN on K and an isomorphism of GA with the semi-direct product
GA ∼
= K n GN .
Let Ks denote the Q-group scheme K with this GN -action.
On the other hand, we have seen (proposition 2.7.1) that writing A = Sym∗ E ⊗ N gives
Sym∗ E a flat nilpotent connection
Γ : Sym∗ E → N + ⊗ Sym∗ E
0
and an isomorphism of HN
(B̄N (A)) with H 0 (B̄(Sym∗ E)) as Hopf algebras in Conn0N .
Replacing N with its 1-minimal model, and noting that Conn0N ∼ Conn0N {1} we have
the canonical structure of H 0 (B̄(Sym∗ E)) as a Hopf algebra in the category of co-modules
over the co-Lie algebra QN = γN (remark 1.14.8). But this category is equivalent to the
category of representations of GN , giving us another action of GN on K.
55
Theorem 2.8.3 The action of GN on K = Spec H 0 (B̄(Sym∗ E)) induced by the splitting s
is the same as the action given by the flat nilpotent N -connection Γ on Sym∗ E. In other
words, there is an isomorphism
0
Ks ∼
(B̄N (A))
= Spec HN
as Q-group schemes with GN -action.
Proof It suffices to check that the two co-actions of the co-Lie algebra γN are the same, in
fact, it suffices to check that the two co-actions of γN on the co-Lie algebra γSym∗ E of K are
the same.
By Quillen’s theorem (theorem 1.15.2(2)), we can identify the co-Lie algebras γA , γN and
γSym∗ E with QA, QN and QSym∗ E, respectively. Since we are assuming A and N are both
generalized nilpotent, QA, QN and QSym∗ E are the respective co-Lie algebras
dA : A1 → Λ2 A1 , dN : N 1 → Λ2 N 1 , dE : E 1 → Λ2 E 1 .
On the level of co-Lie algebras, the splitting s is just the decomposition of A1 = (Sym∗ E⊗N )1
as
A1 = (Sym∗ E)1 ⊕ N 1 .
The co-action of N 1 on A1 determined by the splitting s is therefore given by dA followed
by the projection of Λ2 A1 on N 1 ⊗ A1 via the isomorphism
Λ2 A1 = Λ2 ((Sym∗ E)1 ⊕ N 1 ) ∼
= Λ2 (Sym∗ E)1 ⊕ N 1 ⊗ (Sym∗ E)1 ⊕ Λ2 N 1 .
This induces the co-action of N 1 on (Sym∗ E)1 by taking the composition
d
A
(Sym∗ E)1 → A1 −→
Λ2 A1 → N 1 ⊗ (Sym∗ E)1 .
Via our identifications, this gives us the co-action of γN on γSym∗ E determined by the section
s.
On the other hand, the flat nilpotent connection Γ on Sym∗ E giving the isomorphism of
0
HN
(B̄N (A)) with H 0 (B̄(Sym∗ E)) in Conn0N is just the restriction of dA to Sym∗ E followed
by the projection of A = N ⊗ Sym∗ E to N + ⊗ Sym∗ E. However, by reasons of degree, the
restriction of dA to (Sym∗ E)1 = E 1 decomposes as
dA : E 1 → Λ2 E 1 ⊕ N 1 ⊗ E 1
from which it follows that Γ : E 1 → N 1 ⊗ E 1 is the same as the co-action defined by s.
3
Motives over a base
This section summarizes the material we need from the work of Cisinski-Déglise [10].
56
3.1
Effective motives over a base
We summarize the main points of the construction of the category DM eff (S) of effective
motives over S from [10]; we will describe the category DM (S) of motives over S in the next
subsection. Although S is allowed to be a quite general scheme in [10], we restrict ourselves
to the case of a base-scheme S that is separated, smooth and essentially of finite type over
a field. We let SchS denote the category of finite type separated S-schemes and let Sm/S
denote the full subcategory of SchS consisting of smooth S-schemes.
For X, Y ∈ Sm/S, define the group of finite S-correspondences cS (X, Y ) as the free
abelian group on the integral closed subschemes W ⊂ X ×S Y with W → X finite and
surjective over an irreducible component of X.
For X, Y, Z in Sm/S, let pXY , pY Z and pXZ be the evident projections from X ×S Y ×S Z.
One checks that the formula
W ◦ W 0 := pXZ∗ (p∗XY (W ) · p∗Y Z (W 0 )) ∈ cS (X, Z)
(3.1.1)
where · is the intersection product on X ×S Y ×S Z, is well-defined for all W ∈ cS (X, Y ),
W 0 ∈ cS (Y, Z); this follows from the fact that supp (W ) ×S Z ∩ X ×S supp (W 0 ) is finite over
X and each irreducible component of this intersection dominates a component of X. This
is called the composition of correspondences.
We start with the category SmCor(S). Objects are the same as Sm/S, morphisms are
HomSmCor(S) (X, Y ) := cS (X, Y )
with composition law given by the formula (3.1.1). Sending a morphism f : X → Y in
Sm/S to the graph of f , Γf ⊂ X ×S Y , defines an embedding iS : Sm/S → SmCor(S).
Note that SmCor(S) is an additive category, with direct sum induced by disjoint union.
Define the abelian category of presheaves with transfer on Sm/S, PST(S), as the category
of additive presheaves of abelian groups on SmCor(S). We have the representable presheaves
tr
Ztr
S (Z) for Z ∈ Sm/S defined by ZS (Z)(X) := cS (X, Z) and pull-back maps given by the
composition of correspondences. The full subcategory Shtr
Nis (S) of PST(S) has objects the
presheaves P such that the restriction P ◦ iS of P to a presheaf on Sm/S is a sheaf for the
tr
Nisnevich topology. For instance, the presheaves Ztr
S (Z) are in ShNis (S).
tr
Both PST(S) and ShNis (S) are Grothendieck abelian categories, with set of generators
given by the objects Ztr (X), X ∈ Sm/S.
For an additive category A, we let C(A) denote the category of unbounded complexes
over A. One gives the category C(Shtr
Nis (S)) the model structure of [9, example 1.6, theorem
1.7], that is, cofibrations are generated by maps of the form
σX [n] : Ztr (X)[n] → DX [n]; X ∈ Sm/S, n ∈ Z,
where DX is the cone on the identity map Ztr (X) → Ztr (X), and σX : Ztr (X) → DX is
the canonical map. Here “generated” means that the class of cofibrations is the smallest
collection of morphisms in C(Shtr
Nis (S)) containing the maps σX [n] and closed under pushouts, transfinite compositions and retracts. The weak equivalences are the quasi-isomophisms
(for the Nisnevich topology) and the fibrations are as usual the morphisms having the right
lifting property with respect to acyclic cofibrations. We denote this model structure by
57
tr
C(Shtr
Nis (S))Nis . In particular, the homotopy category of C(ShNis (S))Nis is equivalent to the
(unbounded) derived category D(Shtr
Nis (S)).
The operation
0
0
tr
tr tr
Ztr
S (X) ⊗S ZS (X ) := ZS (X ×S X )
extends to a tensor structure ⊗tr
S making PST(S) a tensor category: one forms the canonical
left resolution L(F) of a presheaf F by taking the canonical surjection
L0 (F) :=
M
φ0
→F
Ztr
S (X) −
X∈Sm/S,s∈F (X)
setting F1 := ker φ0 and iterating, giving the canonical resolution of F in terms of representable presheaves
L(F) → F := . . . → L1 (F) → L0 (F) → F → 0.
(3.1.2)
One then defines
tr
F ⊗tr
S G := H0 (L(F) ⊗S L(G))
noting that L(F) ⊗tr
S L(G) is defined since both complexes are degreewise direct sums of representable presheaves. One makes Shtr
Nis (S) a tensor category by taking the sheaf associated
to the presheaf tensor product; we also denote this tensor product by ⊗tr
S , using the context
to distinguish the presheaf and sheaf tensor products.
tr
Note that the objects Ztr
S (X) of ShNis (S) are weakly flat in the sense of [9, §2.1] and that
tr
tr
{ZS (X), X ∈ Sm/S} is a set of weakly flat generators of Shtr
Nis (S), closed under ⊗S . Thus,
by [9, proposition 2.3, proposition 2.8], the usual extension of ⊗tr
S to a tensor product on
tr
tr
C(ShNis (S)) makes C(ShNis (S))Nis a closed symmetric monoidal model category, and ⊗tr
S
defines a left-derived tensor product
tr
tr
⊗LS : D(Shtr
Nis (S)) × D(ShNis (S)) → D(ShNis (S)),
which makes D(Shtr
Nis (S)) a triangulated tensor category.
Definition 3.1.1 ([9, example 3.15]) DM eff (S) is the localization of the triangulated
category D(Shtr
Nis (S)) with respect to the localizing category generated by the complexes
1
eff
tr
eff
Ztr
(X
×
A
)
→
Ztr
S
S (X), X ∈ Sm/S. Denote by mS (X) the image of ZS (X) in DM (S).
Remark 3.1.2 The following facts are direct consequences of [9, proposition 3.5]:
1. DM eff (S) is a triangulated tensor category with tensor product ⊗S induced from the
tensor product ⊗LS via the localization map
eff
QS : D(Shtr
Nis (S)) → DM (S),
eff
eff
and satisfying meff
S (X) ⊗S mS (Y ) = mS (X ×S Y ).
tr
2. C(Shtr
Nis (S)) has a model category structure C(ShNis (S))A1 , defined as the Bousfield localtr
1
tr
ization of C(Shtr
Nis (S))Nis with respect to the set of complexes {ZS (X × A ) → ZS (X), X ∈
tr
eff
Sm/S}, and the homotopy category of C(ShNis (S))A1 is equivalent to DM (S).
58
eff
3. Let DM∞
(S) ⊂ D(Shtr
Nis (S)) be the full subcategory consisting of complexes C which
1
are A -homotopy invariant, that is, the map
p∗ : Hn (XNis , C) → Hn (X × A1Nis , C)
eff
is an isomorphism for all X and n. Then DM∞
(S) is a triangulated subcategory of
tr
tr
eff
D(ShNis (S)), and the inclusion DM∞ (S) → D(ShNis (S)) admits a left adjoint
eff
LA1 : D(Shtr
Nis (S)) → DM∞ (S),
eff
which descends via the localization functor D(Shtr
Nis (S)) → DM (S) to define an equivalence
eff
LA1 : DM eff (S) → DM∞
(S)
of triangulated categories.
3.2
T tr -spectra and the category of motives
We now recall the construction of the category DM (S). This is given by “inverting” tensor
product with the Lefschetz motive, done via the category of symmetric T tr -spectra
Remark 3.2.1 Hovey [19] has formed a general machine for the construction of model
structures on categories of spectra over a model category M with respect to an endofunctor
T . Some of his results require the technical assumption that M be weakly finitely generated.
This property of C(Shtr
Nis (S))A1 does not appear to be directly addressed in either [9] or [10],
however, the arguments of [13, lemma 2.15, corollary 2.16] do show this. The main point
is that the Brown-Gersten property and A1 -homotopy invariance for a schemewise fibrant
complex in C(Shtr
Nis (S)) is implied by having the RLP with respect to the finite complexes
corresponding to an elementary Nisnevich square, or a projection A1 × X → X. We will use
the weak finite generation of C(Shtr
Nis (S))A1 without further mention in the sequel.
Definition 3.2.2 Let T tr be the presheaf with transfers
i
∞∗
1
T tr := coker(Ztr
−
→ Ztr
S (S) −
S (P ))
tr
tr
⊗S n
and let ZS (1) be the image in DM eff (S) of T tr [−2]. We often write Ztr
S (n) for (T [−2])
tr
eff
and ZS (n) for the image of ZS (n) in DM (S).
Note that, as a summand of the cofibrant object ZStr (P1 )), T tr is cofibrant.
Let SptT tr (S) be the category of T tr spectra in C(Shtr
Nis (S))A1 with the stable model
structure: Objects are sequence E := (E0 , E1 , . . .), En ∈ C(Shtr
Nis (S)), with bonding maps
tr
n : En ⊗tr
→ En+1 .
S T
Morphisms are given by sequences of maps in C(Shtr
Nis (S)) which strictly commute with the
respective bonding maps. We will describe the model structure below.
tr
We let SptS
spectra in C(Shtr
T tr (S) be the category of symmetric T
Nis (S))A1 with the
stable model structure. Objects are sequences E := (E0 , E1 , . . .), En ∈ C(Shtr
Nis (S)), with
En endowed with an action of the symmetric group Sn , together with bonding maps
tr
n : En ⊗tr
→ En+1 .
S T
59
One requires in addition that, for all n ≥ 0, m ≥ 1, the iterated bonding map
n ⊗id(T tr )⊗m
n+m−1
tr ⊗m−1
tr ⊗m
−−−−−−−−→ En ⊗tr
→ . . . → En+m−1 ⊗ T tr −−−−→ En+m
En ⊗tr
S (T )
S (T )
is Sn ×Sm equivariant, with respect to the standard inclusion Sn ×Sm ⊂ Sn+m . Morphisms
are given by sequences of maps f = {fn } in C(Shtr
Nis (S)) which strictly commute with the
respective bonding maps, and with fn being Sn -equivariant for each n.
The model structure on the category of T tr -spectra is defined by following the construction of Hovey [19]. For an object A ∈ C(Shtr
Nis (S)), and integer i ≥ 0, we have the object
A{−i} of SptT tr (S), with A{−i}i+n = A ⊗ (T tr )⊗n , and A{−i}n = 0 for n < i; sending
A to A{−i} defines a functor (−){−i}. The projective model structure on SptT tr (S) has
generating cofibrations the maps of the form f {−i} with f a cofibration in C(Shtr
Nis (S)),
and with weak equivalences and fibrations being those maps f = {fn } with each fn a weak
equivalence, resp. fibration. We let SptT tr (S)proj denote this model category.
Next, one defines the notion of a T tr -Ω spectrum, this being a T tr -spectrum E =
(E0 , E1 , . . .) such that each En is fibrant in C(Shtr
Nis (S))A1 , and such that the map En →
tr
Hom(T , En+1 ) adjoint to n is a weak equivalence in C(Shtr
Nis (S))A1 . A stable weak equivalence f : A → B is a map in SptT tr (S) such that the induced map
f ∗ : HomH(SptT tr (S)proj ) (B, E) → HomH(SptT tr (S)proj ) (A, E)
is an isomorphism for all T tr -Ω spectra E. The model category SptT tr (S)s is the Bousfield
localization of the model category SptT tr (S)proj with respect to stable weak equivalences.
In the symmetric setting, one does exactly the same, except that we use a symmetric
version A{−i}S of A{−i}. Explicitly,
tr
tr ⊗n
A{−i}S
,
n+i := Sn+i ×Sn A ⊗ (T )
with the evident bonding maps. This gives us the model category SptS
T tr (S)s with the stable
model structure.
Definition 3.2.3 The “big” category of triangulated motives over S, DM (S), is the homo0
topy category of SptS
T tr (S)s . We write DM (S) for the homotopy category of SptT tr (S)s .
