SOME REMARKS ON THE INTEGRAL HODGE REALIZATION
OF VOEVODSKY’S MOTIVES
VADIM VOLOGODSKY
Abstract. We construct a functor from the triangulated category of Voevodsky’s motives to the derived category of mixed Hodge structures enriched with
integral weight filtration and prove that our Hodge realization functor commutes
with the Albanese functor. This extends the previous results of Barbieri-Viale,
Kahn and the author ([BK], [Vol]) for motives with rational coefficients.
1. Introduction
eff
(C; Z) be the triangulated category of Voevodsky’s motives over C with
Let DMgm
integral coefficients. We have the realization functor ([Hu1], [Hu2], [DG])
(1.1)
eff
RZHodge : DMgm
(C; Z) → Db (M HS Z )
to the bounded derived category of polarizable mixed Z-Hodge structures. Recall
that, a mixed Z-Hodge structure consists of a finitely generated abelian group VZ
together with a weight filtration on the rational vector space VQ = VZ ⊗ Q and a
Hodge filtration on the C-vector space VC satisfying certain compatibility conditions.
On the other hand, it is shown by Gillet and Soulé ([GS]) that the integral homology
of every complex algebraic variety carries a canonical integral weight filtration. The
goal of this paper is to construct a realization functor from the triangulated category
of Voevodsky’s motives to a certain derived category of mixed Hodge structures with
the weight filtration defined integrally.
An enriched mixed Z-Hodge structure V = (W· ⊂ VZ , F · ⊂ VC ) consists of a
finitely generated abelian group VZ together with a weight filtration on VZ and a
Hodge filtration on VC such that the triple (VZ , W· ⊗Q, F · ) constitutes a mixed Hodge
structure in the sense of Deligne. An enriched Hodge structure is called effective if
0
F 1 = 0, W0 = VZ and GrW
VZ is torsion free; V is called polarizable if (VZ , W· ⊗Q, F · )
is polarizable. We denote the category of enriched effective polarizable mixed Hodge
Z,en
Z,en
structures by M HSeff
. The category M HSeff
is not abelian, but it carries a
natural exact structure. We postulate that a sequence
0→V0 →V1 →V2 →0
is exact if, for every i, the sequence of pure integral Hodge structures
i
i
i
0 → GrW
V 0 → GrW
V 1 → GrW
V2 →0
Z,en
splits. Let Db (M HSeff
) be the corresponding bounded derived category ([N]). We
Hodge
lift the functor RZ
to a triangulated functor
(1.2)
Z,en
eff
(C; Z) → Db (M HSeff
).
R̂ZHodge : DMgm
2000 Mathematics Subject Classification. Primary 14F42, 14C25; Secondary 14C22, 14F05.
Key words and phrases. Algebraic cycles, Voevodsky motives, Hodge theory.
1
2
VADIM VOLOGODSKY
Composing R̂ZHodge with the functor
Gr n
Z,en
W
Db (M HSeff
) −→
K b (HSnZ )
to the homotopy category of complexes of pure weight n Hodge structures, we recover
the weight complexes constructed earlier by Bondarko ([Bon]).
The construction of the functor R̂ZHodge is very simple: the hard part of the job is
already done in ([BV], [Bon], [Vol]). In fact, we show that if B is an abelian category
of homological dimension ≤ 1 then every DG enriched functor from the category of
Voevodsky’s motives over a field k of characteristic 0 to the derived category of B
eff
canonically lifts to a functor from DMgm
(k; Z) to a certain filtered category Db (F B).
