Chow motives without projectivity, II
arXiv:1705.10502v2 [math.AG] 22 Jun 2017
by
Jörg Wildeshaus
Université Paris 13
Sorbonne Paris Cité
LAGA, CNRS (UMR 7539)
F-93430 Villetaneuse
France
[email protected]
June 22, 2017
Abstract
The purpose of this article is to provide a simplified construction of
the intermediate extension of a Chow motive, provided a condition on
absence of weights in the boundary is satisfied. We give a criterion,
which guarantees the validity of the condition, and compare our new
construction to the theory of the interior motive established earlier.
Keywords: weight structures, boundary motive, interior motive, motivic intermediate extension, intersection motive.
Math. Subj. Class. (2010) numbers: 14F42 (14C25, 14F20, 14F32, 18E30, 19E15).
1
Contents
0 Introduction
2
1 Rigidification of the intermediate extension
7
2 Motivic intermediate extension and interior motive
12
3 A criterion on absence of weights in the boundary
20
0
Introduction
The aim of this article is to extend the main results from [W2] to the context
of motives over a base scheme X, taking into account and relying on the
substantial progress the motivic theory has undergone since the writing of
[loc. cit.].
As far as our aim is concerned, this progress concerns two main points:
(1) the construction of the triangulated category DMB ,c (X) of motives over
X (generalizing the Q-linear version of Voevodsky’s definition for X equal to
a point, i.e., to the spectrum of a field), together with the formalism of six
operations, (2) the construction of a weight structure on DMB ,c (X), compatible with the six operations.
As in [W2], the focus of our study is the absence of weights, and the
guiding principle remains that absence of weights in motives associated to
a boundary allows for the construction of a privileged extension of a given
(Chow) motive. Even over a point, our approach via relative motives yields
a new criterion (Theorem 3.8) on absence of weights in the boundary.
Let us be more precise. For a scheme U, which is separated and of finite
type over a field k (assumed to be of characteristic zero, to fix ideas), the
boundary motive ∂Mgm (U) of U [W1] fits into an exact triangle
(∗)
u
c
∂Mgm (U) −→ Mgm (U) −→ Mgm
(U) −→ ∂Mgm (U)[1] .
c
Here, Mgm (U) and Mgm
(U) denote the motive and the motive with compact
support, respectively, of U, as defined by Voevodsky [V]. Assuming in addic
tion that U is smooth over k, the objects Mgm (U) and Mgm
(U) are of weights
≤ 0 and ≥ 0, respectively. The axioms imposed on a weight structure then
formally imply that the morphism u factors over a motive, which is pure of
weight zero, in other words, u factors over a Chow motive over k. However,
such a factorization is by no means unique (for example, the motive of any
2
smooth compactification of U provides such a factorization). In this context,
which is the one studied in [W2], the “boundary” is the boundary motive
∂Mgm (U), and any factorization of u through a Chow motive is an “extension”.
The above-mentioned progress, and more particularly, point (1) allows
for what one might call “geometrical realizations” of the exact triangle (∗).
Indeed, any open immersion j : U ֒→ X gives rise to an exact triangle
(∗∗)
m
i∗ i∗ j∗ 1U [−1] −→ j! 1U −→ j∗ 1U −→ i∗ i∗ j∗ 1U
of motives over X. Here, we denote by i the closed immersion of the complement Z of U into X, and by 1U the structural motive over U. Provided
that the structure morphism d of X is proper, the direct image d∗ of (∗∗),
i.e., the exact triangle
d m
∗
d∗ i∗ i∗ j∗ 1U [−1] −→ d∗ j! 1U −→
d∗ j∗ 1U −→ d∗ i∗ i∗ j∗ 1U ,
is isomorphic to the dual of (∗).
Thanks to point (2), relative motives are endowed with weights. Independently of properness of the morphism d, the motives j! 1U and j∗ 1U over
X are of weights ≤ 0 and ≥ 0, respectively. Again, the axioms for weight
structures imply that m factors over a motive of weight zero. In this relative
context, the “boundary” is the motive i∗ i∗ j∗ 1U over X (or equivalently, the
motive i∗ j∗ 1U over Z), and any factorization of m through a Chow motive
is an “extension” of 1U (note that unless X is smooth, too, the structural
motive 1X is in general not pure of weight zero). If d is proper, then the
functor d∗ is weight exact. Applying it to a factorization of m through a
Chow motive over X, one therefore obtains a factorization of u through a
Chow motive over k.
More generally, any Chow motive MU over U yields an exact triangle
m
i∗ i∗ j∗ MU [−1] −→ j! MU −→ j∗ MU −→ i∗ i∗ j∗ MU ,
j! MU and j∗ MU are of weights ≤ 0 and ≥ 0, respectively, and therefore, the
morphism m factors over a Chow motive over X. It is this context of relative
motives which seems to be best adapted to our study. The above mentioned
guiding principle relates absence of weights 0 and 1 in the boundary i∗ j∗ MU
to the existence of a privileged extension of MU to a Chow motive over X,
which is minimal in a precise sense among all such extensions. Furthermore,
we are able to describe the sub-category of Chow motives over X arising
as such extensions. Our first main result is Theorem 1.2; it states that
restriction j ∗ from X to U induces an equivalence of categories
∼
j ∗ : CHM(X)i∗ w≤−1,i! w≥1 −−
→ CHM(U)∂w6=0,1 ,
3
where the left hand side denotes the full sub-category of Chow motives M
over X such that i∗ M is of weights at most −1, and i! M is of weights at least
1, and the right hand side denotes the full sub-category of Chow motives MU
over U such that i∗ j∗ MU is without weights 0 and 1.
It turns out that Theorem 1.2 is best proved in the abstract setting of
triangulated categories C(U), C(X) and C(Z) related by gluing, and equipped
with weight structures compatible with the gluing. This is the setting of
Section 1. The proof of Theorem 1.2 relies on Construction 1.3, which relates
factorizations of m : j! MU → j∗ MU , for objects MU of the heart C(U)w=0 of
the weight structure on C(U), to weight filtrations of i∗ j∗ MU . Theorem 1.2
establishes an equivalence
∼
j ∗ : C(X)w=0,i∗ w≤−1,i! w≥1 −−
→ C(U)w=0,∂w6=0,1 ,
where source and target are defined in obvious analogy with the motivic
situation. We can thus define the restriction of the intermediate extension to
the category C(U)w=0,∂w6=0,1
j!∗ : C(U)w=0,∂w6=0,1 ֒−→ C(X)w=0
as the composition of the inverse of the equivalence of Theorem 1.2, followed
by the inclusion C(X)w=0,i∗ w≤−1,i! w≥1 ֒−→ C(X)w=0 (Definition 1.4).
Theorem 1.2 allows to provide important complements for the existing
theory. First (Remark 1.6), the functor j!∗ is compatible with the theory developed in [W5, Sect. 2] when the additional hypothesis enabling the set-up of
the latter, namely semi-primality of the category C(Z)w=0 , is satisfied. Note
that the functoriality properties of the theory from [loc. cit.] are intrinsically
incomplete as the target of the intermediate extension is only a quotient of
C(X)w=0 . Definition 1.4 can thus be seen as providing a rigidification of the
intermediate extension on the sub-category C(U)w=0,∂w6=0,1 of C(U)w=0 . This
observation has rather useful consequences. Indeed, when C(Z)w=0 is semiprimary, then by our very Definition 1.4, it is possible to read off i∗ j!∗ MU
and i! j!∗ MU whether or not MU belongs to C(U)w=0,∂w6=0,1 . In particular,
the non-rigidified intermediate extension j!∗ MU from [W5, Sect. 2] is rigid a
posteriori, if j!∗ MU belongs to the full sub-catoegry C(X)w=0,i∗ w≤−1,i! w≥1 of
C(X)w=0 . Furthermore (Theorem 1.7), for MU ∈ C(U)w=0,∂w6=0,1 , the interval [α, β] ⊃ [0, 1] of weights avoided by i∗ j∗ MU can be determined directly
from i∗ j!∗ MU and i! j!∗ MU .
Second, in the motivic context discussed further above, Theorem 1.2 provides a criterion on absence of weights in the boundary, provided that the
structure morphism d of X is proper. More generally, if d is any proper
morphism with source X, and MU ∈ CHM(U)∂w6=0,1 , then thanks to weight
exactness of d∗ , the motive d∗ i∗ i∗ j∗ MU is still without weights 0 and 1. This
4
means that condition [W2, Asp. 2.3] is satisfied for the morphism
d∗ m : d∗ j! MU −→ d∗ j∗ MU .