Remarks 3.2.4
1. The homotopy categories of SptS
T tr (S)s and SptT tr (S)s are triangulated categories [10,
proposition 3.4, definition 3.8, §4.12, §6.9]. In addition, one can define additive categories of
tr
S
T tr -spectra and symmetric T tr -spectra Spt(Shtr
Nis (S)) and Spt (ShNis (S)), so that
tr
S
S
∼
∼
C(Spt(Shtr
Nis (S))) = SptT tr (S); C(Spt (ShNis (S))) = SptT tr (S),
giving SptT tr (S) and SptS
T tr (S) a dg structure.
tr
tr
tr
tr ⊗n
2. Sending A ∈ C(Shtr
, . . .) defines
Nis (S)) to the sequence (A, A ⊗ T , . . . , A ⊗ (T )
functors
tr
Σ∞
T : C(ShNis (S)) → SptT tr (S)
tr
S
Σ∞
T : C(ShNis (S)) → SptT tr (S)
60
(the symmetric version uses the permutation action on (T tr )⊗n and the trivial action on A),
left-adjoint to the projection (E0 , . . .) 7→ E0 . These induce an adjoint pair of exact functors
on the homotopy categories
eff
Σ∞
t : DM (S) o
eff
Σ∞
t : DM (S) o
/
/
DM (S)0 : Ωt
DM (S) : Ωt
(see [10, §4.12]).
3. Forgetting the action of the symmetric groups defines a functor u : SptS
T tr (S) →
SptT tr (S). By [10, theorem 6.10], this induces an equivalence of triangulated categories
u : DM (S)0 → DM (S).
We let
mS : Sm/S → DM (S)
be the composition
meff
Σ∞
t
Sm/S −−S→ DM eff (S) −−→
DM (S).
We will use the following fundamental result from [10].
Theorem 3.2.5 ([10, section 10.4]) Suppose that S is in Sm/k for a field k, take X in
Sm/S, and let mk (X), mS (X) denote the motives of X in DM (k), DM (S), respectively.
Then there is a natural isomorphism
HomDM (S) (mS (X), ZS (n)[m]) ∼
= HomDM (k) (mk (X), Zk (n)[m])
3.3
Tensor product in SptS
T tr (S)
S
Let C = C(Shtr
Nis (S)), and let C be the category of sequences E = (E0 , E1 , . . .), with En an
object of C endowed with an Fn -action; morphisms are sequences f = {fn } of morphisms in
C, with fn Sn -equivariant.
For E = (E0 , E1 , . . .), F = (F0 , F1 , . . .) in C S , one defines
tr
∼
(E^
⊗tr
S F )n := ⊕p+q=n,α:{1,...,p}q{1,...,q}−
→{1,...,n} Ep ⊗S Fq ,
where α runs over all bijections of sets. Using the evident operation of Sn on the set of
∼
bijections {1, . . . , p} q {1, . . . , q} −
→ {1, . . . , n}, the Sp × Sq action on Ep ⊗tr Fq induces
S
tr
^
an Sn -action on (E^
⊗tr
S F )n , giving us the object E ⊗S F of C . This defines a symmetric
monoidal structure on C S .
tr
Let Sym(T tr ) be the sequence n 7→ (T tr )⊗ n . Then Sym(T tr ) is a commutative monoid
tr
S
object in C S , and SptS
T tr (S)) is just the category of (right) Sym(T )-modules in C . Thus,
(see [21, lemmas 2.2.2 and 2.2.8]) the symmetric monoidal structure on C S induces a canonical
tr
symmetric monoidal structure on SptS
T tr (S), which we denote by ⊗S .
tr
By [19, theorem 8.11], the symmetric monoidal operation ⊗S defines a tensor operation
⊗S on the homotopy category DM (S), making DM (S) a triangulated tensor category. In
tr
addition, the suspension spectra Σ∞
t (ZS (X)) are flat (in the sense of [9, proposition 6.35]),
and we have
61
Proposition 3.3.1 ([19, theorem 8.10]) The functor
− ⊗ T tr : DM (S) → DM (S)
is an equivalence.
3.4
Motives with Q-coefficients
tr
We replace the category Shtr
Nis (X) with the category of sheaves of Q-vector spaces ShNis (S)Q ,
tr
1
eff
giving us the derived category D(ShNis (S)Q ) and the A -localization DM (S)Q . This latter
category is the homotopy category of the model category C(Shtr
Nis (S)Q )A1 , defined exactly
tr
as C(ShNis (S))A1 .
tr
We have the evident Q-linearization functors, e.g., from Shtr
Nis (S) to ShNis (S)Q , which we
denote as M 7→ MQ , and we have isomorphisms
Hom? (M, N ) ⊗ Q ∼
= Hom?Q (MQ , NQ ).
tr
tr
and Ztr
We write TQtr and Qtr
S (n) in C(ShNis (S)Q ), and write QS (n)
S (n) for the image of T
eff
for the image of Qtr
S (n) in DM (S)Q .
We have the model categories of TQtr -spectra and TQtr -symmetric spectra, SptTQtr (S) and
0
SptS
TQtr (S), with homotopy categories DM (S)Q and DM (S)Q , respectively.
One can also easily compare spectra and symmetric spectra: send E = (E0 , E1 , . . .) in
SptTQtr (S) to the same sequence E = (E0 , E1 , . . .) with the same bonding maps; we denote
this functor as
ι : SptTQtr (S) → SptS
T tr (S).
Q
The homtopy inverse sends a sequence E = (E0 , E1 , . . .) in SptS
T tr (S) to the sequence of
Q
S∗ -invariants E S∗ := (E0 , E1 , E2S2 , . . .),
?S∗ : SptS
T tr (S) → SptTQtr (S).
Q
Since E S∗ is a summand of E in SptS
TQtr (S), these operations give well-defined functors
on the homotopy categories.
Proposition 3.4.1 The functors
ι : DM (S)0Q → DM (S)Q
?S∗ : DM (S)Q → DM (S)0Q
are inverse equivalences.
Proof We use throughout the motivic model structures, without putting this explicitly into
the notation. We recall the proof of the equivalence of DM (S)0 with DM (S) as given by
[19, theorems 10.1, 10.3]. This is done by comparing the model categories SptT tr (SptS
T tr (S))
S
tr
and SptT tr (SptT tr (S)). Indeed, ⊗T is a equivalence on the respect homotopy categories,
by [19, theorem 8.10] for SptS
T tr (S) and by [19, theorem 10.3] for SptT tr (S) (this is where
62
one uses the fact that the cyclic permutation of T tr ⊗ T tr ⊗ T tr is homotopic to the identity).
Thus, by [19, theorems 5.1 and 9.1], the infinite suspension functors
S
S
Σ∞
T tr : SptT tr (S) → SptT tr (SptT tr (S))
S
ΣS∞
T tr : SptT tr (S) → SptT tr (SptT tr (S))
also induce equivalences on the homotopy categories. The equivalence DM (S)0 ∼ DM (S)
is then induced by the isomorphism τ
S
∼
SptT tr (SptS
T tr (S)) = SptT tr (SptT tr (S))
defined by “exchanging indices”: an object Y of the left-hand category is a doubly indexed
collection of objects of C(Shtr
Nis (S)), Y = {Ym,n }, where Sn acts on Ym,n , the two bonding
tr
maps Ym,n ⊗ T → Ym+1,n and Ym,n ⊗ T tr → Ym,n+1 are Sn -equiviariant, and the `-fold
iterated bonding map in the second variable is Sn × S` equivariant. SptS
T tr (SptT tr (S)) has
a similar description, with the symmetric variable being the first one, so sending Y = {Ym,n }
0
0
:= Yn,m defines the isomorphism.
}, with Ym,n
to Y 0 = {Ym,n
S
We apply our functor ?S∗ to SptTQtr (SptS
TQtr (S)) and SptTQtr (SptTQtr (S)), giving the commutative diagram
SptTQtr (SptS
T tr (S))
Q
/
τ
SSS
SSS
SSS
SS
SptT tr (?S∗ ) SSS)
Q
SptS
T tr (SptTQtr (S))
kk
kkk
kkSk
k
k
k ? ∗
ku kk
Q
(3.4.1)
SptTQtr (SptTQtr (S))
The composition
Σ∞tr
T
?S∗
SptTQtr (S) −−−→ SptS
−→ SptTQtr (SptTQtr (S))
T tr (SptTQtr (S)) −
Q
Q
sends E to the TQtr -spectrum
(E, E ⊗ TQtr , . . . , E ⊗ [TQtr⊗n ]Sn , . . .).
But the inclusion of the summand [TQtr⊗n ]Sn in TQtr⊗n induces an isomorphism in DM eff (S)
(this follows from lemma 4.2.1 below), hence the evident map
S∗
(Σ∞
→ Σ∞
T tr E)
T tr E
Q
Q
is a weak equivalence in SptTQtr (SptTQtr (S)). As
ΣTQtr : SptTQtr (S) → SptTQtr (SptTQtr (S))
induces an equivalence of homotopy categories [19, theorems 5.1 and 10.3], we see that
each map in the diagram (3.4.1) induces an equivalence between the respective homotopy
categories. Combining this with the commutative diagram
?S∗
SptS
T tr (S)
Q
SptT tr (?S∗ )◦Σ∞tr
Q
T
/
SptTQtr (S)
Σ∞tr
T
Q
SptTQtr (SptTQtr (S))
SptTQtr (SptTQtr (S))
Q
finishes the proof.
63
We will use proposition 3.4.1 to simplify the computation of tensor products in DM (S).
tr
We have the functor − ⊗ Q : Shtr
Nis (S) → ShNis (S)Q , with P ⊗ Q the sheaf associated to
the presheaf Y 7→ P (Y ) ⊗Z Q; clearly − ⊗ Q extends to exact tensor functors
− ⊗ Q : DM eff (S) → DM eff (S)Q ; − ⊗ Q : DM (S) → DM (S)Q .
For n ∈ Z, we let ZS (n) denote the Tate object Σnt (mS (S))[−2n], and set QS (n) := ZS (n)⊗Q.
3.5
Geometric motives
eff
Let k be a perfect field. We recall the category of effective geometric motives DMgm
(k), from
eff
[15, chapter V], and the category of geometric motives DMgm (k) := DMgm (k)[⊗Z(1)−1 ],
eff
(k), represented by the complex [P1 ] → [Spec k] with [P1 ]
with Z(1) the Tate object of DMgm
in degree 2. Let
eff
ι : DMgm
(k) → DMgm (k)
be the canonical functor. We have the functor
eff
eff
Mgm
: Sm/k → DMgm
(k)
inducing the functor Mgm : Sm/k → DMgm (k).
This has been extended in [9, example 5.5]
g eff (S) be the localization of trianguDefinition 3.5.1 Let S be a smooth k-scheme. Let DM
gm
lated category K b (SmCor(S)) with respect to the thick subcategory generated by complexes
of the form
(iU ∗ ,−iV ∗ )
j
+j
U∗
V∗
−−−
→ [U ∪ V ], for U, V open subschemes of some
(a) [U ∩ V ] −−−−−−→ [U ] ⊕ [V ] −−
Y ∈ Sm/S.
p∗
→ [Y ] for Y ∈ Sm/S.
(b) [Y × A1 ] −
The maps in (a) are the evident open immersions, and the map p in (b) is the projection.
eff
g eff
DMgm
(S) is by definition the pseudo-abelianization of DM
gm (S).
eff
By [1], DMgm
(S) has a canonical structure of a triangulated tensor category, so that the
eff
canonical functor π : K b (SmCor(S)) → DMgm
(S) is an exact tensor functor.
Cisinski-Déglise [10, definition 10.2] use the same approach to define category of geometric
motives over S
eff
DMgm (S) := DMgm
(S)[− ⊗ ZS (1)−1 ],
where
i
∞
ZS (1) := Cone([S] −→
[P1S ])[−2].
eff
Let ι : DMgm
(S) → DMgm (S) be the canonical functor, let
eff
eff
Mgm
: Sm/S → DMgm
(S)
be the functor induced by the graph embedding Sm/S → SmCor(S) and let
Mgm : Sm/S → DMgm (S)
eff
be the composition ι ◦ Mgm
.
64
Remark 3.5.2 Sending Y ∈ Sm/S to the representable presheaf with transfers Ztr
S (Y )
evidently extends to an exact tensor functor
eff
eff
ieff
S : DMgm (S) → DM (S).
eff
∼
As − ⊗ ZS (1) is invertible on DM (S) and ieff
S (ZS (1)) = ZS (1), iS extends canonically to an
exact tensor functor
iS : DMgm (S) → DM (S),
giving us the commutative diagram of exact tensor functors
eff
DMgm
(S)
ι
ieff
S
DMgm (S)
iS
/ DM eff (S)
/
Σ∞tr
T
DM (S)
Cisinsk-Déglise show that the horizontal maps in this diagram are fully faithful embeddings, extending Voevodsky’s embedding theorem [15, chapter V, theorem 3.2.6].
Theorem 3.5.3 ([10, §10.2]) The functors
eff
eff
ieff
S : DMgm (S) → DM (S).
and
iS : DMgm (S) → DM (S)
are full embeddings.
Remark 3.5.4 Voevodsky ([15, chapter V, theorem 3.4.1] and [38]) has also shown that the
canonical functor
eff
ι : DMgm
(k) → DMgm (k)
is a full embedding. The analog of this result for arbitrary S ∈ Sm/k appears to be unknown
at present, however, a partial result follows from theorem 3.2.5.
3.6
Tate motives
eff
We write ZS (n) for ZS (1)⊗n in DMgm
(S) or DMgm (S), and QS (n) for the image of ZS (n)
eff
in the Q-linearizations DMgm (S)Q or DMgm (S)Q . In DMgm (S) and DMgm (S)Q , we have
the objects ZS (n), QS (n) for n < 0 as well. We have used the same notations for the
corresponding objects in DM eff (S), DM (S), DM eff (S)Q and DM (S)Q , but the context will
make the meaning clear.
Definition 3.6.1 The triangulated category of mixed Tate motives over S, DMTgm (S), is the
smallest full triangulated subcategory of DMgm (S)Q containing the objects QS (n), n ∈ Z,
and closed under isomorphism in DMgm (S)Q . Similarly, let DMT(S) be the smallest full
triangulated subcategory of DM (S)Q containing the objects QS (n), n ∈ Z, and closed under
isomorphism in DM (S)Q .
65
Since QS (n) ⊗ QS (m) ∼
= QS (n + m), DMTgm (S) and DMT(S) are tensor subcategories of
DMgm (S)Q and DM (S)Q , respectively.
Proposition 3.6.2 The restriction of the Q-extension of
iS : DMgm (S) → DM (S)
to DMTgm (S) defines an equivalence
iS : DMTgm (S) → DMT(S)
of triangulated tensor categories.
Proof This is an immediate consequence of fact that iS (ZS (n)) ∼
= ZS (n), together with
theorem 3.5.3.
Just as for the case of motives over a field, the category DMT(S) admits a canonical
weight filtration, and, in case S satisfies the Beilinson-Soulé vanishing conjectures, a tstructure with heart generated by the Tate objects QS (n). In fact, the results of [31] apply
directly, so we will content ourselves here with giving the relevant definitions.
Definition 3.6.3 Let Wn DMT(S) denote the full triangulated subcategory of DMT(S) generated by the Tate motives QS (−a) with a ≤ n. Let W[n,m] DMT(S) be the full triangulated
subcategory of DMT(S) generated by the Tate motives QS (−a) with n ≤ a ≤ m, and let
W >n DMT(S) be the full triangulated subcategory of DMT(S) generated by the Tate motives
QS (−a) with a > n.