In the second part of the paper we apply the above construction to study the
motivic Albanese functor
eff
LAlbZ : DMgm
(k; Z) → Db (M1 (k))
from the category of Voevodsky’s motives over a subfield k ⊂ C to the derived category of Deligne’s 1-motives, introduced by Barbieri-Viale and Kahn ([BK]). Let
M HS1Z be the category of torsion free polarizable mixed Z-Hodge structures of type
{(0, 0), (0, −1), (−1, 0), (−1, −1)}. This category has two natural exact structures induced by the embeddings of M HS1Z to the abelian category M HS Z and to the exact
Z,en
category M HSeff
. Abusing notation, denote by Db (M HS1Z ) and Db (M HS1Z,en )
the corresponding derived categories. We show that the functor Db (M HS1Z,en ) ,→
Z,en
Db (M HSeff
) has a left adjoint and denote by
Z,en
LAlbZ : Db (M HSeff
) → Db (M HS1Z )
its composition with the projection Db (M HS1Z,en ) → Db (M HS1Z ). We remark that
Z 1
).
the Hodge Albanese functor LAlbZ does not factor through the category Db (M HSeff
Our main result, Theorem 1, is the commutativity of the following diagram
(1.3)
eff
DMgm
(k; Z)
 Hodge
yR̂Z
LAlbZ
Db (M
1 (k))
 Hodge
yTZ
Z,en
Db (M HSeff
)
LAlbZ
Db (M HS1Z ),
−→
−→
where TZHodge is the Hodge realization functor defined by Deligne ([D3]). The commutativity of (1.3) can be viewed as a derived version of Deligne’s conjecture on
1-motives.
Notation. For DG categories C, C 0 we denote by T (C, C 0 ) the category of DG
quasi-functors ([Dri], §16.1). If F : C → C 0 is a DG quasi-functor we will write
Ho F : Ho C → Ho C 0 for the corresponding functor between the homotopy categories.
We denote by →
C (resp. ←
C
−
−) the DG ind-completion (resp. the DG proj-completion) of
C ([Dri], §14 or [BV] §1.6). Abusing notation, we use the same letter F to denote the
quasi-functors →
C →→
C 0, ←
C
→←
C−0 induced by F ∈ T (C, C 0 ). We write F ! : →
C0 → →
C
−
−
−
−
−
∗
0
and F : ←
C− → ←
C
− for the right and left adjoint functors respectfully.
Acknowledgments. I would like to thank Leonid Positselski for helpful conversations related to the subject of this paper.
1We denote by M HS Z a full subcategory of M HS Z formed by mixed Hodge structures with
eff
F 1 = 0.
SOME REMARKS ON THE INTEGRAL HODGE REALIZATION OF VOEVODSKY’S MOTIVES 3
2. Enriched Hodge structures.
Let A be a commutative ring, B an A-linear abelian category, and let F B be the
category of finitely filtered objects of B. That is, objects of F B are pairs (V, W· V ),
where V ∈ B and W· V is a increasing filtration on V , such that, for sufficiently large
i, we have Wi V = V , W−i V = 0. Morphisms between (V, W· V ) and (V 0 , W· V 0 ) form
a subgroup of HomB (V, V 0 ) consisting of those f : V → V 0 that carry Wi V to Wi V 0 ,
for every i. We introduce an exact structure on F B postulating that a sequence
0 → (V 0 , W· V 0 ) → (V 1 , W· V 1 ) → (V 2 , W· V 2 ) → 0
is exact if, for every i, the sequence
i
i
i
0 → GrW
V 0 → GrW
V 1 → GrW
V2 →0
b
splits. We denote by Ddg
(F B) the corresponding derived DG category and by Db (F B)
the triangulated derived category ([N]). For an object V of B, let V {i} be the object
of F B with Wj V {i} equal to V if j ≥ −i and to 0 otherwise. The following result is
due to Positselski ([P], Example in §8).
Lemma 2.1. If n ≤ j − i the forgetful functor Db (F B) → Db (B) induces an isomorphism
HomDb (F B) (V {i}, V 0 {j}[n]) ' HomDb (B) (V, V 0 [n]).
If n > j − i then HomDb (F B) (V {i}, V 0 {j}[n]) is trivial.