The principal aim of Section 2 is to spell out the consequences for our situation of the general theory developed in [W2, Sect. 2], given the validity of
[W2, Asp. 2.3], and to relate them to the restriction of the intermediate extension to CHM(U)∂w6=0,1 (Theorems 2.4–2.6). Let us mention Theorem 2.5
in particular: the Chow motive d∗ j!∗ MU behaves functorially with respect
to both (d ◦ j)! MU and (d ◦ j)∗ MU . Theorem 2.5 applies in particular to
endomorphisms “of Hecke type”; we shall use this observation elsewhere. In
case the proper morphism d equals the structure morphism of X, the Chow
motive d∗ j!∗ MU is defined to be the intersection motive of U relative to X
with coefficients in MU (Definition 2.7). Given the state of the literature,
it appeared useful to spell out the isomorphism between the dual of the interior motive [W2, Sect. 4] and the intersection motive. The comparison
results from Proposition 2.8 onwards contain the earlier mentioned isomorphism between the dual of the exact triangle
(∗)
u
c
∂Mgm (U) −→ Mgm (U) −→ Mgm
(U) −→ ∂Mgm (U)[1]
and the exact triangle
d m
∗
d∗ i∗ i∗ j∗ 1U [−1] −→ d∗ j! 1U −→
d∗ j∗ 1U −→ d∗ i∗ i∗ j∗ 1U .
A concrete difficulty arises when the defining condition of CHM(U)∂w6=0,1
needs to be checked for a concrete object of CHM(U): given a Chow motive
MU over U, how to determine whether or not the motive i∗ j∗ MU is without
weights 0 and 1? Section 3 gives what we think of as the optimal answer
that can be given to date. Combining key results from [W6] and [W5], we
prove Theorem 3.4, which we consider as our second main result: assume
that the (generic) ℓ-adic realization Rℓ,U (MU ) of MU is concentrated in a
single perverse degree, and that the motive i∗ j∗ MU is of Abelian type [W5].
Then whether or not MU belongs to CHM(U)∂w6=0,1 can be read off the perverse cohomology sheaves of i∗ j!∗ Rℓ,U (MU ) and of i! j!∗ Rℓ,U (MU ). If MU ∈
CHM(U)∂w6=0,1 , then the precise interval [α, β] ⊃ [0, 1] of weights avoided
by i∗ j∗ MU can be determined from i∗ j!∗ Rℓ,U (MU ) and i! j!∗ Rℓ,U (MU ).
Chow motives have a tendancy to be auto-dual up to a shift and a twist;
this is in any case true for the applications we have in mind. Given that
the criterion from Theorem 3.4 is compatible with duality, one may therefore hope that the verification of a certain half of that criterion, for example
the half concerning i∗ j!∗ Rℓ,U (MU ), is sufficient, when MU is auto-dual. This
hope is made explicit in Corollary 3.6; we think of this result as potentially
very useful for applications (see e.g. [W7]). For the sake of completeness, we
combine Corollary 3.6 with the comparison from Section 2; the result is the
earlier mentioned Theorem 3.8.
5
Part of this work was done while I was enjoying a délégation auprès du
CNRS, to which I wish to express my gratitude.
Conventions: Throughout the article, F denotes a finite direct product
of fields of characteristic zero. We fix a base scheme B, which is of finite
type over some excellent scheme of dimension at most two. By definition,
schemes are B-schemes which are separated and of finite type (in particular,
they are excellent, and Noetherian of finite dimension), morphisms between
schemes are separated morphisms of B-schemes, and a scheme is nilregular
if the underlying reduced scheme is regular.
We use the triangulated, Q-linear categories DMB ,c (X) of constructible
Beilinson motives over X [CD2, Def. 15.1.1], indexed by schemes X (always
in the sense of the above conventions). In order to have an F -linear theory
at one’s disposal, one re-does the construction, but using F instead of Q as
coefficients [CD2, Sect. 15.2.5]. This yields triangulated, F -linear categories
DMB ,c (X)F satisfying the F -linear analogues of the properties of DMB ,c (X).
In particular, these categories are pseudo-Abelian (see [H, Sect. 2.10]). Furthermore, the canonical functor DMB ,c (X) ⊗Q F → DMB ,c (X)F is fully faithful [CD2, Sect. 14.2.20]. As in [CD2], the symbol 1X is used to denote the
unit for the tensor product in DMB ,c (X)F . We shall employ the full formalism of six operations developed in [loc. cit.]. The reader may choose to
consult [H, Sect. 2] or [W3, Sect. 1] for concise presentations of this formalism.
Beilinson motives can be endowed with a canonical weight structure,
thanks to the main results from [H] (see [B1, Prop. 6.5.3] for the case
X = Spec k, for a field k of characteristic zero). We refer to it as the
motivic weight structure. Following [W3, Def. 1.5], the category CHM(X)F
of Chow motives over X is defined as the heart DMB ,c (X)F,w=0 of the motivic
weight structure on DMB ,c (X)F . It equals the pseudo-Abelian completion of
CHM(X)Q ⊗Q F .
When we assume a field k to admit resolution of singularities, then it will
be in the sense of [FV, Def. 3.4]: (i) for any separated k-scheme X of finite
type, there exists an abstract blow-up Y → X [FV, Def. 3.1] whose source
Y is smooth over k, (ii) for any pair of smooth, seperated k-schemes X, Y of
finite type, and any abstract blow-up q : Y → X, there exists a sequence of
blow-ups p : Xn → . . . → X1 = X with smooth centers, such that p factors
through q. We say that k admits strict resolution of singularities, if in (i),
for any given dense open subset U of the smooth locus of X, the blow-up
q : Y → X can be chosen to be an isomorphism above U, and such that
arbitrary intersections of the irreducible components of the complement Z
of U in Y are smooth (e.g., Z ⊂ Y a normal crossing divisor with smooth
irreducible components).
6
1
Rigidification of the intermediate extension
Throughout this section, let us fix three F -linear pseudo-Abelian triangulated
categories C(U), C(X) and C(Z), the second of which is obtained from the
others by gluing. This means that the three categories are equipped with six
exact functors
C(U)
C(U) o
C(U)
j!
/
j∗
j∗
/
C(X)
i∗
C(X) o
i∗
C(X)
i!
C(Z)
/
/
C(Z)
C(Z)
satisfying the axioms from [BBD, Sect. 1.4.3]. We assume that C(U), C(X)
and C(Z) are equipped with weight structures w (the same letter for the
three weight structures), and that the one on C(X) is actually obtained from
the two others in a way compatible with the gluing, meaning that the left
adjoints j! , j ∗ , i∗ and i∗ respect the categories C(•)w≤0 , and the right adjoints
j ∗ , j∗ , i∗ and i! respect the categories C(•)w≥0 . In particular, we have a fully
faithful functor
i∗ : C(Z)w=0 ֒−→ C(X)w=0 ,
and a functor
j ∗ : C(X)w=0 −→ C(U)w=0 .
According to [W5, Prop. 2.5], the latter is full and essentially surjective. We
shall need to understand its restriction to a certain sub-category of C(X)w=0 .
Recall [W2, Def. 1.10] that an object M is said to be without weights
m, . . . , n, or to avoid weights m, . . . , n, for integers m ≤ n, if it admits a
weight filtration avoiding weights m, . . . , n, i.e. [W2, Def. 1.6], if there is an
exact triangle
M≤m−1 −→ M −→ M≥n+1 −→ M≤m−1 [1]
with M≤m−1 of weights at most m − 1, and M≥n+1 of weights at least n + 1.
Definition 1.1. (a) Denote by C(U)w=0,∂w6=0,1 the full sub-category of
C(U)w=0 of objects MU such that i∗ j∗ MU is without weights 0 and 1.
(b) Denote by C(X)w=0,i∗ w≤−1,i! w≥1 the full sub-category of C(X)w=0 of objects M such that i∗ M ∈ C(Z)w≤−1 and i! M ∈ C(Z)w≥1 .
Theorem 1.2. The restriction of j ∗ to C(X)w=0,i∗ w≤−1,i! w≥1 induces an
equivalence of categories
∼
j ∗ : C(X)w=0,i∗ w≤−1,i! w≥1 −−
→ C(U)w=0,∂w6=0,1 .
The proof of Theorem 1.2 relies on the following.