Lemma 3.6.4 For S ∈ Sm/k there is a natural isomorphism
HomDMT(S) (QS (a), QS (b)[m]) ∼
= H m (S, Q(b − a))
Proof Clearly, we have
HomDMT(S) (QS (a), QS (b)[m]) ∼
= HomDM (S) (ZS (a), QS (b)[m])
∼
= HomDM (S) (ZS (0), QS (b − a)[m])
∼
= HomDM (S) (mS (S), ZS (b − a)[m]) ⊗ Q
By theorem 3.2.5, we have
HomDM (S) (mS (S), ZS (b − a)[m]) ∼
= HomDM (k) (mk (S), Zk (b − a)[m])
and by theorem 3.5.3 we have
HomDM (k) (mk (S), Zk (b − a)[m]) ∼
= HomDMgm (k) (Mgm (S), Z(b − a)[m])
=: H m (S, Z(b − a)).
Lemma 3.6.5 DMT(S) is a rigid tensor triangulated category.
66
Proof The unit 1 for the tensor operation is QS (0). It suffices to check that the generators
QS (n) of DMT(S) admit a dual (see e.g. [30, part I, IV.1.2]). Setting QS (n)∨ = QS (−n),
with maps δ : 1 → QS (n)∨ ⊗ QS (n), : QS (n) ⊗ QS (n)∨ → 1 being the canonical isomorphisms shows that QS (n) has a dual.
Theorem 3.6.6 1. (Wn DMT(S), W >n DMT(S)) is a t-structure on DMT(S) with heart
consisting of 0-objects.
2. Denote the truncation functors for the t-structure (Wn DMT(S), W >n DMT(S)) by
Wn : DMT(S) → Wn DMT(S) ⊂ DMT(S)
W >n : DMT(S) → W >n DMT(S) ⊂ DMT(S).
Then
(a) Wn and W >n are exact
(b) Wn is right adjoint to the inclusion Wn DMT(S) → DMT(S) and W >n is left adjoint
to the inclusion W >n DMT(S) → DMT(S).
(c) For each n < m there is an exact functor
W[n+1,m] : DMT(S) → W[n+1,m] DMT(S) ⊂ DMT(S)
and a natural distinguished triangle
Wn → Wm → W[n+1,m] → Wn [1].
(d) DMT(S) = ∪n∈Z Wn DMT(S) = ∪n∈Z W >n DMT(S).
Proof By lemma 3.6.4, we have an isomorphism
HomDM (S)Q (QS (a), QS (b)[m]) ∼
= H m (S, Q(b − a))


for b < a
0
= 0
for b = a, m 6= 0


Q · id for b = a, m = 0.
Thus, [31, lemma 1.2] applies to prove the theorem.
We denote the exact functor W[n,n] : DMT(S) → W[n,n] DMT(S) by grW
n and the category
W
W[n,n] DMT(S) by grn DMT(S).
Remark 3.6.7 Since
(
0
HomDMT(S) (QS (−n), QS (−n)[m]) =
Q · id
for m 6= 0
for m = 0,
b
the category grW
n DMT(S) is equivalent to D (Q). Thus, we can define the Q-vector space
H n (grW
n M ) for M in DMT(S).
67
Definition 3.6.8 1. We say that S satisfies the Beilinson-Soulé vanishing conjectures if
H m (S, Q(n)) = 0 for m ≤ 0 and n 6= 0.
2. Let DMT(S)≤0 be the full subcategory of DMT(S) with objects those M such that
≥0
be the full subcategory of
H m (grW
n M ) = 0 for all m > 0 and all n ∈ Z. Let DMT(S)
m
W
DMT(S) with objects M such that H (grn M ) = 0 for all m < 0 and all n ∈ Z. Let
MT(S) := DMT(S)≤0 ∩ DMT(S)≥0 .
Theorem 3.6.9 Suppose S satisfies the Beilinson-Soulé vanishing conjectures. Then
1. (DMT(S)≤0 , DMT(S)≥0 ) is a non-degenerate t-structure on DMT(S) with heart MT(S)
containing the Tate motives QS (n), n ∈ Z.
2. MT(S) is equal to the smallest abelian subcategory of MT(S) which contains the QS (n),
n ∈ Z, and which is closed under extensions in MT(S).
3. The tensor operation in DMT(S) restricted to MT(S) makes MT(S) a rigid Q-linear
abelian tensor category.
4. The functor ⊕n grW
n : MT(S) → VecQ is a fiber functor, making MT(S) a neutral Tannakian category.
Proof By lemma 3.6.4, the assumption that S satisfies the Beilinson-Soulé vanishing conjectures implies that
(
0 for b > a, m ≤ 0
HomDMT(S)Q (QS (a), QS (b)[m]) =
0 for b = a, m 6= 0
With this, the result follows from [31, theorem 1.4, proposition 2.1].
4
Cycle algebras
Bloch’s cycle complex z p (S, ∗) is defined using cycles on S × ∆n , where ∆n is the algebraic
n-simplex
X
∆n := Spec k[t0 , . . . , tn ]/(
ti − 1).
i
One can also use cubes instead of simplices to define the various versions of the cycle complexes. The major advantage is that the product structure for the cubical complexes is
easier to define and, with Q-coefficients, one can construct cycle complexes which have a
strictly commutative and associative product. This approach is used by Hanamura in his
construction of a category of mixed motives, as well as in the construction of categories of
Tate motives by Bloch [3], Bloch-Kriz [2], Kriz-May [26] and Joshua [25].
We combine the cubical version with the strictly functorial constructions of FriedlanderSuslin-Voevodsky to give a functorial version of the cycle complex. This allows us to extend
the representation theorem of Spitzweck to give a description of mixed Tate motives over a
smooth base in terms of cell modules over a cycle algebra.
68
4.1
Cubical complexes
We recall the definition of the cubical version of the Suslin-complex C∗Sus from [15, Chap.
V].
Let (1 , ∂1 ) denote the pair (A1 , {0, 1}),P
and (n , ∂n ) the
n-fold product of (1 , ∂1 ).
P
n
n
Explicitly, n = An , and ∂n is the divisor i=1 (xi = 0) + i=1 (xi = 1), where x1 , . . . , xn
are the standard coordinates on An . A face of n is a face of the normal crossing divisor
∂n , i.e., a subscheme defined by equations of the form xi1 = 1 , . . . , xis = s , with the j in
{0, 1}. If a face F has codimension m in n , we write dim F = n − m.
For ∈ {0, 1} and j ∈ {1, . . . , n} we let ιj, : n−1 → n be the closed embedding defined
by inserting an in the jth coordinate. We let πj : n → n−1 be the projection which
omits the jth factor.
Definition 4.1.1 Let S be a noetherian scheme and let F be presheaf on Sm/S. Let
Cncb (F) be the presheaf
Cncb (F)(S) := F(S × n )/
n
X
πj∗ (F(S × n−1 )),
j=1
and let C∗cb (F) be the complex with differential
dn =
n
X
(−1)
j−1
F (ιj,1 ) −
j=1
We refer to the subgroup
written degn.
Pn
j=1
n
X
(−1)j−1 F (ιj,0 ).
j=1
πj∗ (F(S × n−1 )) of F(S × n ) as the degenerate elements,
If F is a Nisnevich sheaf, then C∗cb (F) is a complex of Nisnevich sheaves, and if F is
a presheaf (resp. Nisnevich sheaf) with transfers, then C∗cb (F) is a complex of presheaves
(resp. Nisnevich sheaves) with transfers. We extend the construction to complexes of sheaves
(with transfers) by taking the total complex of the evident double complex.
For a presheaf F on Sm/S and Y ∈ Sm/S, let
CnAlt (F)(Y ) ⊂ Cncb (F)(Y )Q = F(Y × n )Q /degn
be the Q-subspace consisting of the alternating elements of F(Y × n )Q with respect to the
action of the symmetric group Sn on n , i.e., the elements x satisfying
(id × σ)∗ (x) = sgn(σ) · x
for all σ ∈ Sn . Here Sn acts on n = An by permuting the coordinates. Y 7→ CnAlt (F)(Y )
evidently forms a sub-presheaf of Cncb (F)Q , which we denote by CnAlt (F); in fact the CnAlt (F)
form a subcomplex C∗Alt (F) ⊂ C∗cb (F)Q . We extend this to complexes of presheaves by
taking the total complex of the evident double complex.
Remark 4.1.2 Following Bloch [3], one can define the alternating complex as a subcomplex
of F(Y × ∗ )Q , i.e., without taking the quotient by the degenerate cycles. For this, one
extends the action of Sn on n to an action of the semi-direct product (Z/2)n n Sn where
69
Z/2 acts on 1 by sending t to 1 − t. The sign representation of Sn extends to a sign
representation (Z/2)n nSn → {±1}, and the subcomplex of F(Y ×∗ )Q which is alternating
with respect to these extended sign representations is isomorphic to our complex C∗Alt (F)
via the projection F(Y × ∗ )Q → F(Y × ∗ )Q /degn.
The arguments of e.g. [29, section 2.5] show
Lemma 4.1.3 Let F be a complex of presheaves on Sm/S.
1. There is a natural isomorphism C∗Sus (F) ∼
= C∗cb (F) in the derived category of presheaves
on Sm/S. If F is a complex of presheaves with transfer, we have an isomorphism
C∗Sus (F) ∼
= C∗cb (F) in the derived category D(PST(S)).
2. The inclusion C∗Alt (F)(Y ) ⊂ C∗cb (F)Q (Y ) is a quasi-isomorphism for all Y ∈ Sm/S.
Remark 4.1.4 One can define a cubical version of Bloch’s cycle complex, following the
pattern of definition 4.1.1. That is, define z q (S, n)cb to be the free abelian group on the
codimension q subvarieties W ⊂ S × n such that W ∩ S × F has codimension q for every
face F ⊂ n , and let z q (S, n)cb be the quotient of z q (S, n)cb by the “degenerate” cycles
coming from z q (S, n − 1)cb by pull-back. This gives us the complex z q (S, ∗)cb , which is
quasi-isomorphic to the simplicial version z q (S, ∗) defined in [4].
Taking the subgroups of alternating cycles gives us the subcomplex
z q (S, ∗)Alt ⊂ z q (S, ∗)cb
Q,
quasi-isomorphic to z q (S, ∗)cb
Q.
tr
Call F ∈ C(Shtr
Nis (S)) quasi-fibrant with respect to some model structure on C(ShNis (S))
if the map F → F fib to a fibrant model is quasi-isomorphism of presheaves, that is, for each
Y ∈ Sm/S, the map on sections
F(Y ) → F fib (Y )
is a quasi-isomorphism of complexes.
cb
Lemma 4.1.5 Let F be in C(Shtr
Nis (S)). Suppose that C∗ (F) satisfies Nisnevich excision.
tr
cb
Then C∗ (F) is quasi-fibrant in model category C(ShNis (S))A1 .
Proof Let C∗cb (F) → C∗cb (F)f be a fibrant model for C∗cb (F) in the model category
cb
C(Shtr
Nis (S))Nis . Since C∗ (F) satisfies Nisnevich excision, the map of complexes
C∗cb (F)(Y ) → C∗cb (F)f (Y )
is a quasi-isomorphism for every Y ∈ Sm/S. Thus, C∗cb (F) is quasi-fibrant in the model
category C(Shtr
Nis (S))Nis .
tr
In addition, since the homotopy category of C(Shtr
Nis (S))Nis is equivalent to D(ShNis (S)),
we have isomorphisms for every Y ∈ Sm/S and n ∈ Z:
cb
HomD(Shtr
(Ztr
S (Y ), C∗ (F)[n])
Nis (S))
∼
= HomD(Shtr (S)) (Ztr (Y ), C cb (F)f [n])
S
HomK(Shtr
(Ztr
S (Y
Nis (S))
H n (C∗cb (F)f (Y ))
H n (C∗cb (F)(Y )).
Nis
∼
=
∼
=
∼
=
70
∗
), C∗cb (F)f [n])
On the other hand, for every F, the cubical complex construction C∗cb (F) is homotopy
invariant as a complex of presheaves, i.e.,
C∗cb (F)(Y ) → C∗cb (F)(Y × A1 )
is a quasi-isomorphism for each Y ∈ Sm/S. Thus
cb
1
cb
HomD(Shtr
(Ztr
(Ztr
S (Y ), C∗ (F)[n]) → HomD(Shtr
S (Y × A ), C∗ (F)[n])
Nis (S))
Nis (S))
is an isomorphism for all Y ∈ Sm/S, i.e., C∗cb (F) is A1 -local. Thus C∗cb (F)f is also A1 cb
local, hence C∗cb (F)f is quasi-fibrant in C(Shtr
Nis (S))A1 , and thus C∗ (F) is quasi-fibrant in
C(Shtr
Nis (S))A1 as well.
Example 4.1.6 Let W be a finite type k-scheme. We recall the presheaf with transfers
zq.fin (W ) (also denoted zequi (W, 0) in [15]) on Sm/k. For Y ∈ Sm/k, zq.fin (W )(Y ) is defined
to be the free abelian group on integral closed subschemes Z ⊂ Y ×k W such that Z → Y
is quasi-finite and dominant over a component of Y . The presheaf zq.fin (W )(Y ) is in fact a
Nisnevich sheaf.
It follows from [15, chapter V, theorem 4.2.2(4)] and lemma 4.1.3 that one has a natural
isomorphism for Y ∈ Sm/k
cb
Hn (C∗cb (zq.fin (W ))(Y )) ∼
= H−n
Nis (Y, C∗ (zq.fin (W ))),
and hence C∗cb (zq.fin (W )) satisfies Nisnevich excision as a complex of presheaves on Sm/S.
Thus C∗cb (zq.fin (W )) is quasi-fibrant in C(Shtr
Nis (S))A1 .
1
Denote by Ztr
S (P /∞) the sheaf defined by the exactness of the split exact sequence
i
∞∗
1
tr
1
0 → Ztr
−
→ Ztr
S −
S (P ) → ZS (P /∞) → 0
1
r
tr
1
tr
Of course, Ztr
S (P /∞) = ZS (1)[2]. Similarly, let ZS ((P /∞) ) be defined by the exactness of
P
1 r−1
⊕rj=1 Ztr
)
S ((P )
j ij,∞∗
1
r
−−−−−→ Ztr ((P1 )r ) → Ztr
S ((P /∞) ) → 0
1
r
where ij,∞ : (P1 )r−1 → (P1 )r inserts ∞ in the jth spot. Thus Ztr
S ((P /∞) ) is isomorphic to
Ztr
S (r)[2r].
1
Remark 4.1.7 We used the notation T tr for Ztr
S (P /∞) in the context of “Tate spectra”
(definition 3.2.2); we introduce this new notation to make clear the relation with the sheaf
zq.fin (A1 ).
4.2
The cycle cdga in DM eff (S)
tr
For Y ∈ Sm/k, we denote Ztr
Spec k (Y ) by Z (Y ).
The symmetric group Σq acts on Ztr ((P1 /∞)q ) by permuting the coordinates in (P1 )q .
We let N (q) ⊂ C∗Alt (Ztr ((P1 /∞)q ) be the subsheaf of symmetric sections with respect to this
action. This defines N (q) as an object of C(Shtr
Nis (k)Q ).
71
Lemma 4.2.1 The inclusion N (q) ⊂ C∗Alt (Ztr ((P1 /∞)q )) is a quasi-isomorphism of complexes of presheaves on Sm/k.