Let k be a field of characteristic 0. The following construction is inspired by
eff
the work of Bondarko ([Bon]). Let DMeff
gm,et (k; A) (resp. DMgm (k; A)) be the DG
category of étale (resp. Nisnevich) geometric Voevodsky’s motives ([Vol], §2; [BV]),
and let
b
R : DMeff
gm,et (k; A) → Ddg (B)
be a DG quasi-functor ([Dri]). Assuming that the homological dimension of B is less
or equal than 1 we shall construct a commutative diagram
(2.1)
DMeff
gm
(k; A)
α
y
−→
R̂
b
Ddg
(F B)


yΦ
R
b
Ddg
(B),
DMeff
gm,et (k; A) −→
where α is the “change of topology” functor, Φ is the forgetful functor, and R̂ is a
DG quasi-functor defined below.
Let P ⊂ DMeff
gm (k; A) be a full DG subcategory formed by those motives that are
eff
(k; A) to direct summands of motives of smooth projective variisomorphic in DMgm
eties. Recall (see, e.g. [BV] §6.7.3) that for every X, Y ∈ P the complex Hom(X, Y )
has trivial cohomology in positive degrees. It follows that the restriction of R to P
factors as follows
R̂
P
b
b
P −→
Ddg
(B)trunc → Ddg
(B).
b
b
Here Ddg
(B)trunc is a DG category having the same objects as Ddg
(B) and morphisms
defined by the formula
0
HomDdg
(V, V 0 ) = τ≤0 HomDdg
b (B)
b (B) (V, V ),
trunc
where τ≤i C · denotes the canonical truncation of a complex C · in degree i.
4
VADIM VOLOGODSKY
b
Lemma 2.2. The pretriangulated completion ([BV], §1) of the category Ddg
(B)trunc
b
is homotopy equivalent to Ddg
(F B).
b
Proof. Let Q be the full DG subcategory of Ddg
(F B) formed by objects V {i}[i], with
0
V ∈ B, i ∈ Z . Since, by Lemma 2.1, the complex HomDdg
b (F B) (V {i}[i], V {j}[j]) has
trivial cohomology groups in positive degrees the restriction of the forgetful functor
to Q factors as
b
b
Q → Ddg
(B)trunc → Ddg
(B).
b
Since Ddg
(F B) is strongly generated by Q we get a quasi-functor
(2.2)
b
b
Ddg
(F B) → (Ddg
(B)trunc )pretr ,
which is a homotopically fully faithful by the Lemma 2.1. As B has homological
b
dimension ≤ 1, every complex in Ddg
(B)trunc is homotopy equivalent to a finite direct
sum of objects in image of Q. It follows that (2.2) is a homotopy equivalence.
Using the above lemma we construct the functor R̂ in the diagram (2.1) as foleff
lows. According to ([Bon], §6 or [BV], §1.5.6) the category DMgm
(k; A) is strongly
generated by P. Thus, the quasi-functor functor R̂P extends uniquely to a functor
b
pretr
b
DMeff
' Ddg
(F B).
gm (k; A) → (Ddg (B)trunc )
The composition of the above quasi-functors is our R̂.
Observe that, for the motive M (X) of a smooth projective variety, we have an
isomorphism in the homotopy category of complexes over B
(2.3)
i
GrW
(R̂(M (X)) ' H i R(M (X))[−i].
Let us also remark that for every smooth projective varieties X and Y the forgetful
functor Φ yields a quasi-isomorphism
∼
(2.4) HomDdg
b (F B) (R̂(M (X)), R̂(M (Y ))) −→ τ≤0 HomD b (B) (R(M (X)), R(M (Y ))).
dg
Throughout the rest of the paper A denotes a subring of Q. We apply the above
construction to the Hodge realization DG quasi-functor
(2.5)
Hodge
b
A
RA
: DMeff
gm,et (C; A) → Ddg (M HS ).