7
Construction 1.3. Let MU ∈ C(U)w=0 , and consider the morphism
m : j! MU −→ j∗ MU
in C(X). There is a canonical choice of cone of m, namely, the object
i∗ i∗ j∗ MU . Any weight filtration of i∗ j∗ MU
c−
c+
δ
C≤0 −→ i∗ j∗ MU −→ C≥1 −→ C≤0 [1]
(with C≤0 ∈ C(Z)w≤0 and C≥1 ∈ C(Z)w≥1) yields a diagram, which we shall
denote by the symbol (1), of exact triangles
0
i∗ C≥1 [−1]
/
i∗ C≥1 [−1]
i∗ δ[−1]
j! MU
i∗ C≤0
j! MU
m
0
/
j! MU [1]
/
j! MU [1]
i∗ i j∗ MU
/
i∗ c +
i∗ C≥1
i∗ C≥1
/
i∗ c −
∗
j∗ MU
/
0
/
0
/
which according to axiom TR4’ of triangulated categories [BBD, Sect. 1.1.6]
can be completed to give a diagram, denoted by (2)
0
/
i∗ C≥1 [−1]
i∗ C≥1 [−1]
j! MU
/
/
/
/
j∗ MU
/
/
j! MU [1]
/
j! MU [1]
i∗ C≤0
m
0
M
0
i∗ δ[−1]
j! MU
/
i∗ c −
i∗ i∗ j∗ MU
i∗ c +
i∗ C≥1
i∗ C≥1
/
0
with M ∈ C(X). By the second row, and the second column of diagram (2),
the object M is simultaneously an extension of objects of weights ≤ 0, and an
extension of objects of weights ≥ 0. It follows easily (c.f. [B1, Prop. 1.3.3 3])
that M belongs to both C(X)w≤0 and C(X)w≥0 , and hence to C(X)w=0 .
Applying the functors j ∗ , i∗ , and i! to (2), we see that j ∗ M = MU ,
∼
i∗ M −−
→ C≤0 ,
Proof of Theorem 1.2.
∼
and C≥1 [−1] −−
→ i! M .
For MU ∈ C(U)w=0 and a weight filtration
C≤0 −→ i∗ j∗ MU −→ C≥1 −→ C≤0 [1]
of i∗ j∗ MU , apply Construction 1.3 to get an extension M ∈ C(X)w=0 of MU
to X. From the isomorphisms
∼
i∗ M −−
→ C≤0
∼
and C≥1 [−1] −−
→ i! M ,
8
we see that M ∈ C(X)w=0,i∗ w≤−1,i! w≥1 if and only if
C≤0 ∈ C(Z)w≤−1
and C≥1 ∈ C(Z)w≥2 .
In particular, we see that any object in C(U)w=0,∂w6=0,1 admits a pre-image
under j ∗ in C(X)w=0,i∗ w≤−1,i!w≥1 .
Conversely, any object M from C(X)w=0 fits into a diagram of type (2)
0
/
i∗ i! M
j! j ∗ M
/
/
/
/
j∗ j ∗ M
i∗ i∗ M
/
j! j ∗ M[1]
/
j! j ∗ M[1]
/
i∗ i∗ j∗ j ∗ M
i∗ i! M[1]
i∗ i! M[1]
0
/
M
j! j ∗ M
0
i∗ i! M
/
0
Its third column shows that j ∗ maps C(X)w=0,i∗ w≤−1,i! w≥1 to C(U)w=0,∂w6=0,1 .
The restriction of j ∗ therefore yields a well-defined functor
j ∗ : C(X)w=0,i∗ w≤−1,i! w≥1 −→ C(U)w=0,∂w6=0,1 ,
which is full [W5, Prop. 2.5] and essentially surjective. It remains to show
that it is faithful, i.e., that a morphism f in C(X)w=0,i∗ w≤−1,i!w≥1 such that
j ∗ f = 0, equals zero.
Thus, let M, M ′ ∈ C(X)w=0,i∗ w≤−1,i! w≥1, f : M → M ′ , and assume that
∗
j f = 0. This means that the composition of f with the adjunction morphism
M ′ → j∗ j ∗ M ′ is zero. Given the exact localization triangle
i∗ i! M ′ −→ M ′ −→ j∗ j ∗ M ′ −→ i∗ i! M ′ [1] ,
the morphism f factors through i∗ i! M ′ . Now M is of weight zero, while
i! M ′ , and hence i∗ i! M ′ , is of strictly positive weights. By orthogonality, any
morphism M → i∗ i! M ′ is zero.
q.e.d.
Definition 1.4. Let the F -linear pseudo-Abelian triangulated categories
C(U), C(X) and C(Z) be related by gluing, and equipped with weight structures w compatible with the gluing. Define the restriction of the intermediate
extension to the category C(U)w=0,∂w6=0,1
j!∗ : C(U)w=0,∂w6=0,1 ֒−→ C(X)w=0
as the composition of the inverse of the equivalence of Theorem 1.2, followed
by the inclusion C(X)w=0,i∗ w≤−1,i! w≥1 ֒−→ C(X)w=0 .
Remark 1.5. Assume that contravariant auto-equivalences
∼
D • : C( • ) −−
→ C( • ) , • ∈ {U, X, Z}
are given, that they are compatible with the gluing (e.g., DX ◦ j∗ ∼
= j! ◦ DU )
and with the weight structures (e.g., DX C(X)w≥0 = C(X)w≤0). From the
9
isomorphisms DZ ◦ i∗ ◦ j∗ ∼
= i∗ ◦ j∗ [−1], it follows
= i! ◦ j! ◦ DU and i! ◦ j! ∼
first that the functor DU respects the category C(U)w=0,∂w6=0,1 . Similarly, the
functor DX respects the category C(X)w=0,i∗ w≤−1,i! w≥1 .
It then follows formally from Definition 1.4, and from DU ◦ j ∗ ∼
= j ∗ ◦ DX
that
DX ◦ j!∗ ∼
= j!∗ ◦ DU .
In other words, the restriction of the intermediate extension to C(U)w=0,∂w6=0,1
is compatible with local duality.
Remark 1.6. (a) Assume that some composition of morphisms
i∗ N −→ M −→ i∗ N
gives the identity on i∗ N, for some object N of C(Z)w=0 . Then the adjunction
properties of i∗ , i∗ and i! show that N is a direct factor of both i! M and i∗ M,
and that the restriction of the composition i! M → i∗ M of the adjunction
morphisms to this direct factor is the identity.
We obtain that objects in C(X)w=0,i∗ w≤−1,i! w≥1 do not admit non-zero
direct factors belonging to the image of C(Z)w=0 under the functor i∗ . This
justifies Definition 1.4: the intermediate extension of MU ∈ C(U)w=0,∂w6=0,1 is
indeed an extension of MU not admitting non-zero direct factors belonging
to the image of i∗ .
(b) More generally, for M ∈ C(X)w=0,i∗ w≤−1,i! w≥1 and N ∈ C(Z)w=0 , any
morphism M → i∗ N is zero, and so is any morphism i∗ N → M.
(c) Following [W5, Def. 1.6 (a)], denote by g the two-sided ideal of C(X)w=0
generated by
HomC(X)w=0 (M, i∗ N) and
HomC(X)w=0 (i∗ N, M) ,
for all objects (M, N) of C(X)w=0 × C(Z)w=0 , such that M admits no nonzero direct factor belonging to the image of i∗ . Denote by C(X)uw=0 the
quotient of C(X)w=0 by g. From (b), we see that g(M, M ′ ) = 0, for any
M, M ′ ∈ C(X)w=0,i∗ w≤−1,i! w≥1 . Thus, the quotient functor C(X)w=0 −→
→
C(X)uw=0 induces an isomorphism on C(X)w=0,i∗ w≤−1,i! w≥1 .
(d) From the above, we see that Definition 1.4 is compatible with the theory
developed in [W5, Sect. 2] when the hypotheses enabling the set-up of the
latter are satisfied. More precisely, assume in addition that C(Z)w=0 is semiprimary [AK, Déf. 2.3.1]. Then
j!∗ : C(U)w=0 ֒−→ C(X)uw=0
is defined on the whole of C(U)w=0 [W5, Def. 2.10]. Parts (a) and (c) of the
present Remark, and [W5, Summ. 2.12 (a) (1)] show that the diagram
C(U)w=0,∂w6
=0,1


j!∗
/
C(X)w=0
_
C(U)w=0

j!∗
10
/
C(X)uw=0
is commutative. Thus, j!∗ : C(U)w=0,∂w6=0,1 ֒→ C(X)w=0 from Definition 1.4 is
indeed the restriction of j!∗ : C(U)w=0 ֒→ C(X)uw=0 from [W5, Sect. 2], when
the latter is defined, to C(U)w=0,∂w6=0,1 .
Theorem 1.7. Let MU ∈ C(U)w=0 .
(a) Assume that the category C(Z)w=0 is semi-primary. Then the object MU
belongs to C(U)w=0,∂w6=0,1 if and only if j!∗ MU belongs to C(X)w=0,i∗ w≤−1,i!w≥1 .