Proof Fix X ∈ Sm/k. We have the sequence of maps
C∗ (Ztr ((P1 /∞)q ))(X) → C∗ (zq.fin (Aq ))(X) → z q (X × Aq , ∗),
the first map induced by the inclusion Aq ⊂ (P1 )q , the second by the inclusion of the
quasi-finite cycles on X × Aq × ∆n to the cycles in good position on X × Aq × ∆n . Both
maps are quasi-isomorphisms: for the first, use the localization sequence of [15, chapter IV,
corollary 5.12] together with [15, chapter IV, theorem 8.1]; for the second, use the duality
theorem [15, chapter IV, theorem 7.4] and Suslin’s comparison theorem [15, chapter VI,
theorem 3.1].
Passing to the cubical versions, tensoring with Q and taking the alternating subcomplexes, it follows from lemma 4.1.3 and remark 4.1.4 that we have the sequence of quasiisomorphisms
C∗Alt (Ztr ((P1 /∞)q ))(X) → C∗Alt (zq.fin (Aq ))(X) → z q (X × Aq , ∗)Alt .
As the pull-back by the projection p : X × Aq → X
z q (X, ∗)Alt → z q (X × Aq , ∗)Alt
is also a quasi-isomorphism by the homotopy property for Bloch’s higher Chow groups [4,
theorem 2.1], Sq acts trivially on z q (X × Aq , ∗)Alt , in D− (Ab), and thus Sq acts trivially
on the cohomology of the complex C∗Alt (Ztr ((P1 /∞)q ))(X). Since C∗Alt (Ztr ((P1 /∞)q ))(X)
is a complex of Q-vector spaces, it follows that N (q)(X) → C∗Alt (Ztr ((P1 /∞)q ))(X) is a
quasi-isomorphism, as claimed.
For X, Y ∈ Sm/k, the external product of correspondences gives the associative external
product
cb
cb
Cncb (Ztr ((P1 /∞)q )(X) ⊗ Cm
(Ztr ((P1 /∞)p ))(Y ) → Cn+m
(Ztr ((P1 /∞)p+q ))(X ×k Y ).
Taking X = Y and pulling back by the diagonal X → X ×k X gives the cup product of
complexes of sheaves
∪ : C∗cb (Ztr ((P1 /∞)p )) ⊗ C∗cb (Ztr ((P1 /∞)q )) → C∗cb (Ztr ((P1 /∞)p+q )).
Taking the alternating projection with respect to the ∗ and symmetric projection with
respect to the A∗ yields the associative, commutative product
· : N (p) ⊗ N (q) → N (p + q),
which makes N := Q ⊕ ⊕r≥1 N (r) into an Adams graded cdga object in C(ShNis (k)Q ). By
abuse of notation, we write N (0) for the constant presheaf Q.
Definition 4.2.2 For S ∈ Sm/k, we let NS (q) denote the restriction of N (q) to SmCor(S);
similarly define the Adams graded cdga object in C(ShNis (S)Q ):
NS = Q ⊕ ⊕q≥1 NS (q).
72
Taking sections of N on S gives us the Adams graded cdga N (S). In fact, NS is a
presheaf of Adams graded cdgas over N (S), where for f : X → S in Sm/S, the algebra
structure comes from the pull-back map
f ∗ : N (S) → NS (X) = N (X).
Remark 4.2.3 We will show in §4.3 how to make NS into an Adams graded cdga in
C − (Shtr
Nis (S)Q ), that is, we will extend the product map defined above to an associated
graded-commutative product
· : NS (p) ⊗tr
S NS (q) → N (p + q).
4.3
Products and internal Hom in Shtr
Nis (S)
It is convenient to give a more abstract construction of the product on N , using canonical
products on internal Hom complexes.
tr
For F ∈ Shtr
Nis (S) and X ∈ Sm/S, let Hom(ZS (X), F) denote the sheaf
Hom(Ztr
S (X), F)(W ) := F(X ×S W ).
For fixed F, sending X to Hom(Ztr
S (X), F) extends to a functor
op
Hom(Ztr
→ Shtr
S (−), F) : SmCor(S)
Nis (S).
Extend the definition of Hom(−, F) to small direct sums by setting
Y
Hom(Ztr
Hom(⊕α Ztr
S (Sα ), F).
S (Sα ), F) :=
α
For G ∈ Shtr
Nis (S), we have the canonical left resolution (3.1.2)
. . . → L1 (G) → L0 (G) → G → 0.
One defines Hom(G, F) as the kernel of
Hom(L0 (G), F) → Hom(L1 (G), F).
tr
b
We extend Hom(G, F) to F ∈ C(Shtr
Nis (k)), G ∈ C (ShNis (k)) by taking the extended total
complex of the evident double complex, giving the bi-functor
tr
tr
Hom(−, −) : C b (Shtr
Nis (k)) × C(ShNis (k)) → C(ShNis (k)).
Concretely,
Hom(G, F)n := ⊕m∈Z Hom(G m , F m+n );
the sum is finite since G is in C b (Shtr
Nis (k)).
The isomorphism
tr
tr
∼
Hom(Ztr
S (W ), Hom(ZS (X), F)) = Hom(ZS (X), F)(W ) = F(X ×k W )
tr
tr
∼
= Hom(Ztr
S (X ×k W ), F) = Hom(ZS (X) ⊗ ZS (W ), F)
73
tr
extends to give an adjunction of Hom complexes, for G ∈ C b (Shtr
Nis (k)), F, H ∈ C(ShNis (k)),
HomC(Shtr
(H, Hom(G, F)) ∼
(G ⊗tr
= HomC(Shtr
S H, F).
Nis (k))
Nis (k))
This in turn formally gives an adjunction (for G, H ∈ C b (Shtr
Nis (k)))
Hom(H, Hom(G, F)) ∼
= Hom(G ⊗tr
S H, F).
Similarly, we have a canonical map
tr
− ⊗ idH : Hom(G, F) → Hom(G ⊗tr
S H, F ⊗ H)
which, via the adjunction
tr
Hom(Hom(G, F) ⊗tr
S H, Hom(G, F ⊗S H))
∼
= Hom(Hom(G, F), Hom(H, Hom(G, F ⊗tr
S H)))
tr
∼
= Hom(Hom(G, F), Hom(G ⊗tr
S H, F ⊗S H)),
gives a canonical product map
tr
Hom(G, F) ⊗tr
S H → Hom(G, F ⊗S H).
The identity map on Hom(A, B) gives by adjunction the evaluation map
evA : A ⊗tr
S Hom(A, B) → B
The map
tr
tr
tr
tr
evA ⊗tr
S evC : A ⊗S Hom(A, B) ⊗S C ⊗S Hom(C, D) → B ⊗S D
gives, via the adjunction
tr
tr
tr
Hom(A ⊗tr
S Hom(A, B) ⊗S C ⊗S Hom(C, D), B ⊗S D)
tr
tr
∼
= Hom(Hom(A, B) ⊗tr
S Hom(C, D), Hom(A ⊗S C, B ⊗S D)),
the external product map
tr
tr
Hom(A, B) ⊗tr
S Hom(C, D) → Hom(A ⊗S C, B ⊗S D).
Taking A = C = Z tr (X) and pulling back by the diagonal
tr
δX : Ztr (X) → Ztr (X ×S X) = Ztr (X) ⊗tr
S Z (X)
gives us the cup product map
tr
tr
tr
tr
∪F ,G : Hom(Ztr
S (X), F) ⊗S Hom(ZS (X), G) → Hom(ZS (X), F ⊗S G),
defined for all F, G ∈ C(Shtr
Nis (S)).
Given F, G ∈ C(Shtr
(S)),
we can restrict F and G to complexes of Nisnevich sheaves on
Nis
Sm/S, where we have the usual tensor product and internal Hom of sheaves, with natural
maps (of complexes of sheaves on Sm/S)
sh
F ⊗sh G → F ⊗tr
S G, Hom(G, F) → Hom (G, F).
We note that the respective adjunction isomorphisms are compatible with the restriction
maps from HomC(Shtr
to HomC(ShNis (S)) and the above comparison maps. In particular,
Nis (S))
the various products described above are compatible with their counterparts for Nisnevich
sheaves on Sm/S.
74
Remark 4.3.1 In general, the functor Hom(Ztr
S (X), −) does not transform quasi-isomorphisms to quasi-isomorphisms, so to get a well defined functor
tr
D(Shtr
Nis (S)) → D(ShNis (S)),
one needs to pass to the right-derived functor RHom(Ztr
S (X), −). However, if a complex
tr
F ∈ C(ShNis (S)) satisfies Nisnevich excision, then the canonical map
tr
Hom(Ztr
S (X), F) → RHom(ZS (X), F)
In the examples of interest, we will usually apply
is an isomorphism in D(Shtr
Nis (S)).
tr
Hom(ZS (X), −) to complexes satisfying Nisnevich excision, so we will suppress the use
of the derived version RHom(Ztr
S (X), −) in order that we may have a concrete model on the
level of complexes.
Example 4.3.2 Take S ∈ Sm/k. For F ∈ C(Shtr
Nis (S)) we have
n
Cncb (F) = Hom(Ztr
S ( ), F).
Thus, the product map
C∗cb (F) ⊗ C∗cb (G) → C∗cb (F ⊗ G) → C∗cb (F ⊗tr
S G)
extends to a product
cb
cb
tr
C∗cb (F) ⊗tr
S C∗ (G) → C∗ (F ⊗S G)
via the external products
n
tr
tr
m
Hom(Ztr
S ( ), F) ⊗S Hom(ZS ( ), G)
n
tr tr
m
tr
→ Hom(Ztr
S ( ) ⊗S Z ( ), F ⊗S G)
n+m
= Hom(Ztr
), F ⊗tr
S (
S G).
1
m
tr
1
n+m
tr
1
n
tr
), we thus have the associative
As Ztr
S ((P /∞) ) ⊗S ZS ((P /∞) ) = Z ((P /∞)
product
1
n
tr
cb
tr
1
m
cb
tr
1
n+m
C∗cb (Ztr
)).
S ((P /∞) )) ⊗S C∗ (ZS ((P /∞) )) → C∗ (ZS ((P /∞)
Applying the appropriate alternating and symmetric projections, we have the commutative
and associative product
NS (n) ⊗tr
S NS (m) → NS (n + m)
making NS an Adams graded cdga object of C(Shtr
Nis (S)Q ).
Passing to the derived category, and composing with the canonical natural transformation
− ⊗L − → − ⊗tr −,
makes NS an Adams graded commutative ring object of D(Shtr
Nis (S)Q ).
Thus, we may apply the localization functor
eff
D(Shtr
Nis (S)Q ) → DM (S)Q
to the product defined above, giving us the product map
µn,m : NS (n) ⊗ NS (m) → NS (n + m)
in DM eff (S)Q , making NS an Adams graded commutative ring object of DM eff (S)Q .
75
Lemma 4.3.3 In DM eff (S)Q , we have a canonical isomorphism
QS (r) → NS (r)
giving a commutative diagram
QS (n) ⊗ QS (m)
QS (n + m)
NS (n) ⊗ NS (m)
µn,m
/
NS (n + m),
of isomorphisms in DM eff (S)Q .
tr
1
tr
tr
tr
tr
Proof By definition Ztr
S (1)[2] = ZS (P /∞). As ZS (W ) ⊗S ZS (X) = ZS (W ×S X), we
tr
1
n
thus have Ztr
S (n)[2n] = ZS ((P /∞) ).
tr
Take F ∈ ShNis (k). By [15, the proof of proposition 3.2.3, chap. V] and lemma 4.1.3,
the canonical map F = C0cb (F) → C∗cb (F) becomes an isomorphism after applying the
eff
cb
localization functor RC∗ : D− (Shtr
Nis (k)) → DM− (k). Thus, the cone of F → C∗ (F) is
tr
1
in the localizing subcategory of D− (Shtr
Nis (k)) generated by the complexes Z (X × A ) →
Ztr (X), X ∈ Sm/k.
Let p : S → Spec k be the structure morphism. Sending (f : X → S) ∈ Sm/S to
pf : X → Spec k defines the functor
p : Sm/S → Sm/k.
Noting that X ×S Z is a closed subscheme of p(X)×k p(Z), we see that p extends to a faithful
functor
p : SmCor(S) → SmCor(k),
inducing the exact restriction functor
tr
p∗ : Shtr
Nis (k) → ShNis (S).
∗ cb
cb ∗
We note that, for F ∈ Shtr
Nis (k), we have p C∗ (F) = C∗ (p F). Furthermore, we
∗
tr
tr
have p (Z (X)) = ZS (X ×k S). Thus, the fact that Cone(F → C∗cb (F)) is in the localizing
tr
1
tr
subcategory of D− (Shtr
Nis (k)) generated by the complexes Z (X ×A ) → Z (X), X ∈ Sm/k,
implies that Cone(p∗ F → C∗cb (p∗ F)) is in the localizing category generated by the complexes
1
tr
∗
cb ∗
Ztr
S (X × A ) → ZS (X), X ∈ Sm/S. Thus p F → C∗ (p F) becomes an isomorphism after
eff
applying the localization functor D(Shtr
Nis (S)) → DM (S).
As a particular case, the map
1
n
cb
1
n
ZStr (n)[2n] = Ztr
S ((P /∞) ) → C∗ ((P /∞) )
induces an isomorphism
ZS (n)[2n] → C∗cb ((P1 /∞)n )
in DM eff (S). By lemma 4.2.1, composing this map with the canonical projection
C∗cb ((P1 /∞)n )) ⊗ Q → NS (n)[2n]
76
induces an isomorphism
QS (n) → NS (n)
in DM eff (S)Q .
It follows directly from the definition of the products µn,m that the diagram
Ztr
S (n + m)
tr tr
Ztr
S (n) ⊗S ZS (m)
NS (n) ⊗tr
S sNS (m)
µn,m
/
NS (n + m)
commutes in C(Shtr
Nis (S)). Applying the localization functor, we see that
QS (n) ⊗ QS (m)
QS (n + m)
NS (n) ⊗ NS (m)
µn,m
/
NS (n + m),
commutes in DM eff (S)Q .
4.4
Equi-dimensional cycles
We consider the case S = Spec k.
Definition 4.4.1 Let X be in Sm/k, r ≥ 0 an integer. The sheaf zequi (X, r) has sections
over T ∈ Sm/k the free abelian group on the integral closed subschemes W ⊂ T ×k X with
W → T dominant and of pure relative dimension r over some irreducible component of T .
Acting by correspondences in the evident manner defines zequi (X, r) as an object in Shtr
Nis (k).
For r = 0, we have the evident map
Ztr (X) → zequi (X, 0)
which is an isomorphism if X is proper over k. Similarly, if f : Z → X is a dominant
equi-dimensional morphism of relative dimension d, the pull-back of cycles from T ×k X to
T ×k U defines a map
f ∗ : zequi (X, r) → zequi (Z, r + d).
If f : Z → X is proper, we have the push-forward map
f∗ : zequi (Z, r) → zequi (X, r)
and if j : U → X is an open immersion with closed complement i : W → X, the sequence
j∗
i
∗
0 → zequi (W, r) −
→
zequi (X, r) −
→ zequi (U, r)
is exact.
Taking products of cycles gives the pairing
: zequi (X, r)(T ) ⊗ zequi (X 0 , r0 )(T 0 ) → zequi (X ×k X 0 , r + r0 )(T ×k T 0 );
for T = T 0 , one pulls back by the diagonal T → T ×k T to define the pairing
∪X,X 0 (T ) : zequi (X, r)(T ) ⊗ zequi (X 0 , r0 )(T ) → zequi (X ×k X 0 , r + r0 )(T ).