constructed in ([Vol], §2). By Lemma 1.1 from [Bei] the category M HS A of polarizable mixed A-Hodge structures has cohomological dimension 1. Consider the full
subcategory M HS A,en of F (M HS A ) whose objects (called enriched mixed Hodge
structures) are polarizable mixed A-Hodge structures V equipped with a filtration
i
W· V such that GrW
V is pure of weight i (in particular, the filtration W· V ⊗A Q coincides with the weight filtration on V ⊗A Q). The category M HS A,en inherits an exact
structure from F (M HS A ). An enriched mixed Hodge structure (V, W· ) ∈ M HS A,en
0
is called effective if F 1 = 0, W0 V = V and GrW
V is torsion free. Denote by
A,en
A,en
M HSeff ⊂ M HS
the full subcategory of effective enriched mixed Hodge structures.
A,en
Lemma 2.3. The functor Db (M HSeff
) → Db (F (M HS A )) is a fully faithful embedding.
SOME REMARKS ON THE INTEGRAL HODGE REALIZATION OF VOEVODSKY’S MOTIVES 5
Proof. As M HS A has homological dimension 1, Lemma 2.1 implies that, for every
objects V, V 0 ∈ F (M HS A ) and every i > 1, we have
HomF (M HS A ) (V, V 0 [i]) = 0.
A,en
Since the subcategory M HSeff
→ F (M HS A ) is closed under extensions the lemma
follows from ([P], Proposition A.7).
Hodge
The lemma together with formula (2.3) imply that the functor R̂A
from the
A,en
b
diagram (2.1) factors through Ddg (M HSeff ) yielding the enriched Hodge realization
functor
A,en
b
DMeff
gm (C; A) → Ddg (M HSeff ).
Hodge
Notation: If k is a subfield of C, we will write, by abuse of notation, R̂A
for the
composition
f∗
Hodge
A,en
f
ef f
b
R̂A
: DMef
gm (k; A) −→ DMgm (C; A) → Ddg (M HSef f )
(2.6)
where f ∗ is the base change functor.
Hodge
Hodge
Remark 2.4. Our category M HS Q,en is just M HS Q and R̂Q
= RQ
. On the
other hand, the integral realization functor R̂ZHodge does not even factor through the
category of étale motives. Indeed, the functor R̂ZHodge takes the motive Z/m(n) =
m
m
cone(Z(n) −→ Z(n)) to the complex Z(n) −→ Z(n) of enriched Hodge structures of
Hodge
weight −2n and, in particular, R̂Z
(Z/m(n)) is not isomorphic to R̂ZHodge (Z/m(n0 ))
Z,en
in Db (M HSeff
), unless n = n0 .
Hodge
Remark 2.5. Composing Ho(R̂A
) with the functor
·
M
A,en GrW
Db (M HSeff
) −→
K b (HSiA ),
i≤0
b
(HSiA )
where K
is the homotopy category of complexes of pure weight i Hodge structures, we recover the Hodge realization of the Gillet-Soulé weight complexes ([GS]).
Let
(2.7)
eff
γ∗M : d≤1 DMeff
gm (k; A) ,→ DMgm (k; A)
be the smallest full homotopy Karoubian DG subcategory of DMeff
gm (k; A) that contains motives of smooth varieties of dimension 0 and 1.
0
ef f
f
Lemma 2.6. For every M ∈ DMef
gm (C; A) and M ∈ d≤1 DMgm (C; A) the functor
Hodge
R̂A
induces a quasi-isomorphism
∼
HomDMef
(M, M 0 ) −→ HomDb
f
gm (C;A)
A,en
dg (M HSef f )
Hodge
Hodge
(R̂A
(M ), R̂A
(M 0 )).