(b) Assume that MU belongs to C(U)w=0,∂w6=0,1. Let α ≤ 0 and β ≥ 1 two
integers. Then i∗ j∗ MU is without weights α, α + 1, . . . , β if and only if
i∗ j!∗ MU ∈ C(Z)w≤α−1
and i! j!∗ MU ∈ C(Z)w≥β .
Proof. The “if” part from (a) is Theorem 1.2, and its “only if” part is
Remark 1.6 (d).
As for (b), note that by definition, Construction 1.3 for MU and a weight
fitration
C≤−1 −→ i∗ j∗ MU −→ C≥2 −→ C≤−1 [1]
of i∗ j∗ MU avoiding weights 0 and 1 yields the extension M = j!∗ MU of MU
to X. The claim thus follows from the isomorphisms
∼
i∗ j!∗ MU −−
→ C≤−1
∼
and C≥2 [−1] −−
→ i! j!∗ MU .
q.e.d.
Remark 1.8. (a) As the attentive reader will have remarked already,
the formalism could have been developed on larger sub-categories of C(X)
and C(U), at the cost of losing its inherent auto-duality. More precisely,
define full sub-categories C(U)w=0,∂w6=0 and C(U)w=0,∂w6=1 of C(U)w=0 , and
C(X)w=0,i∗ w≤−1 and C(X)w=0,i! w≥1 of C(X)w=0 in the obvious way. Then as
in Theorem 1.2,
∼
j ∗ : C(X)w=0,i∗ w≤−1 −−
→ C(U)w=0,∂w6=0
and
∼
j ∗ : C(X)w=0,i! w≥1 −−
→ C(U)w=0,∂w6=1 ,
allowing to define the restrictions j!∗ of the intermediate extension to both
C(U)w=0,∂w6=0 and C(U)w=0,∂w6=1 . Local duality as in Remark 1.5 exchanges
the two constructions. Parts (a), (c), and (d) (but not (b)) of Remark 1.6
apply with the most obvious modifications.
(b) Even if it will not be needed in the sequel of this article, it is worthwhile
to spell out the modified version of Theorem 1.7. Let MU ∈ C(U)w=0 . Then
MU ∈ C(U)w=0,∂w6=0 ⇐⇒ j!∗ MU ∈ C(X)w=0,i∗ w≤−1
and
MU ∈ C(U)w=0,∂w6=1 ⇐⇒ j!∗ MU ∈ C(X)w=0,i!w≥1 ,
11
provided that C(Z)w=0 is semi-primary. More interestingly, part (b) of Theorem 1.7 can be separated into two statements: let α ≤ 0 and β ≥ 1. Assume
that MU ∈ C(U)w=0,∂w6=0 or MU ∈ C(U)w=0,∂w6=1 . Then i∗ j∗ MU is without
weights α, α + 1, . . . , 0 if and only if
i∗ j!∗ MU ∈ C(Z)w≤α−1 ,
and i∗ j∗ MU is without weights 1, 2, . . . , β if and only if
i! j!∗ MU ∈ C(Z)w≥β .
2
Motivic intermediate extension and interior motive
The purpose of the present section is to connect Section 1 to the theory developed in [W2, Sect. 2]. The main results are Theorems 2.4, 2.5 and 2.6;
these concern the motivic intermediate extension, and are formal analogues
of the main results from [W2, Sect. 4] on the interior motive, defined and
studied in [loc. cit.]. When the base scheme B is the spectrum of a field admitting strict resolution of singularities, then the analogy is not just formal:
Corollary 2.10 establishes an isomorphism between the dual of the interior
motive and the direct image of the motivic intermediate extension under the
structure morphism, provided the latter is proper and that weights 0 and 1
are avoided.
Let X be a scheme (in the sense of our Introduction), and j : U ֒→ X an
open immersion with complement i : Z ֒→ X. Thanks to localization [CD2,
Sect. 2.3], and to compatibility of the motivic weight structure with gluing
[H, Thm. 3.8], Theorem 1.2 applies to the categories C(•) = DMB ,c (•)F ,
for • = U, X, Z. We write CHM(U)F,∂w6=0,1 for DMB ,c (U)F,w=0,∂w6=0,1 , and
CHM(X)F,i∗ w≤−1,i!w≥1 for DMB ,c (X)F,w=0,i∗w≤−1,i! w≥1. In this motivic context, Remark 1.6 (d) says that the functor j!∗ from Definition 1.4 equals the
restriction to CHM(U)F,∂w6=0,1 of the motivic intermediate extension, when
the latter is defined; note that this is the case in the context of motives of
Abelian type, studied in [W5, Sect. 5] (see our Section 3).
Proposition 2.1. Let d : X → D a proper morphism of schemes, and
MU a Chow motive over U belonging to CHM(U)F,∂w6=0,1 . Denote by m the
morphism j! MU → j∗ MU .
(a) The image
d m
∗
d∗ j∗ MU −→ d∗ i∗ i∗ j∗ MU −→ d∗ j! MU [1]
d∗ j! MU −→
12
under d∗ of the triangle
m
j! MU −→ j∗ MU −→ i∗ i∗ j∗ MU −→ j! MU [1]
in DMB ,c (D)F is exact.
(b) The motive d∗ i∗ i∗ j∗ MU ∈ DMB ,c (D)F is without weights 0 and 1.
Proof.
(a): Indeed, the triangle
m
j! MU −→ j∗ MU −→ i∗ i∗ j∗ MU −→ j! MU [1]
is exact.
(b): The morphism d ◦ i being proper, we have (d ◦ i)! = (d ◦ i)∗ = d∗ i∗ .
Thus, d∗ i∗ is weight exact. It therefore transforms any weight filtration
avoiding weights 0 and 1 into the same kind of weight filtration.
q.e.d.
Thus, [W2, Asp. 2.3] is satisfied, with C = DMB ,c (D)F , u = d∗ m :
d∗ j! MU → d∗ j∗ MU , and C = d∗ i∗ i∗ j∗ MU [−1]. Consequently, the theory
developed in [W2, Sect. 2] applies.
Definition 2.2 (cmp. [W2, Def. 2.1]). Denote by DMB ,c (D)F,w≤0,6=−1 the
full sub-category of DMB ,c (D)F,w≤0 of objects without weight −1, and by
DMB ,c (D)F,w≥0,6=1 the full sub-category of DMB ,c (D)F,w≥0 of objects without
weight 1.
Proposition 2.3 ([W2, Prop. 2.2]). The inclusions
ι− : CHM(D)F ֒−→ DMB ,c (D)F,w≤0,6=−1
and
ι+ : CHM(D)F ֒→ DMB ,c (D)F,w≥0,6=1
admit a left adjoint
Gr0 : DMB ,c (D)F,w≤0,6=−1 −→ CHM(D)F
and a right adjoint
Gr0 : DMB ,c (D)F,w≥0,6=1 −→ CHM(D)F ,
respectively. Both adjoints map objects (and morphisms) to the term of
weight zero of a weight filtration avoiding weight −1 and 1, respectively. The
compositions Gr0 ◦ι− and Gr0 ◦ι+ both equal the identity on CHM(D)F .
Theorem 2.4. Let d : X → D a proper morphism.
(a) The essential image of the restriction of the functor d∗ j! to the subcategory CHM(U)F,∂w6=0,1 is contained in DMB ,c (D)F,w≤0,6=−1, inducing a
functor
(d ◦ j)! = d∗ j! : CHM(U)F,∂w6=0,1 −→ DMB ,c (D)F,w≤0,6=−1 .
13
More precisely, if MU ∈ CHM(U)F is such that i∗ j∗ MU avoids weights α, α+
1, . . . , β, for integers α ≤ 0 and β ≥ 1, then
d∗ i∗ i∗ j!∗ MU [−1] −→ d∗ j! MU −→ d∗ j!∗ MU −→ d∗ i∗ i∗ j!∗ MU
is a weight filtration of d∗ j! MU avoiding weights α − 1, α, . . . , −1.
(b) The essential image of the restriction of the functor d∗ j∗ to the subcategory CHM(U)F,∂w6=0,1 is contained in DMB ,c (D)F,w≥0,6=1, inducing a functor
(d ◦ j)∗ = d∗ j∗ : CHM(U)F,∂w6=0,1 −→ DMB ,c (D)F,w≥0,6=1 .
More precisely, if MU ∈ CHM(U)F is such that i∗ j∗ MU avoids weights α, α+
1, . . . , β, for integers α ≤ 0 and β ≥ 1, then
d∗ j!∗ MU −→ d∗ j∗ MU −→ d∗ i∗ i! j!∗ MU [1] −→ d∗ j!∗ MU [1]
is a weight filtration of d∗ j∗ MU avoiding weights 1, 2, . . . , β.