77
Lemma 4.4.2 The pairings ∪X,X 0 (T ) extend to a pairing
∪X,X 0 : zequi (X, r) ⊗tr zequi (X 0 , r0 ) → zequi (X ×k X 0 , r + r0 ).
Proof Take W ∈ zequi (X, r)(T ), W 0 ∈ zequi (X 0 , r0 )(T 0 ). We let
φW W 0 : Ztr (T ×k T 0 ) → zequi (X ×k X 0 , r + r0 )
be the map corresponding to W W 0 ∈ zequi (X ×k X 0 , r + r0 )(T ×k T 0 ). Thus we have the
map
M
⊕φW W 0 :
Ztr (T ×k T 0 ) → zequi (X ×k X 0 , r + r0 ),
T ∈Sm/k,W ∈zequi (X,r)(T )
T 0 ∈Sm/k,W 0 ∈zequi (X 0 ,r0 )(T 0 )
i.e., a map
˜ : L0 (zequi (X, r)) ⊗tr L0 (zequi (X 0 , r0 )) → zequi (X ×k X 0 , r + r0 ).
∪
˜ descends to a map on the quotient H0 (L(zequi (X, r)) ⊗tr
It is a simple matter to check that ∪
L(zequi (X 0 , r0 ))), giving the desired pairing.
5
N (S)-modules and motives
We relate the category of Tate motives over S ∈ Sm/k to the derived category of dg modules
over the cycle algebra N (S).
5.1
The contravariant motive
We define a functor
hS : Sm/S op → DM (S)
as follows: For X → S in Sm/S we have the internal Hom presheaf on SmCor(S)
Sus
tr
tr
Hom(Ztr
S (X), C∗ (ZS (n)[2n]))(W ) := C∗ (ZS (n)[2n])(X ×S W ).
The multiplication
tr
tr tr
Ztr
S (n)[2n] ⊗S ZS (1)[2] → ZS (n + 1)[2n + 2]
together with the canonical map T tr := Ztr (1)[2] → C∗ (Ztr
S (1)[2]) gives rise to the bonding
maps
tr
tr tr
tr
Hom(Ztr
→ Hom(Ztr
S (X), C∗ (ZS (n)[2n])) ⊗S T
S (X), C∗ (ZS (n + 1)[2n + 2])).
tr
tr
⊗ n
Noting that Ztr
, the commutativity constraints for the tensor strucS (n)[2n] = (ZS (1)[2])
tr
tr
ture define a Sn action on Hom(ZS (X), C∗ (Ztr
specS (n)[2n])), giving us the symmetric T
S
trum hS (X) ∈ SptT tr (S):
tr
tr
tr
hS (X) := (Hom(Ztr
S (X), C∗ (ZS )), . . . , Hom(ZS (X), C∗ (ZS (n)[2n])), . . .).
78
Using the action of correspondences on Ztr
S (X), one sees immediately that hS extends to
a functor
hS : SmCor(S)op → SptS
T tr (S),
which in turn extends to
C b (hS ) : C b (SmCor(S)op ) → SptS
T tr (S).
Passing to the respective homotopy categories gives the exact functor
K b (hS ) : K b (SmCor(S))op → DM (S).
Lemma 5.1.1 The functor K b (hS ) : K b (SmCor(S))op → DM (S) descends to an exact
functor
eff
op
heff
→ DM (S).
gm : DMgm (S)
Proof By [15, chapter IV, theorem 8.1], the natural map
H m (C∗Sus (Ztr (n)[2n])(X)) → Hm (XNis , C∗Sus (Ztr (n)[2n]))
is an isomorphism for all X ∈ Sm/k and all m. Thus, by the Mayer-Vietoris property for
hypercohomology, the total complex associated to the following term-wise exact sequence of
complexes
C∗Sus (Ztr (n)[2n])(T ×S (U ∩ V ))
→ C∗Sus (Ztr (n)[2n])(T ×S U ) ⊕ C∗Sus (Ztr (n)[2n])(T ×S V )
→ C∗Sus (Ztr (n)[2n])(T ×S (U ∪ V ))
is acyclic for all U, V as in (a), for all T ∈ Sm/S and for all n ≥ 0. Thus K b (hS ) maps the
complexes in (a) to an object ∼
= 0 in DM (S).
Sus
tr
Similarly, since C∗ (ZS (n)[2n]) is in DM−eff (k) ⊂ D− (Shtr
Nis (k)), the map
p∗ : C∗Sus (Ztr (n)[2n])(T ×S X) → C∗Sus (Ztr (n)[2n])(T ×S X × A1 )
is a quasi-isomorphism for all T, X ∈ Sm/k and all n ≥ 0. Thus K b (hS ) maps the complexes
in (b) to an object ∼
= 0 in DM (k), giving us the exact functor
eff
op
g
h̃eff
→ DM (S).
gm : DM gm (S)
As arbitrary direct sums exist in DM (S), that category is pseudo-abelian, hence h̃eff
gm extends
canonically to the pseudo-abelian hull
eff
op
heff
→ DM (S).
gm : DMgm (S)
Lemma 5.1.2 heff
gm is a lax tensor functor. If S = Spec k, with k a perfect field admitting
resolution of singularities, then heff
gm is a tensor functor.
79
Proof We have the pairing
tr
Sus
tr
Sus
tr
C∗Sus (Ztr
S (n)[2n]) ⊗S C∗ (ZS (m)[2m]) → C∗ (ZS (n + m)[2(n + m)])
induced by the identity pairing
tr tr
tr
Ztr
S (n)[2n] ⊗S ZS (m)[2m] → ZS (n + m)[2(n + m)]).
Thus, for X, X 0 ∈ Sm/k, we have the pairing
Sus
tr
tr
tr
0
Sus
tr
Hom(Ztr
S (X), C∗ (ZS (n)[2n])) ⊗S Hom(ZS (X ), C∗ (ZS (m)[2m])
0
Sus
tr
→ Hom(Ztr
S (X ×S X ), C∗ (ZS (n + m)[2(n + m)])), (5.1.1)
giving rise to the commutative diagram
/
tr
tr
HX (n) ⊗tr
S T ⊗S HX 0 (m)
HX (n + 1) ⊗tr
S HX 0 (m)
/
tr
HX×S X 0 (n + m) ⊗tr
S T
HX×X 0 (n + m + 1)
where
Sus
tr
HX (n) := Hom(Ztr
S (X), C∗ (ZS (n)[2n]))
0
Sus
tr
HX 0 (m) := Hom(Ztr
S (X ), C∗ (ZS (m)[2m])
0
Sus
tr
HX×X 0 (l) := Hom(Ztr
S (X ×S X ), C∗ (ZS (l)[2l]))
Replacing X and X 0 with arbitrary objects in C b (SmCor(k)), this yields the natural transformation
ψM,N : hS (M ) ⊗ hk (N ) → hS (M ⊗ N ),
making heff
gm a lax tensor functor.
To show that heff
gm is a tensor functor in case S = Spec k, it suffices to show that ψX,X 0
is an isomorphism for X, X 0 ∈ Sm/k. For this, it suffices to show that the pairing (5.1.1)
induces an isomorphism in DM−eff (k)
Hom(Ztr (X), C∗Sus (Ztr (n)[2n])) ⊗ Hom(Ztr (X 0 ), C∗Sus (Ztr (m)[2m])
→ Hom(Ztr (X ×k X 0 ), C∗Sus (Ztr (n + m)[2(n + m)])).
for n, m sufficiently large.
To see this, take S, T, X ∈ Sm/k, and let dX = dimk X. We have the map
ρX : C∗Sus (zequi (S, r))(X × T ) → C∗Sus (zequi (S × X, r + dX ))(T )
which sends a cycle W on X × T × ∆p × S of relative dimension r over X × T , to the
same cycle, now of relative dimension r + dX over T . By [15, chapter IV, theorem 8.1], the
canonical map
H n (C∗Sus (zequi (S, r))(T )) → Hn (TNis , C∗Sus (zequi (S, r))Nis )
80
is an isomorphism for every n and r ≥ 0. Thus, it follows from [15, chapter IV, theorem 8.2]
that the map ρX is a quasi-isomorphism for all r ≥ 0.
Noting that C∗Sus (zequi ((P1 /∞)n , 0)) = C∗Sus (Z(n)[2n]), we have the quasi-isomorphism
ρX : Hom(Ztr (X), C∗Sus (Z(n)[2n])) → C∗Sus (zequi ((P1 /∞)n × X, dX ))
Finally, by [15, chapter IV, theorem 8.3(2)], the pull-back by the projection X × (P1 )n →
X × (P1 )n−1 induces a natural quasi-isomorphism
C∗Sus (zequi ((P1 /∞)n−1 × X, m − 1))(T ) ∼
= C∗Sus (zequi ((P1 /∞)n × X, m))(T )
for all n, m ≥ 1. Thus, for n ≥ dX we have the diagram of quasi-isomorphisms
/ C Sus (z
Hom(Ztr (X), C∗Sus (Z(n)[2n]))
∗
equi ((P
1
/∞)n × X, dX ))
O
C∗Sus (zequi ((P1 /∞)n−dX × X, 0))
and similarly for S. Thus, these quasi-isomorphisms give isomorphisms in DM−eff (k)
Hom(Ztr (X), C∗Sus (Z(n)[2n])) ∼
= C∗Sus (zequi ((P1 /∞)n−dX × X, 0))
Hom(Ztr (S), C Sus (Z(m)[2m])) ∼
= C Sus (zequi ((P1 /∞)m−dS × S, 0))
∗
tr
Hom(Z
∗
Sus
(S ×k X), C∗ (Z(n + m)[2(n + m)]))
∼
= C∗Sus (zequi ((P1 /∞)n+m−dS −dX × S ×k
X, 0))
for m ≥ dS , n ≥ dX . One checks that these isomorphisms are compatible with the pairings
Hom(Ztr (S), C∗Sus (Ztr (n)[2n])) ⊗ Hom(Ztr (X), C∗Sus (Ztr (m)[2m])
→ Hom(Ztr (S ×k X), C∗Sus (Ztr (n + m)[2(n + m)]))
C∗Sus (zequi ((P1 /∞)n−dX × X, 0)) ⊗ C∗Sus (zequi ((P1 /∞)m−dS × S, 0))
→ C∗Sus (zequi ((P1 /∞)n+m−dS −dX × S ×k X, 0))
But by [15, chapter V, proposition 4.1.7], this last pairing is an isomorphism in DM−eff (k),
completing the proof.
eff
op
Lemma 5.1.3 Take S ∈ Sm/k and consider the functor heff
→ DM (S).
gm : DMgm (S)
1. There is a natural isomorphism
∼ eff
heff
gm (M (1)) = hgm (M )(−1).
eff
op
2. The functor heff
→ DM (S) extends to an exact lax tensor functor
gm : DMgm (S)
hgm : DMgm (S)op → DM (S).
3. If S = Spec k, and k is a perfect field admitting resolution of singularities, then hgm is a
tensor functor.
81
Proof By lemma 5.1.2, heff
gm is a lax tensor functor, and is a tensor functor if S = Spec k.
eff
Since DMgm (S) = DMgm (S)[⊗Z(1)−1 ] and ⊗Z(1) is invertible on DM (S), it suffices to
prove (1).
Since heff
gm is a lax tensor functor, we have the natural map
eff
eff
ψM : heff
gm (M ) ⊗ hgm (ZS (1)) → hgm (M (1)).
eff
eff
(X), for X ∈ Sm/S, it suffices to show
(S) is generated by the motives Mgm
As DMgm
∼
(a) heff
gm (ZS (1)) = ZS (−1).
(b) ψMgm
eff (X) is an isomorphism for all X ∈ Sm/S.
tr
For (a), by definition, heff
gm (ZS (1)[2]) is represented by the T -spectrum with nth term
tr
1
Sus
Hom(ZS (P /∞), C∗ (ZS (n)[2n])).This presheaf on Sm/S is isomorphic to the restriction
of the presheaf Hom(Ztr (S × P1 /S × ∞), C∗Sus (Z(n)[2n])) on Sm/k. Similarly, ZS (−1)[−2]
is represented by the T tr -spectrum with nth term the restriction to Sm/S of the presheaf
Hom(Ztr (S), C∗Sus (Z(n − 1)[2(n − 1)])), with bonding maps induced by the multiplication
C∗Sus (Z(n − 1)[2(n − 1)]) ⊗tr Z(1)[2] → C∗Sus (Z(n)[2n]).
As in the proof of lemma 5.1.2, we have the diagram of quasi-isomorphisms of presheaves
on Sm/k (for n ≥ 1)
Hom(Ztr (S × P1 /S × ∞), C∗Sus (Z(n)[2n]))
XXXXXX
XXXXXX
XXXXXX
XXXXXX
,
Hom(Ztr (S, C∗Sus (zequi ((P1 /∞)n × (P1 /∞), 1)))
O
Hom(Ztr (S, C∗Sus (zequi ((P1 /∞)n−1 , 0)))
Hom(Ztr (S, C∗Sus (Z(n − 1)[2(n − 1)])),
compatible with bonding maps, proving (a).
eff
tr
For (b), heff
gm (Mgm (X)) is represented by the T -spectrum with nth term the presheaf
Sus
eff
eff
tr
Hom(Ztr
S (X), C∗ (ZS (n)[2n])) and hgm (Mgm (X)(1)[2]) is represented by the T -spectrum
tr
1
Sus
with nth term Hom(ZS (X × P /X × ∞), C∗ (ZS (n)[2n])). We note that the presheaf
Sus
Hom(Ztr
S (X), C∗ (ZS (n)[2n])) is as above the restriction to SmCor(S) of the presheaf
Hom(Ztr (X), C∗Sus (Z(n)[2n])) on SmCor(k), where we consider X as in Sm/k via the composition X → S → Spec k.
1
Sus
Similarly, Hom(Ztr
S (X × P /X × ∞), C∗ (ZS (n)[2n])) is the restriction to SmCor(S) of
tr
1
Sus
the presheaf Hom(Z (X × P /X × ∞), C∗ (Z(n)[2n])) on SmCor(k). The same proof as
for (a), replacing S with X, proves (b).
We call hgm the dual motive functor. Recall from theorem 3.5.3 the full tensor embedding
iS : DMgm (S) → DM (S).
Our terminology for hgm is justified by
82
Proposition 5.1.4 Let k be a perfect field admitting resolution of singularities. There is a
natural isomorphism of hgm : DMgm (k)op → DM (k) with the composition
∨
i
S
DMgm (k)op −
→ DMgm (k) −
→
DM (k).
c
Proof For X ∈ Sm/k, we denote C∗ (zequi (X, 0)) by C∗c (X) and let Mgm
(X) denote the
eff
c
image of C∗ (X) in DM− (k).
For X ∈ Sm/k of dimension d, one has the dual motive Mgm (X)∨ in DMgm (k), since
eff
(k) and the image
k admits resolution of singularities. Also, Mgm (X)∨ (d)[2d] is in DMgm
∨
eff
c
of Mgm (X) (d)[2d] in DM− (k) is canonically isomorphic to Mgm (X) (see [15, chapter V,
c
tr
section 4.3]). Letting Σ∞
spectrum
t Mgm (X)(−d)[−2d] denote the T
(0, . . . , 0, C∗c (X), C∗c (X)(1)[2], . . .)
with d − 1 0’s, we see that in DM (k), Σ∞
t Mgm (X) has a dual, namely, the object represented
∞
c
by Σt Mgm (X)(−d)[−2d].