Proof. By formula (2.4) it suffices to prove that for every smooth projective variety
X and a smooth projective curve C the canonical morphism
Hodge
Hodge
HomDMeff
(M (X), M (C)) → τ≤0 HomDdg
(M (X)), RA
(M (C))
b (M HS A ) (R
A
gm (C;A)
is a quasi-isomorphism. In turn, the above is equivalent to showing that the morphism
(2.8)
Hodge
HomDMeff
(M (X × C), A(1)) → τ≤2 HomDdg
(M (X × C)), A(1))
b (M HS A ) (R
A
gm (C;A)
6
VADIM VOLOGODSKY
is a quasi-isomorphism. In fact, the complex at the left-hand side of (2.8) computes
∗−1
∗
the groups HZar
(X × C, OX×C
) ⊗ A that are mapped isomorphically via (2.8) to
∗
Deligne’s cohomology H (X × C, A(1)D ) for ∗ ≤ 2.
A,en
Remark 2.7. Unless A = Q, Lemma 2.6 does not hold with M HSef
f replaced by
A
M HSef f .
Let M HS1A be the full subcategory of the category M HS A that consists of torsionfree objects of type {(0, 0), (0, −1), (−1, 0), (−1, −1)}. The category M HS1A has two
natural exact structures induced by the embedding into the abelian category M HS A
A,en
b
b
and into the exact category M HSeff
. Denote by Ddg
(M HS1A ), Ddg
(M HS1A,en ) the
corresponding derived DG categories.
Lemma 2.8. The functor
A,en
b
b
Ddg
(M HS1A,en ) → Ddg
(M HSef
f )
A,en
induced by the embedding M HS1A ,→ M HSef
f is homotopically fully faithful and has
a left adjoint DG quasi-functor
(2.9)
A,en
A,en
b
b
Ddg
(M HSef
).
f ) → Ddg (M HS1
A,en
Proof. The first claim follows from the fact that the subcategory M HS1A ⊂ M HSeff
0
is closed under extensions and from the vanishing of HomDb (M HS A,en ) (V, V [i]), for
eff
A,en
i > 1 and V, V 0 ∈ M HSeff
. For the existence of a left adjoint functor it suffices to
A,en
check that, there exists a set S of generators of Db (M HSeff
) such that, for every
V ∈ S the functor
ΦV : Db (M HS1A,en ) → M od(A),
b
ΦV (V 0 ) = HomDb (M HS A,en ) (V, V 0 )
eff
(M HS1A,en ).
is representable by an object of D
We take for S the set of pure enriched
Hodge structures. If V has weight < −2, then ΦV is 0. Let V be a pure polarizable
Hodge structure of weight −2. There exists a unique decomposition V ⊗A Q into the
direct sum of a Hodge-Tate substructure and a substructure P ⊂ V ⊗A Q that has no
Hodge-Tate subquotients. Then ΦV is representable by Im(V → V ⊗A Q/P ). Next,
every pure weight −1 Hodge structure V is the direct sum of a torsion free Hodge
structure Vf and Hodge structures of the form A/mA, m ∈ Z (with W−1 (A/mA) =
A/mA, W−2 (A/mA) = 0). The functor ΦVf is representable by Vf and the functor
m
ΦA/mA is representable by the complex A(1) −→ A(1). Finally, every effective weight
0 Hodge structure is already in M HS1A .
b
b
A
Remark 2.9. Unless A = Q, the embedding Ddg
(M HS1A ) → Ddg
(M HSef
f ) does
not have a left adjoint functor.
We define the Hodge Albanese functor
A,en
b
b
LAlbA : Ddg
(M HSeff
) → Ddg
(M HS1A ).
b
to be the composition of the adjoint functor (2.9) and the projection Ddg
(M HS1A,en ) →
b
Ddg
(M HS1A ).