(c) There are canonical isomorphisms of functors
∼
Gr0 ◦(d◦j)! = Gr0 ◦d∗j! −−
→ d∗ j!∗
∼
and d∗ j!∗ −−
→ Gr0 ◦d∗ j∗ = Gr0 ◦(d◦j)∗
on the category CHM(U)F,∂w6=0,1. Their composition equals the value of the
functor Gr0 ◦d∗ at the restriction of the natural transformation m : j! → j∗
to CHM(U)F,∂w6=0,1; in particular,
Gr0 ◦(d∗ ◦ m) : Gr0 ◦(d ◦ j)! −→ Gr0 ◦(d ◦ j)∗
is an isomorphism of functors on CHM(U)F,∂w6=0,1.
Proof. Let MU ∈ CHM(U)F,∂w6=0,1 . By definition (and Theorem 1.2),
the motive j!∗ MU belongs to CHM(X)F,i∗ w≤−1,i! w≥1 . Thus, the exact triangles
i∗ i∗ j!∗ MU [−1] −→ j! MU −→ j!∗ MU −→ i∗ i∗ j!∗ MU
and
j!∗ MU −→ j∗ MU −→ i∗ i! j!∗ MU [1] −→ j!∗ MU [1]
are weight filtrations (of j! MU ) avoiding weight −1, and (of j∗ MU ) avoiding
weight 1, respectively. An analogous statement is therefore true for their
images under the weight exact functor d∗ (recall that d is assumed to be
proper). Together with Proposition 2.3, this shows part (c) of the statement,
as well as the first claims of parts (a) and (b). The second, more precise
claims follow from Theorem 1.7 (b).
q.e.d.
At first sight, it may thus appear that the theory from [W2, Sect. 2]
does not add much to what we get by explicit identification of the weight
filtrations. But then, note the following.
Theorem 2.5. Let d : X → D a proper morphism. Then for any Chow
motive MU ∈ CHM(U)F,∂w6=0,1 , the Chow motive d∗ j!∗ MU ∈ CHM(D)F
14
behaves functorially with respect to both motives (d ◦ j)! MU and (d ◦ j)∗ MU .
In particular, any endomorphism of (d ◦ j)! MU or of (d ◦ j)∗ MU induces an
endomorphism of d∗ j!∗ MU .
Proof.
This follows from the functorial identities from Theorem 2.4 (c).
q.e.d.
Theorem 2.6. Let MU ∈ CHM(U)F,∂w6=0,1 , d : X → D a proper morphism, and assume given a factorization (d ◦ j)! MU → MD → (d ◦ j)∗ MU
of the morphism d∗ m : (d ◦ j)! MU → (d ◦ j)∗ MU through an object MD of
CHM(D)F . Then d∗ j!∗ MU is canonically identified with a direct factor of
MD , admitting a canonical complement.
Proof.
This is [W2, Cor. 2.5].
q.e.d.
The theory applies in particular when d equals the structure morphism
from X to the base scheme B.
Definition 2.7. Assume that X is proper over D = B. Denote by d :
X → B the structure morphism of X. Let MU ∈ CHM(U)F,∂w6=0,1 . We call
d∗ j!∗ MU the intersection motive of U relative to X with coefficients in MU .
Our terminology is motivated by one of the main results of [W5]. It states
that on Chow motives of Abelian type, the cohomological Betti [Ay, Déf. 2.1]
and ℓ-adic realizations [CD3, Sect. 7.2, see in part. Rem. 7.2.25] are compatible with intermediate extensions (of motives, and of perverse sheaves). For
details, we refer to [W5, Thm. 7.2]. Since the realizations are compatible
with direct images, they therefore map d∗ j!∗ MU to intersection cohomology
whenever MU is a Chow motive of Abelian type.
Let us connect the above to the notion of interior motive.
Proposition 2.8. Assume that D = B = Spec k for a field k admitting
strict resolution of singularities. Assume also that the structure morphism
d : X → B is proper, and that its restriction d ◦ j to U ⊂ X is smooth. Let
π : C → U proper and smooth (hence C is smooth over k). Consider the
(smooth) Chow motive π∗ 1C over U. Then the morphism
d∗ m : (d ◦ j)! π∗ 1C −→ (d ◦ j)∗ π∗ 1C
is canonically and CHdC (C×U C)-equivariantly isomorphic (dC := the absolute
dimension of C) to the dual of the morphism
c
u : Mgm (C) −→ Mgm
(C)
in DMgm (k)F [V, pp. 223–224].
A few words of explanation are in order. First, by [CD2, Cor. 16.1.6],
the triangulated category DMB ,c (Spec k)F is identified with the F -linear
15
version of the triangulated category of geometrical motives DMgm (k)F [V,
p. 192] (see [An, Sect. 17.1.3]). Second, the duality in question is the functor
mapping N to
N ∗ := Hom N, 1Spec k .
Third, equivariance under the Chow group CHdC (C ×U C) refers to the following. According to [CD2, Cor. 14.2.14],
EndCHM (U )F π∗ 1C = CHdC (C ×U C) ⊗Z F ,
meaning that the Chow group CHdC (C ×U C) acts on π∗ 1C ∈ DMB ,c (U)F .
Hence the morphism d∗ m is CHdC (C ×U C)-equivariant. As for the action on
c
Mgm (C), on Mgm
(C), and the equivariance of u, we refer to [D, Thm. 5.23]
and [CD1, Ex. 4.12, Ex. 7.15].
Remark 2.9. According to [L, Prop. 5.19, Cor. 6.14], the identification
EndCHM (U )F π∗ 1C = CHdC (C ×U C) ⊗Z F ,
is compatible with composition. Thus, the action of CHdC (C ×U C) on π∗ 1C
is by correspondences in the classical sense.
Proof of Proposition 2.8. The morphism d∗ m coincides with the value
of the natural transformation of functors
(d ◦ j ◦ π)! −→ (d ◦ j ◦ π)∗
on 1C , since both π and d are proper [CD2, Thm. 2.2.14 (2)]. Fix a smooth
compactification C̄ of C over k, write j ′ for the open immersion of C into C̄,
and c for the structure morphism of C̄. Thus, c ◦ j ′ = d ◦ j ◦ π is the structure
morphism of C. Then d∗ m equals c∗ m′ , for the morphism m′ : j!′ 1C → j∗′ 1C
in DMB ,c (C̄)F , and can be factorized as follows:
c∗ adj2
c∗ adj1
d∗ m = c∗ adj1 ◦ c∗ adj2 : c∗ j!′ 1C −→ c∗ 1C̄ −→ c∗ j∗′ 1C ,
where adj1 : 1C̄ → j∗′ (j ′ )∗ 1C̄ = j∗′ 1C and adj2 : j!′ 1C = j!′ (j ′ )∗ 1C̄ → 1C̄ are
the adjunction maps. Now c∗ adj2 and c∗ adj1 are related by duality: we have
∗
∗
c∗ j!′ 1C = c∗ j∗′ (c ◦ j ′ )! 1Spec k = c∗ j∗′ (j ′ )∗ c! 1Spec k ,
∗
c∗ 1C̄ = c∗ c! 1Spec k ,
and under these identifications, c∗ adj2 is dual to the morphism
c∗ adj1! : c∗ c! 1Spec k −→ c∗ j∗′ (j ′ )∗ c! 1Spec k ,
where adj1! denotes the adjunction map c! 1Spec k → j∗′ (j ′ )∗ c! 1Spec k [CD2,
∼
Thm. 15.2.4]. Fix an isomorphism α : c! 1Spec k −−
→ 1C̄ (dC )[2dC ]; according to [W3, Cor. 3.7], such an isomorphism exists, and it is unique up to
multiplication by elements of F ∗ . Via α, the morphism c∗ adj1! is identified
with
c∗ adj1 (dC )[2dC ] : c∗ 1C̄ (dC )[2dC ] −→ c∗ j∗′ (j ′ )∗ 1C̄ (dC )[2dC ] ,
16
and this identification does not depend on the choice of α.
To summarize the discussion so far: the morphism d∗ m equals the composition of c∗ adj1 , preceded by the dual of c∗ adj1 (dC )[2dC ].
As for the morphism u, observe that it, too, can be factorized:
(j ′ )∗
j′
∗
c
c
Mgm (C̄) = Mgm
(C̄) −→ Mgm
u = (j ′ )∗ ◦ j∗′ : Mgm (C) −→
(C) ,
where we denote by j∗′ and (j ′ )∗ the morphisms induced by the open immerc
sion j ′ on the level of Mgm and Mgm
, respectively [V, pp. 223–224]. According
to [V, Thm. 4.3.7 3], the dual of (j ′ )∗ is identified with
j∗′ (−dC )[−2dC ] : Mgm (C)(−dC )[−2dC ] −→ Mgm (C̄)(−dC )[−2dC ] .