The restriction by the open immersion An → (P1 )n induces a quasi-isomorphism of
presheaves
tr
tr
c
n
Hom(Ztr
k (X), C∗ (Zk (n)[2n])) → Hom(ZS (X), C∗ (A )).
By the duality theorem [15, chapter IV, theorem 7.1], the inclusion of complexes of presheaves
Hom(Ztr (X), C∗c (An )) → C∗ (zequi (X × An , d))
is a quasi-isomorphism of complexes of presheaves, as is each morphism in the sequence
C∗c (X × An−d ) → Hom(Ztr (Ad ), C∗c (X × An−d )) → C∗ (zequi (X × An , d))
for all n ≥ d.
c
c
By [15, chapter V, corollary 4.1.8] we have Mgm
(X × An ) ∼
(X)(n)[2n] for all n ≥ 0
= Mgm
eff
Thus we have the canonical isomorphisms in DM− (k):
tr
C∗c (X)(n − d)[2n − 2d] ∼
= C c (X × An−d ) ∼
= Hom(Ztr
k (X), C∗ (Zk (n)[2n])),
for all n ≥ d. One checks that this isomorphism is compatible with the bonding morphisms
c
∨ ∼
for Σ∞
t Mgm (X)(−d)[−2d] and hk (X), giving the desired isomorophism Mgm (X) = hk (X)
in DM (k).
Finally, we may consider the Q-extension of hgm
op
hgm : DMgm
(S)Q → DM (S)Q .
Proposition 5.1.5 The restriction of hgm to DMTgm (S)op defines a tensor functor
hgm : DMTgm (S)op → DMT(S)
with hgm (QS (n)) ∼
= QS (−n).
Proof This follows directly from lemma 5.1.3 and the fact that hgm (QS ) ∼
= QS .
83
5.2
The dual motive and cycle complexes
We let
hS : K(SmCor(S)op ) → DM (S)
be the exact functor induced by the composition
C(hS )
C(SmCor(S)op ) −−−→ SptT tr (S) → DM (S).
We can use the cycle complex construction NS (definition 4.2.2) to define a Q version of
hS . Indeed, for X ∈ Sm/S, set
hS (X)(n) := Hom(Qtr
S (X), NS (n)).
The composition
cb
tr
Ztr
S (1)[2] → C∗ (ZS (1)[2]) → NS (1)
together with the multiplication in NS induces bonding maps
tr tr
tr
n : Hom(Qtr
S (X), NS (n)) ⊗S TS → Hom(QS (X), NS (n + 1)),
giving us the symmetric T tr -spectrum
tr
hS (X) := (Hom(Qtr
S (X), NS (0)), Hom(QS (X), NS (1)), . . .)
(with trivial S∗ -action). Sending X to hS (X) gives an exact functor
hS : K(SmCor(S))op → DM (S)Q .
We have the canonical isomorphism in D(Q)
N (n)(X) ∼
= C∗ (Ztr
S (n)[2n])(X)Q .
This gives an isomorphism (in D(PST(S))Q )
tr
tr
∼
Hom(Qtr
S (X), NS (n)) = Hom(ZS (X), C∗ (ZS (n)[2n]))Q =: hS (X)Q ,
which induces a canonical isomorphism
hS (X) ∼
= hS (X)Q
natural in X, in fact an isomorphism of functors
hS ∼
= hSQ : K(SmCor(S))op → DM (S)Q .
(5.2.1)
Lemma 5.2.1 For each r ≥ 0, the presheaf Hom(Qtr
S (X), NS (r)) is quasi-fibrant in the
tr
tr
model category C(ShNis (S)Q )A1 , that is, Hom(QS (X), NS (r)) satisfies Nisnevich excision
and A1 -homotopy invariance,
Proof This follows from lemma 4.2.1 and the comments in example 4.1.6.
84
5.3
Cell modules and Tate motives
Recall the Adams graded cdga N (S) gotten by evaluating the presheaf N := Nk of Adams
graded cdgas at S ∈ Sm/k. The identity N (S) = NS (S) makes the presheaf NS on Sm/S
a presheaf of N (S) algebras.
For fixed r, sending M ∈ CMN (S) to the weight r summand (M ⊗N (S) NS )(r) of M ⊗N (S)
NS defines a dg functor
MS (r)dg : CMN (S) → C(Shtr
Nis (S)),
and thus an exact functor
MS (r) : KCMN (S) → D(Shtr
Nis (S))
For M ∈ CMN (S) , the multiplication in NS gives us the map in C − (Shtr
Nis (S))
M ⊗N (S) N ⊗tr
S NS (1) → M ⊗N (S) NS ;
restricting to the summand (M ⊗N (S) NS )(r) and composing with the canonical map TQtr →
NS (1) gives us the map in C(Shtr
Nis (S))
r (M )
tr
(M ⊗N (S) NS )(r) ⊗tr
S TQ −−−→ (M ⊗N (S) NS )(r + 1).
Sending M ∈ CMN (S) to the sequence
Mdg
S (M ) := ((M ⊗N (S) NS )(0), (M ⊗N (S) NS )(1), . . .)
with bonding maps r (M ) (and trivial Sn -action) defines the dg functor
S
Mdg
S : CMN (S) → SptT tr (S),
Q
giving the exact functor on the respective homotopy categories
MS : KCMN (S) → DM (S)Q .
Lemma 5.3.1 1. M(N (S)<n>) ∼
= QS (n).
2. There are natural isomorphisms
M(M ⊗ Q(n)) ∼
= M(M ) ⊗ QS (n);
3. The restriction of M to KCMfN (S) is a tensor functor.
Proof For (1), we note that
M(r)dg (N (S)<n>) = NS (r + n).
As the canonical map Qtr
S (r + n) → NS (r + n) induces an isomorphism
QS (r + n) ∼
= NS (r + n)
85
in DM eff (S), we have the canonical isomorphism
M(r)(N (S)<n>) ∼
= QS (n + r)
compatible with the respective bonding maps, proving (1).
(2) follows by noting
M(r)dg (M ⊗ N (S)<n>) = M(r + n)dg (M )
for r + n ≥ 0.
For (3), we have canonical maps in C − (Shtr
Nis (S))
0
(M ⊗N (S) NS ) ⊗tr
S (M ⊗N (S) N )
→ (M ⊗N (S) M 0 ) ⊗N (S) (NS ⊗tr
S NS )
id⊗µ
−−−→ (M ⊗N (S) M 0 ) ⊗N (S) NS
where µ is the multiplication. On the respective Adams graded summands, this induces
(M ⊗N (S) NS )(r) ⊗tr
S (M ⊗N (S) NS )(s)
ρM,M 0 (r,s)
−−−−−−→ ((M ⊗N (S) M 0 ) ⊗N (S) NS )(r + s).
The maps ρM,M 0 (r, s) are compatible with the bonding maps, giving us the natural transformation
ρM,M 0 : M(M ) ⊗ M(M 0 ) → M(M ⊗ M 0 )
in SptS (S)Q , making the functor M a lax tensor functor.
If M = N (S)<a> and M 0 = N (S)<b> it is a simple matter to check that ρM,M 0 is just
the canonical isomorphism
QS (a) ⊗ QS (b) → QS (a + b);
it follows by induction on the length of the weight filtration that ρM,M 0 is an isomorphism
for all M, M 0 ∈ KCMfN (S) .
The following result extends Spitzweck’s representation theorem (see [29, section 5]) from
fields to S ∈ Sm/k.
Theorem 5.3.2 Let S be in Sm/k. There is an exact functor
MS : DN (S) → DM (S)Q
with MS (Q(n)) ∼
= QS (n); MS is a lax tensor functor. In addition
1. The restriction of MS to
f
MfS : DN
(S) → DM (S)Q
f
defines an equivalence of DN
(S) with DMT(S), as tensor triangulated categories, natural in S.
f
2. Mf transforms the weight filtration in DN
(S) to that in DMT(S).
86
3. Suppose that S satisfies the Beilinson-Soulé vanishing conjectures. Then Mf is a functor of triangulated categories with t-structure. In particular, Mf intertwines the respective
truncation functors and induces an equivalence of Tannakian categories
f
H 0 (Mf ) : HN
(S) → MT(S),
f
which identifies DN
(S) with DMT(S).
Proof Using the equivalence DN (S) ∼ KCMN (S) , we just use the functor M : KCMN (S) →
f
f
DM (S)Q to define MS . Similarly, the equivalence DN
(S) ∼ KCMN (S) and lemma 5.3.1
f
∼
proves that the restriction of MS to DN
(S) is a tensor functor with MS (Q(n)) = QS (n).
We have
HomDf
N (S)
(Q(n), Q(m + n)[p]) ∼
= HomKCMf
N (S)
(Q(n), Q(m + n)[p]) ∼
= H p (N (S)(m)).
By lemmas 4.1.3 and 4.2.1, we have
H p (N (S)(m)) ∼
= H p (C∗Sus (Z(m))(S)Q ).
By Voevodsky’s results [15, chapter V, theorem 4.22, proposition 4.2.3], we have
H p (S, Q(m)) ∼
= H p (C∗Sus (Z(m))(S)Q ).
Finally, by theorem 3.2.5 and theorem 3.5.3 we have
H p (S, Q(m)) := HomDMgm (k)Q (Mgm (S)Q , Q(m)[p]) ∼
= HomDM (S)Q (QS (n), QS (n + m)[p]),
giving us the isomorphism
HomDf
N (S)
(Q(n), Q(m + n)[p]) ∼
= HomDM (S)Q (QS (n), QS (n + m)[p]).
It is not hard to check that this isomorphism is induced by the functor MS . By induction
f
on the length of the weight filtration, this shows that MfS gives an equivalence of DN
(S) with
the essential image of MfS , that is, with DMT(S). This proves (1).
f
It is clear that MfS sends the subcategory Wn DN
(S) to Wn DMT(S); this together with
f
(1) proves (2). For (3), the t-structures on DN
(S) , resp. DMT(S) are defined by using the
f
b
equivalence of W[n,n] DN
(S) , resp. W[n,n] DMT(S) with D (Vec(Q)), induced by sending a
vector space V to V ⊗ Q(−n), resp. V ⊗ QS (−n). As this is clearly compatible with MfS ,
we have proved (3).
5.4
Motives and NS -modules
Take S ∈ Sm/k. We begin by defining the category MNS of Adams graded dg modules over
the sheaf (on Sm/S) of cdgas NS .
87
Objects in MNS are Adams graded dg objects in C(Shtr
Nis (S)Q ), that is, (M, dM ), where
∗
∗
M := ⊕r (M (r) , dM (r)), with each (M (r) , dM (r)) ∈ C(Shtr
Nis (S)Q ). In addition, with respect to the Adams grading r and the cohomological grading ∗, M is a bi-graded module
over NS in C(Shtr
Nis (S)Q ), that is, we have module action
m : NS ⊗tr
S M → M
which is a bi-graded map in C(Shtr
Nis (S)Q ). We have the Tate twist operator M 7→ M <s>
on MNS , with M <s>(r) := M (s + r).
Let DNS denote the derived category of MNS , i.e, localize the homotopy category KMNS
with respect to the full subcategory of complexes M such that each M (r) has vanishing cohof
mology sheaves (for the Nisnevich topology). We let DN
be the full triangulated subcategory
S
of DNS generated by the objects NS <n>, n ∈ Z.
As NS is a presheaf of N (S)-algebras, we have an action of MN (S) on MNS : given an
N (S)-module N and an M ∈ MNS , we may form the sheaf tensor product
N ⊗N (S) M.
Restricting to CMN (S) gives the bi-exact functor
⊗N (S) : KCMN (S) × DNS → DNS ;
via the equivalence KCMN (S) → DN (S) , we have the bi-exact functor
⊗LN (S) : DN (S) × DNS → DNS .
Clearly ⊗LN (S) restricts to
f
f
f
⊗LN (S) : DN
(S) × DN → DN .
S
S
We have the exact functor
fS : KMN → DM (S).
M
S
defined by sending an NS -module M to the sequence of Adams graded summands
fS (M ) := (M (0), M (1), . . .)
M
tr
tr
with bonding maps M (n) ⊗tr
S TQ → M (n + 1) given by the multiplication M (n) ⊗S NS (1) →
M (n + 1) and the canonical map TQtr → N (1). If M 0 → M is a quasi-isomorphism of NS f
modules, then M 0 (n) → M (n) is a weak equivalence in C(Shtr
Nis (S)Q )Nis , hence MS descends
to an exact functor
fS : DN → DM (S)Q .
M
S
We note that DNS is pseudo-abelian.
Proposition 5.4.1 There is an exact functor
op
hN
S : DMgm (S)Q → DNS
N
f
such that hS : DMgm (S)op
Q → DM (S)Q is isomorphic to MS ◦ hS . In addition, we have
∼
hN
S (QS ) = NS and
∼ N
hN
S (M (1)) = hS (M ) ⊗NS NS <−1>
for all M ∈ DMgm (S)Q .
88
Proof Take X ∈ Sm/S. Recalling that hS (X)r = Hom(Qtr
S (X), NS (r)), define
tr
hN
S (X)(r) := Hom(QS (X), NS (r))
tr
N
−
giving us the Adams graded object hN
S (X) := ⊕r≥0 hS (X)(r) of C (ShNis (S)). We note that
the multiplication
tr
Hom(Qtr
S (X), NS (r)) ⊗S NS (s) → Hom(QS (X), NS (r + s))
extends canonically to a map of complexes
tr
tr
Hom(Qtr
S (X), NS (r)) ⊗S NS (s) → Hom(QS (X), NS (r + s)),
N
giving hN
S (X) the structure of an NS -module. We thus have the object hS (X) of MNS for
every X ∈ Sm/S.
As hS (X)n is just hN
S (X)(n), it follows from the construction of hS that sending X ∈
N
Sm/S to hS (X) extends to an exact functor
b
op
hN
→ DNS .
S : K (SmCor(S))
By the quasi-isomorphisms established in the proof of lemma 5.1.1, hN
S descends further to
an exact functor
eff
op
hN
→ DNS .
S : DMgm (S)
It follows from the isomorphism
∼ eff
heff
gm (M ⊗ Z(1)) = hgm (M ) ⊗ Z(−1)
established in the proof of lemma 5.1.3 and the isomorphism (5.2.1) that hN
S extends canonically to an exact functor
op
hN
→ DNS
S : DMgm (S)
satisfying
∼ N
hN
S (M (1)) = hS (M ) ⊗NS NS <−1>.
As DNS is a Q-linear category, this functor extends canonically to
op
hN
S : DMgm (S)Q → DNS .
Remark 5.4.2 As a particular case, proposition 5.4.1 tells us that
∼
hN
S (QS (n)) = NS <−n>
for all n ∈ Z and thus hN
S restricts to
f
op
hN
→ DN
.
S : DMTgm (S)
S
fS (N <−n>) ∼
Similarly, it is easy to see that M
= QS (−n) and thus
S
fS : DN → DM (S)Q
M
S
restricts to
fS : Df → DMT(S).
M
N
S
89
We have the global sections functor
Γ(S, −) : C(Shtr
Nis (S)) → C(Ab)
with Γ(S, F) := F(S). Applying Γ(S, −) to each M (r) gives us the global sections functor
Γ(S, −) : MNS → MN (S) .