SOME REMARKS ON THE INTEGRAL HODGE REALIZATION OF VOEVODSKY’S MOTIVES 7
3. The Albanese functor
Let k be a field of finite étale homological dimension embedded into C and let
M1 (k; A) := M1 (k) ⊗ A be the category of 1-motives with coefficients in A. We
consider the exact structure on M1 (k; A) induced by the Hodge realization functor
([D3])
TAHodge : M1 (k; A) → M HS A
b
and denote by Ddg
(M1 (k; A)) the corresponding derived DG category. In [BK],
Barbieri-Viale and Kahn defined the motivic Albanese functor
eff
b
LAlbA : DMeff
gm (k; A) → d≤1 DMgm,et (k; A) ' Ddg (M1 (k; A))
as follows. The composition
D
α
eff
eff
DMeff
gm (k; A) −→ DMgm (k; A) −→ DMgm,et (k; A)
−−−−−−−−−→
−−−−−−−−−−−→
of the duality, D(M ) = Hom(M, A(1)), and the “change of topology” functor factors
through the embedding
M
eff
γ∗et
: d≤1 DMeff
gm,et (k; A) ,→ DMet (k; A).
The functor LAlbA is defined to be the composition
D
et
eff
eff
DMeff
gm (k; A) −→ d≤1 DMgm,et (k; A) −→ d≤1 DMgm,et (k; A).
Here is an equivalent more convenient for us construction of LAlbA .
Lemma 3.1. The composition
γ∗
α
ef f
M
f
ef f
DMef
gm (k; A) −→ d≤1 DMgm (k; A) −→ d≤1 DMgm,et (k; A)
←−−−−−−−−−−−−−
←−−−−−−−−−−−−
of the functor left adjoint to the embedding (2.7) with the “change of topology” functor
is isomorphic to the motivic Albanese functor LAlbA .
Proof. Consider the diagram
(3.1)
DMeff
gm (k; A)
←−−−−−−−−−
D
∗
γM
/ d≤1 DMeff
gm (k; A)
←−−−−−−−−−−−−
D
α
/ d≤1 DMeff
gm,et (k; A)
←−−−−−−−−−−−−−
Det
!
γM
eff
α / d
DMeff
≤1 DMgm,et (k; A)
gm (k; A)
−−−−−−−−−→
−−−6 −−−−−−−−−−→
OOO
mmm
OOO
m
m
m
O
m
O
OOO
mm !
α
'
mmm γM,et
eff
DMgm,et (k; A)
−−−−−−−−−−−→
/ d≤1 DMeff
gm (k; A)
−−−−−−−−−−−−→
The rectangular part of the diagram is commutative. To prove the lemma we will
!
!
show that the lower triangle is commutative: the morphism α◦γM
→ γM,et
◦α induced
∼
M
!
M
!
by γ∗et ◦ α ◦ γM −→ α ◦ γ∗ ◦ γM → α is an isomorphism. Let X be an object of
0
M
!
!
0
DMeff
gm (k; A) and let X = cone(γ∗ ◦ γM (X) → X). We have γM (X ) = 0. Thus, it
−−−−−−−−−→
!
suffices to check that γM,et
◦ α(X 0 ) = 0. In fact, if X 0 is represented by a complex F
of Nisnevich sheaves with transfers with homotopy invariant cohomology, then α(X 0 )
!
is represented by its sheafification F et in étale topology. As γM
(X 0 ) = 0, for every
8
VADIM VOLOGODSKY
curve C, the restriction of F to the small Nisnevich site of C is 0. Thus the restriction
!
of F et to the small étale site of C is 0 i.e., γM,et
◦ α(X 0 ) = 0.
We will need a similar description of the Hodge Albanese functor. Denote by
(3.2)
A,en
A,en
b
b
γ∗H : d≤1 Ddg
(M HSeff
) ,→ Ddg
(M HSeff
),
A,en
b
the smallest full homotopy Karoubian DG subcategory of Ddg
(M HSeff
) that conHodge
tains Hodge structures R̂A
(M (C)) of smooth varieties C of dimension 0 and 1.
A,en
b
Equivalently, d≤1 Ddg
(M HSeff
) is the derived category of the subcategory
A,en
d≤1 M HSeff
⊂ M HS1A,en
−1
formed by those V ∈ M HS1A,en that the Hodge structure GrW
V is a direct summand
of the first homology of some smooth proper curve.