To summarize: the morphism u equals the composition of j∗′ , followed by the
dual of j∗′ (−dC )[−2dC ].
To relate d∗ m and u, observe that
Mgm (C) = (c ◦ j ′ )♯ 1C = c♯ j!′ 1C , Mgm (C̄) = c♯ 1C̄ ,
and under these identifications, the morphism j∗′ equals c♯ adj3 , where adj3 :
j!′ 1C = j!′ (j ′ )∗ 1C̄ → 1C̄ is the adjunction [CD2, Sect. 1.1.34, Sect. 11.1.2,
Cor. 16.1.6]. According to one of the projection formulae
HomV (f♯ M, N) = f∗ HomT (M, f ∗ N)
for f : T → V smooth [CD2, Sect. 1.1.33], the morphism j∗′ = c♯ adj3 is
dual to c∗ adj1 . Thus, the dual of u equals c∗ adj1 , preceded by the dual of
c∗ adj1 (dC )[2dC ], i.e., it equals d∗ m.
It remains to show that the identification of d∗ m and the dual of u is
equivariant under CHdC (C ×U C). Given that d∗ m is the value at π∗ 1C
of a natural transformation of functors,
and that CHdC (C ×U C) ⊗Z F is
identified with EndCHM (U )F π∗ 1C , all one needs to establish is that under
our identification, the action of CHdC (C ×U C) on (d ◦ j)∗ π∗ 1C coincides
with the action of CHdC (C ×U C) on the dual of Mgm (C), and likewise for
c
(d ◦ j)! π∗ 1C = Mgm
(C)∗ . As before, the second compatibility is dual to the
first, up to application of a twist by dC and a shift by 2dC .
As for the identification (d◦j)∗ π∗ 1C = c∗ j∗′ 1C = Mgm (C)∗ , it is compatible
with the action of finite correspondences Z ∈ cU (C, C) by the very definition
of the category of motivic complexes [CD2, Def. 11.1.1]. It remains to cite
[L, Lemma 5.18]: every class in CHdC (C ×U C) can be represented by a cycle
Z belonging to cU (C, C).
q.e.d.
Corollary 2.10. Assume that D = B = Spec k for a field k admitting
strict resolution of singularities, that the structure morphism d : X → B is
proper, and that its restriction d ◦ j to U ⊂ X is smooth. Let π : C → U
proper and smooth, and e ∈ CHdC (C ×U C) ⊗Z F an idempotent. Assume
that the direct factor (π∗ 1C )e of the Chow motive π∗ 1C ∈ CHM(U)F lies in
CHM(U)F,∂w6=0,1 .
(a) The e-part ∂Mgm (C)e of the boundary motive ∂Mgm (C) is without weights
17
−1 and 0. In particular, C and e satisfy assumption [W2, Asp. 4.2], and
therefore, the e-part of the interior motive of C, Gr0 Mgm (C)e is defined
[W2, Def. 4.9].
(b) There is a canonical isomorphism
∗
∼
d∗ j!∗ (π∗ 1C )e −−
→ Gr0 Mgm (C)e .
It is compatible with the factorizations
(d ◦ j)! (π∗ 1C )e −→ d∗ j!∗ (π∗ 1C )e −→ (d ◦ j)∗ (π∗ 1C )e
of d∗ m and
c
Mgm (C)e −→ Gr0 Mgm (C)e −→ Mgm
(C)e
of u under the identification of Proposition 2.8.
Remark 2.11. (a) The hypothesis on strict resolution of singularities
is (implicitly) used twice (apart from the proof of Proposition 2.8). First,
the results from [W2, Sect. 4] were formulated only for such fields. This
comes mainly from the fact that at the time when [W2] was written, the
existence of the motivic weight structure on DMgm (k)F was only established
under that hypothesis. Given the main results from [H], and the comparison
result [CD2, Cor. 16.1.6], relating DMB ,c (Spec k)F and DMgm (k)F , one can
dispose of that restriction on k as far as the weight structure is concerned.
Second, and more seriously, the hypothesis is used for the construction
of the action of CHdC (C ×U C) on the boundary motive ∂Mgm (C) [W4,
Thm. 2.2], and hence for the very definition of ∂Mgm (C)e .
Given Corollary 2.10, the reader should obviously feel free to define the
e-part of the interior motive of C as (d∗ j!∗ (π∗ 1C )e )∗ , in case the field k does
not admit strict resolution of singularities.
(b) Recall from [W2, Def. 4.1 (a)] that there is a ring c1,2 (C, C) (of “bi-finite
correspondences”) acting on the exact triangle
(∗)
c
∂Mgm (C) −→ Mgm (C) −→ Mgm
(C) −→ ∂Mgm (C)[1] .
Denote by c̄1,2 (C, C) the quotient of c1,2 (C, C) by the kernel of this action.
The algebra c̄1,2 (C, C) ⊗Z F is a canonical source of idempotent endomorphisms e of (∗), and it is for such choices that the results in [W2, Sect. 4]
were formulated. However, they remain valid in the present context.
Proof of Corollary 2.10.
According to Proposition 2.8,
d∗ m : (d ◦ j)! (π∗ 1C )e −→ (d ◦ j)∗ (π∗ 1C )e
is identified with the dual of
c
u : Mgm (C)e −→ Mgm
(C)e
Any choice of cone of d∗ m is therefore isomorphic to the shift by [1] of the
dual of any choice of cone of u. But d∗ i∗ i∗ j∗ (π∗ 1C )e is a cone of d∗ m, while
18
∂Mgm (C)e is the shift by [−1] of a cone of u. Thus,
∗
∂Mgm (C)e ∼
= d∗ i∗ i∗ j∗ (π∗ 1C )e .
Thanks to our additional assumption on (π∗ 1C )e , and to Proposition 2.1 (b),
the motive d∗ i∗ i∗ j∗ (π∗ 1C )e is without weights 0 and 1. Part (a) of our claim
then follows from the compatibility of the motivic weight structure with
duality [W3, Thm. 1.12].
The same argument yields that Gr0 (d ◦ j)∗ (π∗ 1C )e and Gr0 Mgm (C)e are
dual to each other. Part (b) thus follows from Theorem 2.4 (c).
q.e.d.
Remark 2.12. An alternative proof could be given by showing that the
exact triangle
d m
∗
d∗ j! (π∗ 1C ) −→
d∗ j∗ (π∗ 1C ) −→ d∗ i∗ i∗ j∗ (π∗ 1C ) −→ d∗ j! (π∗ 1C )[1]
is CHdC (C ×U C)-equivariantly isomorphic to the dual of the exact triangle
u
c
c
Mgm
(C)[−1] −→ ∂Mgm (C) −→ Mgm (C) −→ Mgm
(C) .
To establish that latter result, one would apply techniques similar to the ones
used in the proofs of [W4, Thm. 2.2 and 2.5].
Remark 2.13. (a) Assume that we are in the setting of Corollary 2.10.
In particular, D = B = Spec k, and the (structure) morphism d : X → B is
proper. From Corollary 2.10 (b), and from [W2, Theorems 4.7 and 4.8], it
follows that the Chow motive d∗ j!∗ (π∗ 1C )e , i.e., the e-part of the intersection
motive of U relative to X with coefficients in π∗ 1C , realizes to give the epart of interior cohomology of C. In other words, the natural map from the
e-part of intersection cohomology to cohomology with coefficients in π∗ 1C is
injective, and the map from cohomology with compact support to the e-part
of intersection cohomology with coefficients in π∗ 1C is surjective.
(b) In fact, as is shown in [loc. cit.], the statement from (a) follows from
the more precise fact that the values of the respective cohomological (Hodge
theoretic or ℓ-adic) realization (H m ◦ R)m∈Z on the canonical morphisms
(d ◦ j)! (π∗ 1C )e −→ d∗ j!∗ (π∗ 1C )e
and
d∗ j!∗ (π∗ 1C )e −→ (d ◦ j)∗ (π∗ 1C )e
identify H m ◦ R(d∗ j!∗ (π∗ 1C )e ) with the part of weight m of H m ◦ R((d ◦
j)! (π∗ 1C )e ), and of H m ◦ R((d ◦ j)∗ (π∗ 1C )e ), respectively.