It is not hard to show that category MNS has enough Γ(S, −)-acyclic objects (take for
example the Godement resolution), hence Γ(S, −) admits the right-derived functor
RΓ : DNS → DN (S)
with Γ(S, M ) → RΓ(S, M ) an isomorphism in DN (S) if M satisfies Nisnevich excision.
Finally, we have the evident natural map, for M ∈ DNS ,
φM : RΓ(S, M ) ⊗LN (S) NS → M
5.5
From cycle algebras to motives
Let p : X → S be in Sm/S, giving us the map of cycle algebras
p∗ : N (S) → N (X);
in particular, we may consider N (X) as a dg module over N (S).
Lemma 5.5.1 Suppose that Mgm (X)Q ∈ DMgm (S)Q is in the Tate subcategory DMTgm (S).
f
Then N (X) is in DN
(S) .
Proof Note that
N (X)(r) = Hom(Qtr
S (X), NS (r))(S),
giving us the canonical isomorphism in MN (S)
N (X) ∼
= Γ(S, hN
S (Mgm (X))).
By lemma 5.2.1, the presheaf Hom(Qtr
S (X), NS (r)) satisfies Nisnevich excision, hence the
natural map
N
Γ(S, hN
S (Mgm (X))) → RΓ(S, hS (Mgm (X)))
is an isomorphism in DN (S)
Therefore, the image of N (X) in DN (S) is given by applying the composition of functors
Mgm
hN
RΓ(S,−)
S
Sm/S −−−→ DMgm (S)Q −→
DNS −−−−−→ DN (S)
to X. Thus, if Mgm (X) ∼
= M in DMgm (S)Q , we have the isomorphism
N (X) ∼
= RΓ(S, hN
S (M ))
in DN (S) . Therefore, it suffices to show that RΓ(S, −) ◦ hN
S maps DMTgm (S) into the full
f
subcategory DN (S) of DN (S) .
But by remark 5.4.2,
∼
hN
S (QS (n)) = NS <−n>
f
and RΓ(S, NS <−n>) ∼
= N (S)<−n>. Thus, RΓ(S, −) ◦ hN
S (QS (n)) is in DN (S) ; the general
case follows easily by induction on the length of the weight filtration.
90
f
Since KCMfN (S) → DN
(S) is an equivalence, we thus have
Proposition 5.5.2 Take X ∈ Sm/S. Suppose that Mgm (X)Q ∈ DMgm (S)Q is in the
Tate subcategory DMTgm (S). Then there is a finite N (S)-cell module cmS (X) and a quasiisomorphism of dg N (S)-modules cmS (X) → N (X).
We now suppose that Mgm (X)Q is in DMTgm (S), so that the finite N (S)-cell module
cmS (X) is defined (uniquely up to homotopy equivalence, we fix a choice once and for all).
We proceed to define a natural transformation
ψX : MS (cmS (X)) → hS (X).
Recall that h(X) is the symmetric TQtr spectrum defined by the sequence
h(X)n := Hom(Qtr
S (X), NS (n))
with bonding maps induced by the multiplication in NS and the structure map TQtr → NS (1),
while MS (cmS (X)) is given by the sequence
MS (cmS (X))n := MS (n)(cmS (X)) := (cmS (X) ⊗N (S) NS )(n)
and with bonding maps also given by the multiplication with TQtr → NS (1).
Now take W ∈ Sm/S. Then
Hom(Qtr
S (X), NS (r))(W ) := NS (X ×S W )(r).
Using the external products in NS , we thus have the canonical map of Adams graded complexes
ψ̃(W ) : N (X) ⊗N (S) N (W ) → N (X ×S W ).
The maps ψ̃(W ) clearly define a map of Adams graded complexes of presheaves with transfer
ψ̃X : N (X) ⊗N (S) NS → ⊕r≥0 Hom(Qtr
S (X), NS (r));
restricting to the component of Adams weight r gives the map of complexes of presheaves
with transfer
ψ̃X (r) : [N (X) ⊗N (S) NS ](r) → Hom(Qtr
S (X), NS (r)).
It is easy to see that ψ̃X respects the action (on the right) by NS .
Composing ψ̃X (r) with the structure map
ρ ⊗id
X
cmS (X) ⊗N (S) NS −−
−→ N (X) ⊗N (S) NS
gives us the map
ψX (r) : [cmS (X) ⊗N (S) NS ](r) → Hom(Qtr
S (X), NS (r)).
also respecting the right NS action. Thus, the maps ψX (r) define a map of the symmetric
TQtr -spectrum MS (cmS (X)) to the symmetric TQtr -spectrum hS (X)
ψX : MS (cmS (X)) → hS (X),
as desired.
Our main result is
91
Theorem 5.5.3 Suppose that Mgm (X)Q is in DMTgm (S). Then
ψX : MS (cmS (X)) → hS (X)
is an isomorphism.
Proof We have the diagram
hN
S
DMgm (S)op
OOO
OOO
OO
hS OOO'
/
DNS
fS
M
DM (S)Q
commutative up to natural isomorphism. We have as well the finite version of hN
S ,
f
hN
S : DMTgm (S) → DN
S
and diagram
hN
S
DMTgm (S)op
NNN
NNN
NNN
NNN
hS
'
/
f
DN
S
,
fS
M
DMT(S)
compatible with the first diagram via the inclusions DMTgm (S) → DMgm (S)Q , DMT(S) →
f
DM (S)Q and DN
→ DNS .
S
In particular, we have the isomorphism
fS (hN
hS (X) ∼
=M
S (Mgm (X))).
Similarly, we have the functor
RΓ(S, −) : DNS → DN (S) .
Since RΓ(S, NS <r>) ∼
= Γ(S, NS <r>) = N (S)<r>, it follows that RΓ(S, −) restricts to an
exact functor
f
f
RΓf (S, −) : DN
→ DN
(S) .
S
From the proof of lemma 5.5.1 we have
cmS (X) ∼
= RΓf (S, hN
S (Mgm (X)))
f
in DN
(S) .
For F ∈ MNS , we have the canonical map
φF : Γ(S, F) ⊗N (S) NS → F
inducing the natural map
φLF : RΓ(S, F) ⊗LN (S) NS → F
in DNS .
92
f
For F ∈ DN
⊂ DNS , φLF restricts to the natural transformation
S
φLF : RΓf (S, F) ⊗LN (S) NS → F
f
in DN
. This gives us the natural transformation
S
fS (φL ) : M
fS (RΓf (S, F) ⊗L
f
M
F
N (S) NS ) → MS (F)
in DMT(S). In particular, for M ∈ DMTgm (S) ⊂ DMgm (S)Q , we have the natural transformation
fS (φLN ) : M
fS (RΓf (S, hN (M )) ⊗L
f N
ψM := M
S
N (S) NS ) → MS (hS (M ))
h (M )
S
in DMT(S).
In case M = Mgm (X)Q for some X ∈ Sm/S with Mgm (X)Q ∈ DMTgm (S), we have
f
N
∼
∼
RΓf (S, hN
S (M )) = RΓ (S, hS (Mgm (X))) = cmS (X)
f
in DN
(S) ,
∼
fS (RΓf (S, hN (M )) ⊗L
M
S
N (S) NS ) = MS (cmS (X))
and
fS (hN (M )) ∼
M
= hS (X)
S
in DM (S)Q , and, via these isomorphisms, ψM corresponds to ψX .
Thus, it suffices to show that ψM is an isomorphism for all M ∈ DMTgm (S); as usual,
we reduce to the case of M = QS (n) by induction on the length of the weight filtration. For
M = QS (n), we have
hN
S (QS (n)) = NS <−n>
f
RΓf (S, hN
S (QS (n))) = RΓ (S, NS <−n>)
∼
= Γ(S, NS <−n>) = N (S)<−n>
so
L
N
φLhN (QS (n)) : RΓf (S, hN
S (QS (n))) ⊗N (S) NS → hS (QS (n))
f
is already an isomorphism in DN
.
S
5.6
The cell algebra of an S-scheme
We now assume that N (S) is cohomologically connected.
Let p : X → S be in Sm/S with a section s : S → X. We thus have the map of
cycle algebras p∗ : N (S) → N (X) making N (X) a cdga over N (S) with augmentation
s∗ : N (X) → N (S). Let N (X)S {∞} → N (X) be the relative minimal model of N (X) over
N (S). In particular, N (X)S {∞} is a cell module over N (S). In addition, the multiplication
N (X)S {∞} ⊗ N (X)S {∞} → N (X)S {∞}
given by the cdga structure on N (X)S {∞} descends to
µX : N (X)S {∞} ⊗N (S) N (X)S {∞} → N (X)S {∞}.
93
Definition 5.6.1 The motivic cell algebra of X,
caS (X) ∈ CMN (S)
is N (X)S {∞}, considered as a cell module over N (S).
The same construction we used to define the map MS (cmS (X)) → hS (X) gives us the
map in SptS
TQtr (S)
ψX : MS (caS (X)) → hS (X).
(5.6.1)
Theorem 5.6.2 Suppose that Mgm (X)Q is in DMTgm (S) and that X satisfies the BeilinsonSoulé vanishing conjectures. Then
ψX : MS (caS (X)) → hS (X)
is an isomorphism.
Proof Suppose we knew that caS (X) → N (X) is a quasi-isomorphism. As caS (X) is a
generalized nilpotent N (S)-algebra, caS (X) is an N (S)-cell module. Thus, we can take
cmS (X) → N (X) to be caS (X) → N (X), and the proposition follows from theorem 5.5.3.
We now show that caS (X) → N (X) is a quasi-isomorphism.
Recall that the Beilinson-Soulé vanishing conjectures for X are just saying that N (X)
is cohomologically connected. Using the section s : S → X, we see that S also satisfies
the Beilinson-Soulé vanishing conjectures, hence N (S) is cohomologically connected. The
structure map N (X)S {∞} → N (X) is thus a quasi-isomorphism by remark 2.4.11.
6
Motivic π1
We can now put all our constructions together to give a description of the Deligne-Goncharov
motivic π1 in terms of a relative bar construction. In this section, we assume k admits
resolution of singularities.
6.1
Cosimplicial constructions
Fix a base-field k and an S ∈ Sm/k. We have the action of finite sets on SchS by
Y
X A/S :=
X
a∈A
for X ∈ SchS and A a finite set, where
Q
means product over S. As this defines a functor
X ?/S : Setsop
f in → SchS
we have an induced functor (also denoted X ?/S ) from simplicial objects in finite sets to
cosimplicial schemes. In case A is the set {1, . . . , n} we write X n/S for X A/S .
94
Examples 6.1.1 1. We have the simplicial object in finite sets [0, 1]:
[0, 1]([n]) := HomOrd ([n], [1])
giving us the cosimplicial path space of X, X [0,1]/S . The two inclusions i0 , i1 : [0] → [1]
induce the projection
π : X [0,1]/S → X {0,1}/S .
Explicitly, X {0,1}/S is the constant cosimplicial scheme X ×S X. X [0,1]/S has n-cosimplices
X n+2/S with the ith coface map given by the diagonal
(t0 , . . . , tn ) 7→ (t0 , . . . , ti−1 , ti , ti , ti+1 , . . . , tn )
and the codegeneracies given by projections. The structure morphism π is given by the
projection X n+2/S → X 2/S on the first and last factor.
2. Suppose we have sections a, b : S → X, giving the map ib,a : S → X ×S X. The
pointed path space Pb,a (X/S) is
Pb,a (X/S) := S ×ib,a ,π X [0,1]/S .
We write Pa (X/S) for Pa,a (X/S).
In case S = Spec k, we sometimes delete the mention of S in the notation, writing, e.g.,
X A for X A/Spec k .
Remark 6.1.2 Suppose that S and X both satisfy the Beilinson-Soulé vanishing conjectures
and that Mgm (X)Q is in DMTgm (S). Then X n/S also satisfies the Beilinson-Soulé vanishing
conjectures for all n ≥ 1.
Indeed, by theorem 5.6.2, the canonical map
N (X)S {∞} → N (X)
is a quasi-isomorphism.
It follows by induction on the length of the weight filtration for Mgm (X)Q that
⊗L
n
H ∗ (X n/S , Q(∗)) ∼
= H ∗ (X, Q(∗)) H ∗ (S,Q(∗))
and thus, the natural map
L
N (X)⊗N (S) n → N (X n/S )
is a quasi-isomorphism, hence
N (X)S {∞}⊗N (S) n → N (X n/S )
is a quasi-isomorphism. But then N (X n/S ) is cohomologically connected, that is, X n/S
satisfies the Beilinson-Soulé vanishing conjectures.
95
6.2
The motive of a cosimplicial scheme
Let X • : Ord → Sm/S be a smooth cosimplicial S-scheme, [i] 7→ X[i] ∈ Sm/S. Modifying
the construction of Deligne-Goncharov, we define hS (X • ) as an object in DM (S).
Let ZSm/S be the additive category generated by Sm/S: objects are denoted Z(X)
for X ∈ Sm/S, for X irreducible, HomZSm/S (Z(X 0 ), Z(X)) is the free abelian group on
HomSm/S (X 0 , X) and disjoint union is direct sum. The embedding Sm/S → SmCor(S)
extends by Z-linearity to an embedding ZSm/S → SmCor(S).
For a smooth cosimplicial S-scheme X • , let Z(X • ) ∈ C(ZSm/S op ) denote the complex
with
Z(X • )n := Z(X −n )
and with differential the usual alternating sum of the coface maps (in the opposite category). We consider Z(X • ) as an object of C(SmCor(S)op ) via the embedding ZSm/S →
SmCor(S).
The category DM (S) is large enough to define the object hS (X • ) directly.
Definition 6.2.1 For a cosimplicial scheme X • , define hS (X • ) by
hS (X • ) := hS (Z(X • )),
where
hS : K(SmCor(S)op ) → DM (S)
is the exact functor defined in §5.2. Sending X • to hS (X • ) extends to a functor
hS : [Sm/S Ord ]op → DM (S).
We now relate this construction to the ind-object construction of Deligne-Goncharov [12].
For each n, one has the complex C ∗ (∆n , X • ) ∈ C b (ZSm/S) with
C i (∆n , X • ) := ⊕g:[i],→[n] Z(X([i])),
where the sum is over all injective maps g : [i] → [n] in Ord. The boundary
di : C i (∆n , X • ) → C i+1 (∆n , X • )
is defined as follows: For 0 ≤ j ≤ i + 1, we have the coface map δji : [i] → [i + 1] (see
section 1.2). Fix an injection g : [i + 1] → [n]. Define
i,g
δj∗
: C i (∆n , X • ) → C i+1 (∆n , X • )
by projecting C i (∆n , X • ) to the component Z(X[i]) indexed by g ◦ δji followed by the map
X(δji ) : X[i] → X[i + 1]
and then the inclusion Z(X[i + 1]) → C i+1 (∆n , X • ) indexed by g. Set
X
i,g
di :=
sgn(j, g) · δj∗
j,g
96
where sgn(j, g) is the sign of the shuffle permutation of [n] given by (g ◦ δji ([i])c , g ◦ δji ([i])).
Projecting on the factors g with 0 in the image of g defines a map of complexes
πn+1,n : C ∗ (∆n+1 , X • ) → C ∗ (∆n , X • )
giving us a projective system in C b (ZSm/S). Reindexing so that C n is now in degree −n
gives an inductive system in C b (ZSm/S op )
. . . → C∗ (∆n , X • ) → C∗ (∆n+1 , X • ) → . . .