Lemma 3.2. The composition
γ∗
A,en
H
b
A
b
b
A
b
A
Ddg
(M HSef
f ) −→ d≤1 Ddg (M HSef f ) −→ d≤1 Ddg (M HSef f ) ' Ddg (M HS1 )
←−−−−−−−−−−−−− ←−−−−−−−−−
←−−−−−−−−−−−−−−
is isomorphic to the Hodge Albanese functor LAlbA .
Proof. The lemma is implied by the following concrete statement: for every pure
polarizable mixed Hodge structure V of weight −1 there exists a complex V 0 ∈
A,en
A,en
Db (d≤1 M HSeff
) and a morphism V → V 0 in Db (M HSeff
) that becomes an isoA
morphism in Db (M HSeff
). Let X be an abelian variety such that V = H1 (X) ⊗ A.
We can find a direct summand J in the Jacobian of a smooth projective curve and
a surjective morphism X → J with finite kernel K. Pick an embedding of K into
an algebraic torus T and let T 0 = T /K. Consider the quotient G of X ⊕ T by K
diagonally embedded into X ⊕ T . The algebraic group G is an extension of J by
T . The exact sequence 0 → X → G → T 0 → 0 yields a morphism of complexes of
1-motives
X −→ (G → T 0 )
which is an isomorphism in Db (M1 (C; A)). Applying TAHodge we obtain the result.
By adjunction we get a morphism of quasi-functors
(3.3)
(3.4)
Hodge
Hodge
∗
∗
γH
◦ R̂A
→ R̂A
◦ γM
DMeff
gm,et (k; A)

 Hodge
yR̂A
γ∗
M
−→
γ∗
A,en
H
b
Ddg
(M HSeff
) −→
d≤1 DMeff
gm (k; A)
←−−−−−
−−−−−−−
 Hodge
yR̂A
A,en
b
d≤1 Ddg
(M HSeff
).
←−−−−−−−−−−−−−−
b
Composing this morphism with the projection to Ddg
(M HS1A ), we get
←−−−−−−−−−
(3.5)
Hodge
Hodge
LAlbA ◦ R̂A
→ RA
◦ LAlbA .
Theorem 1. The morphism (3.5) is an isomorphism of quasi-functors.
SOME REMARKS ON THE INTEGRAL HODGE REALIZATION OF VOEVODSKY’S MOTIVES 9
Proof. We will prove that morphism (3.3) is an isomorphism. Since the motivic
Albanese functor commutes with an arbitrary base field extension we may assume
that k = C. By Lemma 2.6 the functor
Hodge
A,en
b
R̂A
: d≤1 DMeff
gm (k; A) → d≤1 Ddg (M HSeff )
is a homotopy equivalence. Thus, the right vertical arrow in (3.4) is also a homotopy
equivalence and the adjoint functor
Hodge ∗
A,en
b
(R̂A
) : d≤1 Ddg
(M HSeff
) → d≤1 DMeff
gm (k; A)
←−−−−−−−−−−−−−−
←−−−−−−−−−−−−
is its inverse. Hence, is suffices to check the commutativity of the diagram
(3.6)
∗
γM
/ d≤1 DMeff
gm (k; A)
←−6 −−−−−−−−−−−
m
mmm
Hodge
R̂A
mHmm Hodge ∗
m
m
)
mmm (γ∗ ◦R̂A
A,en
b
Ddg (M HSeff )
DMeff
gm,et (k; A)
which is equivalent to Lemma 2.6.
Using the commutative diagram
(3.7)
b
/ Ddg
(M1 (k; A))
d≤1 DMeff
gm,et (k; A)
RRR
RRR
Hodge
RRR
TA
Hodge RRR
RA
R(
b
(M HS1A )
Ddg
'
from ([Vol], Theorem 3) we obtain our main result (1.3) stated in the introduction.
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[Bei]
[BV]
[Bon]
[D3]
[DG]
[Dri]
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[Hu2]
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