(c) The statement from (b) can be shown without using Corollary 2.10, i.e.,
without any reference to the interior motive, and for arbitrary objects MU
of CHM(U)F,∂w6=0,1 instead of (π∗ 1C )e , by formally imitating the proofs of
[W2, Theorems 4.7 and 4.8]. The latter make essential use of the existence of
weights on the level of realizations; indeed, [W2, Theorems 4.7 and 4.8] should
be seen as sheaf theoretic phenomena: for any (Hodge theoretic or ℓ-adic)
19
sheaf NU which is pure of weight n, and such that i∗ j∗ NU is without weights
n and n + 1, intersection and interior cohomology with coefficients in NU coincide. Thus, the natural map from intersection cohomology to interior cohomology with coefficients in MU is bijective whenever MU ∈ CHM(U)F,∂w6=0,1 .
(d) For the ℓ-adic realization, a relative version of statement (c) holds, provided that morphisms in the image of the realization are strict with respect
to the weight filtration: the morphism d : X → D is still assumed to be
proper, but D may be different from the base scheme, and the latter need
not be a field. For a detailed study of the condition on strictness, we refer to
[B2, Sect. 2]; note that it is satisfied in the situation we are about to study
in Section 3.
(e) An analogue of (d) should hold for the Hodge theoretic realization.
3
A criterion on absence of weights in the
boundary
We keep the geometrical situation of the preceding section: X is a scheme,
and j : U ֒→ X an open immersion with complement i : Z ֒→ X. For a finite
stratification by nilregular locally closed sub-schemes Zϕ of Z, indexed by
ϕ ∈ Φ, recall the definition of the category DMBAb
,c,Φ (Z)F of Φ-constructible
motives of Abelian type over Z [W6, Def. 2.5 (b)]: it is the strict, full, dense,
F -linear triangulated sub-category of DMB ,c (Z)F generated by the images
under π∗ of the objects of DMTS (S(S))F , the category of S-constructible
Tate motives over S(S) [W6, Def. 2.3], where
π : S(S) −→ Z = Z(Φ)
runs through the morphisms of Abelian type [W6, Def. 2.5 (a)] with target
equal to Z = Z(Φ). In particular, π is a morphism of good stratifications
[W6, Def. 2.4].
Definition 3.1. An object M ∈ DMB ,c (Z)F is a motive of Abelian type
over Z if it belongs to the sub-category DMBAb
,c,Φ (Z)F , for a suitable finite
stratification Φ by nilregular locally closed sub-schemes of Z. In this situation, we say that Φ is adapted to M.
Let us now fix a generic point Spec k of the base B. For any scheme Y ,
denote by Rℓ,Y the (generic) ℓ-adic realization [W6, Sect. 3]. Its target is
the F -linear version Dcb (Yk )F of the bounded “derived category” Dcb (Yk ) [E,
Sect. 6] of constructible Qℓ -sheaves on the fibre Yk of Y over Spec k ֒→ B.
According to [CD3, Thm. 7.2.24], the Rℓ,• are compatible with the functors
f ∗ , f∗ , f! , f ! . Furthermore, they are symmetric monoidal; in particular,
Rℓ,Y (1Y ) = Qℓ,Yk ,
20
where Qℓ,Yk denotes the ℓ-adic structure sheaf on Yk .
The following is an immediate consequence of the main result from [W6].
Theorem 3.2. Assume that k is of characteristic zero. Let ℓ be a prime.
Let N ∈ DMBAb
,c,Φ (Z)F . Assume that the generic points of all strata Zϕ ,
ϕ ∈ Φ, lie over Spec k ֒→ B. For ϕ ∈ Φ, denote by iϕ the immersion of Zϕ
into Z.
(a) Let α ∈ Z. Then N lies in DMBAb
,c,Φ (Z)F,w≤α if and only if for all n ∈ Z,
and all ϕ ∈ Φ, the perverse cohomology sheaf
H n i∗ϕ Rℓ,Z (N)
is of weights ≤ n + α.
(b) Let β ∈ Z. Then N lies in DMBAb
,c,Φ (Z)F,w≥β if and only if for all n ∈ Z,
and all ϕ ∈ Φ, the perverse cohomology sheaf
H n i!ϕ Rℓ,Z (N)
is of weights ≥ n + β.
Remark 3.3. (a) The conditions on the generic points of the strata Zϕ
are empty when B is itself the spectrum of a field of characteristic zero.
(b) Recall [B1, Prop. 2.1.2 1] that any additive functor H from a triangulated
category C carrying a weight structure w, to an Abelian category A admits
a canonical weight filtration by sub-functors
. . . ⊂ Wn H ⊂ Wn+1 H ⊂ . . . ⊂ H .
For any m ∈ Z, one defines
Hm : C −→ A , M −→ H X[m] ;
according to the usual convention, the weight filtration of Hm (M) equals the
weight filtration of H(X[m]), i.e., it differs by décalage from the intrinsic
weight filtration of the covariant additive functor Hm .
If k is finitely generated over Q, then there is an instrinsic notion of
weights on those perverse sheaves on Yk , which are in the image of the cohomological realization [B2, Prop. 2.5.1 (II)].
In general, the weights of H ∗Rℓ,Y (M) are by definition those induced by
the weight filtration of the functor H ∗Rℓ,Y (these coincide with the above
when k is finitely generated over Q).
Proof of Theorem 3.2.
and only if for all ϕ ∈ Φ,
The motive N belongs to DMBAb
,c,Φ (Z)F,w≤α if
i∗ϕ N ∈ DMBAb
,c,Φ (Zϕ )F,w≤α .
According to [W6, Thm. 3.4 (b)], the latter condition is equivalent to
is of weights ≤ n + α ,
H n Rℓ,Zϕ i∗ϕ N
21
for all n ∈ Z. But thanks to compatibility of Rℓ,• with i∗ϕ , we have
Rℓ,Zϕ i∗ϕ N = i∗ϕ Rℓ,Z (N) .
This proves part (a) of our claim. Dualizing, we obtain the proof of part (b).
q.e.d.
Together with one of the main compatibility results from [W5], we obtain
the following.
Theorem 3.4. Assume that k is of characteristic zero. Let ℓ be a prime.
Let MU ∈ CHM(U)F , such that Rℓ,U (MU ) is concentrated in a single perverse degree, and such that i∗ j∗ MU ∈ DMB ,c (Z)F is of Abelian type. Let Φ a
stratification of Z adapted to i∗ j∗ MU . Assume that the generic points of all
Zϕ , ϕ ∈ Φ, lie over Spec k ֒→ B. For ϕ ∈ Φ, denote by iϕ the immersion of
Zϕ into Z.
(a) The motive MU belongs to CHM(U)F,∂w6=0,1 if and only if for all n ∈ Z,
and all ϕ ∈ Φ, the following hold: the perverse cohomology sheaf
H n i∗ϕ i∗ j!∗ Rℓ,U (MU )
is of weights ≤ n − 1, and
H n i!ϕ i! j!∗ Rℓ,U (MU )
is of weights ≥ n+ 1. In particular, the intermediate extension j!∗ MU is then
defined up to unique isomorphism, as a Chow motive over X.
(b) Let α ≤ 0 and β ≥ 1 two integers. The motive i∗ j∗ MU is without weights
α, α + 1, . . . , β if and only if for all n ∈ Z, and all ϕ ∈ Φ, the following hold:
the perverse cohomology sheaf
H n i∗ϕ i∗ j!∗ Rℓ,U (MU )
is of weights ≤ n + α − 1, and
H n i!ϕ i! j!∗ Rℓ,U (MU )
is of weights ≥ n + β.
(c) Let α ≤ 0 and β ≥ 1 two integers. Assume that for all n ∈ Z, and all
ϕ ∈ Φ, the following hold: the perverse cohomology sheaf
H n i∗ϕ i∗ j!∗ Rℓ,U (MU )
is of weights ≤ n + α − 1, and
H n i!ϕ i! j!∗ Rℓ,U (MU )
is of weights ≥ n + β. Then d∗ j! MU is without weights α − 1, α, . . . , −1, and
d∗ j∗ MU is without weights 1, 2, . . . , β, for any proper morphism d : X → D.
By slight abuse of notation, we write j!∗ Rℓ,U (MU ) for
j!∗ (Rℓ,U (MU )[s]) [−s] ,
22
if Rℓ,U (MU ) is concentrated in perverse degree s.
Proof of Theorem 3.4. Part (a) follows from part (b) (take α = 0 and
β = 1), and from Definitions 1.1 (a) and 1.4, while part (c) is implied by (b)
and Theorem 2.4 (a) and (b).
As for part (b), note that according to [W6, Thm. 2.10], the heart of
DMBAb
,c,Φ (Z)F is semi-primary. Therefore, Theorem 1.7 can be applied; the
motive i∗ j∗ MU is thus without weights α, α + 1, . . . , β if and only if
i∗ j!∗ MU ∈ C(Z)w≤α−1
and i! j!∗ MU ∈ C(Z)w≥β .