•
Definition 6.2.2 hind
S (X ) is the ind-object of DM (S) defined by the ind-system
n 7→ hS (C∗ (∆n , X • ))
Remark 6.2.3 Suppose that S = Spec k, where k is a perfect field admitting resolution of
singularities. We have the sequence of functors
∨
Mgm
i
Sm/k −−−→ DMgm (k) −
→ DMgm (k) →
− DM (k),
tr
with ∨ the duality involution and i : DMgm (k) → DM (k) the full embedding Σ∞
of
t ◦Z
∨
theorem 3.5.3. We let Hgm : Sm/k → DMgm (k) be the functor X 7→ Mgm (X) and write
Hgm as well for the extension to an exact functor
Hgm : K b (ZSm/k) → DMgm (k).
By proposition 5.1.4 we have a natural isomorphism
hgm ◦ Mgm ∼
= i ◦ Hgm .
ind
(X • ) be the ind-object
For X • a smooth cosimplicial k-scheme, let Hgm
n 7→ Hgm (C∗ (∆n , X • ))
ind
of DMgm (k). Then Hgm
(X • ) is the ind-object associated to X • , as defined in [12, §3.12],
and we have a natural isomorphism of ind-objects of DM (k)
ind
•
i(Hgm
(X • )) ∼
= hind
k (X ).
Taking the sum of the identity maps defines a map
qn : C∗ (∆n , X • ) → Z(X • )
in C(ZSm/S op ), giving a map of the ind-system n 7→ C∗ (∆n , X • ) to Z(X • ).
Lemma 6.2.4 Let F : ZSm/S op → A be an additive functor to a pseudo-abelian category,
closed under filtered inductive limits. Then
lim F (C∗ (∆n , X • )) → F (Z(X • ))
−→
n
is a homotopy equivalence in C(A).
97
For a proof, see [29] or [12, proposition 3.10].
Proposition 6.2.5 We have a natural isomorphism in DM (S)
lim hind
(C∗ (∆n , X • )) ∼
= hS (X • )
−→ S
n
Proof This follows directly from lemma 6.2.4.
Finally, we may replace hS with the functor hS . Sending X • to hS (X • ) := hS (Z(X • ))
extends to the functor
hS : [Sm/S Ord ]op → DM (S)Q ,
the natural isomorphism (5.2.1) hSQ ∼
= hS gives natural isomorphisms
φX • : hS (X • )Q → hS (X • ).
Similarly, we have natural isomorphisms:
hS (C∗ (∆n , X • ))Q → hS ((C∗ (∆n , X • ))
and
lim hS (C∗ (∆n , X • )) ∼
= hS (X • ).
−→
n
6.3
Motivic π1
Let X be a smooth S-scheme with a section x : S → X. This gives us the ind-system in
DM (S)Q
n 7→ hS (C∗ (∆n , Px (X)))Q
as well as the object hS (Px )Q ∈ DM (S)Q with isomorphism
lim hS (C∗ (∆n , Px (X)))Q ∼
= hS (Px )Q .
−→
n
Suppose that Mgm (X)Q is in DMTgm (S). As DMTgm (S) is a tensor subcategory of
DMgm (S)Q and as Mgm (X n/S ) = Mgm (X)⊗n , it follows that Mgm (X n/S )Q is in DMTgm (S)
for all n ≥ 0. Since the individual terms in C∗ (∆n , Px (X)) are all direct sums of selfproducts of X, the motive Mgm (C∗ (∆n , Px (X)))Q is in DMTgm (S) for all n, and thus
hS (C∗ (∆n , Px (X))) = hS (Mgm (C∗ (∆n , Px (X)))Q ) is in DMT(S) for all n.
If S satisfies the Beilinson-Soulé vanishing conjectures, we have the truncation functor
0
Hmot
: DMT(S) → MT(S).
Thus we have the ind-system χ(X, x)∗ in MT(S)
0
n 7→ Hmot
(hS (C∗ (∆n , Px (X)))) := χ(X, x)n .
Suppose that S = Spec k. Deligne-Goncharov [12], following Wojtkowiak [39], note that
the standard structures of product, coproduct and antipode in the classical bar construction
make the ind-system χ(X, x)∗ into an ind-Hopf algebra object in MT(k); we note that the
98
same operations make χ(X, x)∗ into an ind-Hopf algebra object in MT(S) as long as the
ind-system is defined, that is, if S satisfies the Beilinson-Soulé vanishing conjectures and
Mgm (X)Q is in DMTgm (S).
Returning to the case S = Spec k, if X is the complement of a finite set of k-points of
1
Pk , Deligne and Goncharov define π1mot (X, y) to be the dual group scheme object in proMT(k). They also generalize the definition of π1mot (X, y) to the case where X is a smooth
uni-rational variety defined over k and where y is a tangential base-point: they show in
[12, theéorème 4.13] that a suitable object of Deligne’s realization category comes from the
mixed Artin-Tate category MAT(k) (which is larger than MT(k) as it takes into account
trivial motives defined over a finite extension of k). However, in this case, they do not give
a direct construction as a motive in DMgm (k). We extend their definition in the following
direction:
Definition 6.3.1 Suppose that S and X both satisfy the Beilinson-Soulé vanishing conjectures, and that Mgm (X)Q is in DMT(S). Let x : S → X be a section. Define π1mot (X, x) to
be the group scheme object in pro-MT(S) dual to the ind-Hopf algebra object χ(X, x)∗ of
MT(S).
Remark 6.3.2 Deligne-Goncharov work in the geometric category DMTgm (k) rather than
in DMT(k). However, since ik : DMTgm (k) → DMT(k) is an equivalence, we can just as
well work in DMT(k).
6.4
Simplicial constructions
Let A →
− N be an augmented cdga over a cdga N . Recall from section 2.5 the simplicial
version of the relative bar construction
B•pd (A/N , ) := A⊗N [0,1] ⊗A⊗A N .
The total complex associated to the simplicial object n 7→ Bnpd (A/N , ) is the relative bar
pd
complex B̄N
(A, ).
Using the opposite of the construction described in section 6.2, we have the ind-system
of “finite” complexes C∗ (∆n , B•pd (A/N , )), and a homotopy equivalence
pd
lim C∗ (∆n , B•pd (A/N , )) → B̄N
(A, ).
−→
n
Replacing A with its relative minimal model over N (assuming for this that N is cohomologically connected), we have the refined version of the simplicial bar construction, B• (A/N , ),
the associated complex B̄N (A, ), the approximations C∗ (∆n , B• (A/N , )) and the homotopy
equivalence
lim C∗ (∆n , B• (A/N , )) → B̄N (A, ).
−→
n
6.5
The comparison theorem
Take X ∈ Sm/S. with section x : S → X. We apply the construction of the preceeding
section to the augmented cdga N (X) over N (S):
N (X) o
x∗
p∗
99
/
N (S).
Assuming that N (S) is cohomologically connected, we have the relative minimal model
N∞ (X/S) := Nk (X){∞}N (S) , which is an augmented N (S)-algebra via x∗ : N∞ (X/S) →
N (S). The multiplication in N∞ (X/S) gives the natural maps
µn : N∞ (X/S)⊗N (S) n → N (X n/S )
which thus gives natural maps in DM (S)Q
φn (X, x) : MS (C∗ (∆n , B• (N (X)/N (S), x∗ ))) → hS (C∗ (∆n , Px (X)))
and
φ(X, x) : MS (B̄N (S) (N (X), x∗ )) → hS (Px (X))).
The maps φn (X, x) give a map of ind-Hopf algebra objects in DM (X).
Theorem 6.5.1 Suppose that Mgm (X)Q is in DMTgm (S) and X satisfies the BeilinsonSoulé vanishing conjectures. Then both φn (X, x) and φ(X, x) are isomorphisms in DM (S)Q .
Proof Note that the Beilinson-Soulé vanishing conjectures for X imply the vanishing conjectures for S, hence N( S) is cohomologically connected and thus the relative bar complex
B̄N (S) (N (X), x∗ ) is defined.
As φ(X, x) is identified with the filtered homotopy colimit of the maps φn (X, x), it
suffices to show that φn (X, x) is an isomorphism for each n. But on the individual terms
in the complexes defining C∗ (∆n , B• (N (X)/N (S), x∗ )) and C∗ (∆n , Px (X)), φn (X, x) is the
map
φn (X, x)n : MS (N∞ (X/S)⊗N (S) n ) → Hom(Qtr (X n/S ), NS ) = hS (X n/S )
induced by the maps ψX n/S ◦ µn (see(5.6.1) to recall the definition of ψX n/S ).
Since DMTgm (S) is a full tensor subcategory of DMgm (S)Q , closed under isomorphism,
our assumption Mgm (X)Q ∈ DMTgm (S) implies Mgm (X n/S )Q is in DMTgm (S) for all n ≥ 0.
By remark 6.1.2, X n/S satisfies the Beilinson-Soulé vanishing conjectures for all n ≥ 0.
Therefore, it follows from theorem 5.6.2 that ψX n/S is an isomorphism for all n ≥ 0.
In addition, the structure map µ1 is a quasi-isomorphism since N (X) is cohomologically connected. As mentioned in remark 6.1.2, the motivic cohomology of X n/S satisfies
a Künneth formula (over the motivic cohomology of S) for each n. Thus µn is a quasiisomorphism for each n, and hence φn (S, x)n is an isomorphism for each n.
Corollary 6.5.2 Suppose that Mgm (X)Q is in DMT(S) and X satisfies the Beilinson-Soulé
vanishing conjectures. Then we have canonical isomorphisms of ind-Hopf algebras in MT(k),
0
∗
n 7→ [MS (HN
(S) (C∗ (∆n , B• (N (X)/N (X), x ))
H 0 (φn (X,x))
0
−−−−−−−→ Hmot
(hS (C∗ (∆n , Px (X)))].
Proof This follows from theorem 6.5.1 and theorem 5.3.2.
100
6.6
The fundamental exact sequence
Let p : X → S be in Sm/S. We have the exact functor of triangulated tensor categories
p∗ : DM (S) → DM (X); since p∗ (ZS (n)) ∼
= ZX (n), p∗ induces the exact tensor functor
p∗ : DMT(S) → DMT(X).
Similarly, if x : S → X is a section, we have
x∗ : DMT(X) → DMT(S).
Both p∗ and x∗ are compatible with the weight filtrations; we have the analogous functors
on the “geometric” Tate categories DMTgm .
Similarly, the maps p and x induce maps of cdgas
p∗ : N (S) → N (X); x∗ : N (X) → N (S)
and thus exact tensor functors
f
f
f
f
∗
p∗ : DN
(S) → DN (X) , x : DN (X) → DN (S) .
Recall that the equivalence MfS of theorem 5.3.2 is natural in S, so we have natural isomorphisms
f
f
f
MfX ◦ p∗ ∼
= p∗ ◦ MS ; MS ◦ x∗ ∼
= x∗ ◦ M X .
Now suppose that X satisfies the Beilinson-Soulé vanishing conjectures; this property is
inherited by S using the splitting x∗ . Thus we have the functors p∗ and x∗ between the
Tannakian categories MT(X) and MT(S), with p∗ and x∗ respecting the fiber functors grW
∗ .
f
f
∗
∗
Similarly, we have functors p and x for the Tannakian categories HN (X) and HN (S) , ref
f
0
0
specting the fiber functors grW
∗ . Finally, H (MX ) and H (MS ) give an equivalence between
these two structures.
W
Let G(MT(X), grW
∗ ), G(MT(S), gr∗ ) denote the Tannaka groups (more precisely, proW
group schemes over Q) of (MT(X), grW
∗ ) and (MT(S), gr∗ ). We sometimes omit the “basepoint” grW
∗ from the notation.
The functors p∗ and x∗ gives maps of pro-group schemes over Q
W
W
W
p∗ : G(MT(X), grW
∗ ) → G(MT(S), gr∗ )), x∗ : G(MT(S), gr∗ ) → G(MT(X), gr∗ ).
Letting K = ker p∗ , we thus have the split exact sequence
1
/
K
/
G(MT(X), grW
∗ )o
p∗
x∗
/
G(MT(S), grW
∗ )
/
1
of pro-group schemes over Q. Via the splitting x∗ , G(MT(S)) acts by conjugation on K.
Thus the pro-affine Hopf algebra Q[K] is a G(MT(S))-representation. Tannaka duality yields
the corresponding ind object in MT(S), and its dual is a pro-group scheme object in MT(S),
which we denote by Kx . As we have seen above, the Deligne-Goncharov motivic fundamental
group π1mot (X, x), is also a pro-group scheme object in MT(S).
101
Theorem 6.6.1 Let X be in Sm/S with section x : S → X. Suppose that X satisfies
the Beilinson-Soulé vanishing conjectures and that the motive Mgm (X)Q ∈ DMgm (S)Q is in
DMTgm (S). Then there is a natural isomorphism
π1mot (X, x) ∼
= Kx
as pro-group objects in MT(S).
Proof As we have seen above, we may identify G(MT(X)) and G(MT(S)) with the Tannaka
f
f
groups of the categories HN
(X) and HN (S) , respectively. By theorem 1.15.2, this gives an
isomorphism of K with the kernel of the map of pro-groups schemes over Q:
p∗ : Spec (H 0 (B̄(N (X)))) → Spec (H 0 (B̄(N (S))))
induced by
H 0 (B̄(p∗ )) : H 0 (B̄(N (S))) → H 0 (B̄(N (X)))
Similarly, the splitting x∗ becomes identified with
x∗ : Spec (H 0 (B̄(N (S)))) → Spec (H 0 (B̄(N (X)))).
By lemma 2.8.2 and theorem 2.8.3, we have the identification
0
∗
Kx ∼
= Spec (HN
(S) (B̄N (S) (N (X), x )))
f
as group schemes in HN (S) , hence as pro-group schemes in HN
(S) .
But by theorem 6.5.1, the equivalence
f
H 0 (Mf ) : HN
(S) → MT(S)
∗
mot
0
identifies Spec (HN
(S) (B̄N (S) (N (X), x ))) with π1 (X, x), completing the proof.
Corollary 6.6.2 Let k be a number field and S ⊂ P1 (k) a finite set of k-points of P1 . Set
X := P1k \ S and let a ∈ X(k) be a k-point. Then both k and X satisfy the Beilinson-Soulé
vanishing conjectures. Furthermore, there is an isomorphism
π1mot (X, a) ∼
= Ka
as pro-group objects in MT(k).
Proof k satisfies the Beilinson-Soulé vanishing conjectures by Borel’s theorem on the rational K-groups of k [6]. For X, we have the Gysin distinguished triangle
Mgm (X) → Mgm (P1 ) → ⊕y∈S Z(1)[2] → Mgm (X)[1].
Taking motivic cohomology gives the long exact sequence
. . . → ⊕x∈S H p−2 (k, Z(q − 1)) → H p (k, Z(q)) ⊕ H p−2 (k, Z(q − 1))
∂
→ H p (X, Z(q)) →
− ⊕x∈S H p−1 (k, Z(q − 1)) → . . .
Thus the vanishing conjectures for k imply the vanishing conjectures for X. In addition,
since Mgm (P1 ) = Z ⊕ Z(1)[2], the Gysin exact triangle shows that Mgm (X)Q is in DMT(k).
We may therefore apply theorem 6.6.1 to give the isomorphism
π1mot (X, a) ∼
= Ka .
102
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