By Theorem 3.2, this is in turn equivalent to the following: for all n ∈ Z,
and all ϕ ∈ Φ,
H n i∗ϕ Rℓ,Z i∗ j!∗ MU
is of weights ≤ n + α − 1, and
H n i!ϕ Rℓ,Z i∗ j!∗ MU
is of weights ≥ n + β. Thanks to compatibility of Rℓ,• with i∗ and i! , we have
i∗ ◦ Rℓ,X = Rℓ,Z ◦ i∗
and Rℓ,Z ◦ i! = i! ◦ Rℓ,X .
The compatibility of Rℓ,• with j!∗ is the content of [W5, Thm. 7.2 (b)].
q.e.d.
Remark 3.5. (a) Given the full, triangulated sub-category DMBAb
,c,Φ (Z)F
of DMB ,c (Z)F , there is first a maximal choice of full, triangulated subcategory D(U) of DMB ,c (U)F , which glues with DMBAb
,c,Φ (Z)F , to give a
full, triangulated sub-category of DMB ,c (X)F , namely the full sub-category
of objects MU satisfying i∗ j∗ MU ∈ DMBAb
,c,Φ (Z)F (cmp. [W5, Prop 4.1]). Inside D(U), we then find the maximal choice of full, triangulated sub-category
C(U) of DMB ,c (U)F , which glues with DMBAb
,c,Φ (Z)F , to give a full, triangulated sub-category C(X) of DMB ,c (X)F , and which in addition inherits a
weight structure from the motivic weight structure on DMB ,c (U)F , namely
the full triangulated sub-category generated by objects MU of CHM(U)F
satisfying i∗ j∗ MU ∈ DMBAb
,c,Φ (Z)F . The theory from [W5, Sect. 2] and from
the present Section 1 can thus be applied to the triplet of categories C(U),
C(X) ⊂ DMB ,c (X)F , and C(Z) = DMBAb
,c,Φ (Z)F .
(b) In particular, if an object MU of CHM(U)F is such that i∗ j∗ MU ∈
DMBAb
,c,Φ (Z)F , i.e., MU belongs to C(U)w=0 , where C(U) is as in (a), then
(j!∗ MU exists, and) i∗ j!∗ MU and i! j!∗ MU belong to DMBAb
,c,Φ (Z)F . This fact
was implicitly used in the proof of Theorem 3.4.
(c) A priori, the application of [W5, Thm. 7.2] necessitates the validity of
[W5, Asp. 7.1]: the motive MU belongs to the triangulated sub-category
π∗ DMTSU (S(SU ))♮F,w=0 of CHM(U)F , for an extension of π : S(S) → Z to
X. While [W5, Asp. 7.1 (b), (c)] belong to the hypotheses of Theorem 3.4,
23
[W5, Asp. 7.1 (a)] is replaced by the condition that MU belong to the category C(U) from (a). The proof of [W5, Thm. 7.2 (b)] carries over to this
more general context without any modification.
Corollary 3.6. Assume that k is of characteristic zero. Let ℓ be a prime.
Let MU ∈ CHM(U)F , such that Rℓ,U (MU ) is concentrated in a single perverse degree s, and such that Rℓ,U (MU ) is auto-dual up to a shift and a twist:
Dℓ,U Rℓ,U (MU ) ∼
= Rℓ,U (MU )(s)[2s]
(Dℓ,U = ℓ-adic local duality on U). Assume in addition that i∗ j∗ MU is of
Abelian type. Let Φ a stratification of Z adapted to i∗ j∗ MU . Assume that the
generic points of all Zϕ , ϕ ∈ Φ, lie over Spec k ֒→ B. For ϕ ∈ Φ, denote by
iϕ the immersion of Zϕ into Z.
(a) The motive MU belongs to CHM(U)F,∂w6=0,1 if and only if for all n ∈ Z,
and all ϕ ∈ Φ, the following holds: the perverse cohomology sheaf
H n i∗ϕ i∗ j!∗ Rℓ,U (MU )
is of weights ≤ n − 1.
(b) Let β ≥ 1 be an integer. The following are equivalent.
(b1) The motive i∗ j∗ MU is without weights −β + 1, −β + 2, . . . , β.
(b2) For all n ∈ Z, and all ϕ ∈ Φ,
H n i∗ϕ i∗ j!∗ Rℓ,U (MU )
is of weights ≤ n − β.
(b3) For all n ∈ Z, and all ϕ ∈ Φ,
H n i!ϕ i! j!∗ Rℓ,U (MU )
is of weights ≥ n + β.
(c) Let β ≥ 1 be an integer, and assume that for all n ∈ Z, and all ϕ ∈ Φ,
H n i∗ϕ i∗ j!∗ Rℓ,U (MU )
is of weights ≤ n − β. Then d∗ j! MU is without weights −β, −β + 1, . . . , −1,
and d∗ j∗ MU is without weights 1, 2, . . . , β, for any proper morphism d : X →
D.
Proof.
Part (a) is a special case of part (b) (take β = 1, and use the
equivalence (b1) ⇔ (b2)). Similarly, part (c) follows from (b2) ⇔ (b3), and
from Theorem 3.4 (c) (with α = −β + 1).
As for part (b), observe that for all n ∈ Z, and all ϕ ∈ Φ,
H n i!ϕ i! j!∗ Rℓ,U (MU )
is dual to
H −n i∗ϕ i∗ j!∗ DU Rℓ,U (MU ) .
24
By assumption, weight w occurs in
H −n i∗ϕ i∗ j!∗ DU Rℓ,U (MU ) ∼
= H −n i∗ϕ i∗ j!∗ Rℓ,U (MU )(s)[2s]
if and only if weight 2s + w occurs in
H 2s−n i∗ϕ i∗ j!∗ Rℓ,U (MU ) .
Therefore, conditions (b2) and (b3) are equivalent to each other. According
to Theorem 3.4 (b) (with α = −β + 1), each of them is thus equivalent to
(b1).
q.e.d.
Remark 3.7. Applying the variant of the theory from Section 1 sketched
in Remark 1.8, we see that in Corollary 3.6 (b), conditions (b1)–(b3) are also
equivalent to
(b4) The motive i∗ j∗ MU is without weights −β + 1, −β + 2, . . . , 0.
(b5) The motive i∗ j∗ MU is without weights 1, 2, . . . , β.
Clearly condition (b1) implies both (b4) and (b5). We claim that (b4) implies (b2), and that (b5) implies (b3). Indeed, according to Remark 1.8 (b),
condition (b4) implies
i∗ j!∗ MU ∈ C(Z)w≤−β ,
while condition (b5) implies
i! j!∗ MU ∈ C(Z)w≥β .
Now apply Theorem 3.2 (a) and (b).
Together with the comparison result from Section 2, we get the following.
Theorem 3.8. Assume that B = Spec k for a field k of characteristic zero, that the structure morphism d : X → B is proper, and that
its restriction d ◦ j to U ⊂ X is smooth. Let π : C → U proper and
smooth, and e ∈ CHdC (C ×U C) ⊗Z F an idempotent. Let ℓ be a prime.
Assume that Rℓ,U (π∗ 1C )e is concentrated in a single perverse degree s, and
that Rℓ,U (π∗ 1C )e is auto-dual up to a shift and a twist:
Dℓ,U Rℓ,U (π∗ 1C )e ∼
= Rℓ,U (π∗ 1C )e (s)[2s] .
Assume in addition that the motive i∗ j∗ (π∗ 1C )e is of Abelian type. Let Φ
a stratification of Z adapted to i∗ j∗ (π∗ 1C )e . For ϕ ∈ Φ, denote by iϕ the
immersion of Zϕ into Z.
(a) If for all n ∈ Z, and all ϕ ∈ Φ, the perverse cohomology sheaf
H n i∗ϕ i∗ j!∗ Rℓ,U (π∗ 1C )e
is of weights ≤ n − 1, then ∂Mgm (C)e is without weights −1 and 0.
25
(b) If for all n ∈ Z, and all ϕ ∈ Φ, the perverse cohomology sheaf
H n i∗ϕ i∗ j!∗ Rℓ,U (π∗ 1C )e
is of weights ≤ n − 1, then there is a canonical isomorphism
∗
∼
→ Gr0 Mgm (C)e .
d∗ j!∗ (π∗ 1C )e −−
Proof.
Combine Corollaries 3.6 and 2.10.
q.e.d.
Remark 3.9. The condition on auto-duality of Rℓ,U (π∗ 1C )e is satisfied
if the cycle e and its transposition t e have identical images under Rℓ,C .
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