Holmstrom Voevodsky motives

Voevodsky motives

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Voevodsky motives

AG (Algebraic geometry)

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Voevodsky motives

Mixed

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Voevodsky motives

André, chapters 15-17.

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Voevodsky motives

Bondarko: DG realizations of motives… : We describe the Voevodsky category DM of motives in terms of Suslin complexes of smooth projective varieties. We give a description of any triangulated subcategory of DM. We describe ‘truncation’ functors tN for N > 0. t0 generalizes the weight complex of Soulé and Gillet; its target is the homotopy category of Chow motives; it calculates K0(DM). tN give a weight filtration and a ‘motivic descent spectral sequence’ for ‘standard’ (and more general) realizations; these include motivic cohomology (!).

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Brief notes from André

Chapter 15

15.1 Finite correspondences: details omitted.

15.2 A construction of Suslin-Voevodsky: For every $X \in L(k)$, a homomorphism

which is an iso after inverting $char(k)$.

15.3 The category $cL(k)$. Objects are those of $L(k)$, and morphisms are finite correspondences. It is an additive tensor category, and the functor from $L(k)$ is a tensor functor.

Example: The cat $AM(k)_{\mathbb{Z}}$ of finite correspondences between smooth $k$-varieties of dimension 0. Description of this cat in terms of Galois theory.

15.4 Suslin homology

See Suslin homology

Chapter 16: Voevodsky’s mixed motives

From the additive cat $cL(k)$ we can form the cat of bounded complexes, and the homotopy category. Conventions: differentials $\partial: C_n \to C_{n-1}$, and $(C_*[i])_n = C_{n-i}$ with differential multiplied by $(-1)^i$.

The cat of complexes and the homotopy cat are additive cats, and they inherit a tensor structure.

Def: A Mayer-Vietoris complex is a complex of the form $[U \cap V] \to [U] \oplus [V] \to [X]$ in degrees 2,1,0, with differentials $[j']-[i']$ and $[i]+[j]$, using the obvious notation. Call it $MV_{X,U,V}$.

More generally, consider a Nisnevich distinguished square, given by an open inclusion $i: U \to X$ and an etale morphism $f: V \to X$ which is an isomorphism on reduced subschemes away from $U$. Let $W$ be the fiber product; then the diagram analogous to the above is a complex in $cL(k)$, it is denoted $N_{X,U,f}$.

Def: The complex $H_X: [X \times \mathbf{A}^1] \to [X]$, in degrees 1,0, is called the homotopy complex associated to $X$.

Def: Cone.

The homotopy cat is a triangulated cat. Recall that a triangle is distinguished if it is isomorphic in the homotopy cat to a triangle formed from a cone of a morphism.

Recall that any distinguished triangle generate long exact sequences when given as input to a $Hom$ functor, in any argument.

Note compatibility between the tensor structure and the triangulated structure, for example, tensoring a d.t with something gives a d.t.

16.2 The triangulated cat $DM_{gm}^{eff}(k)$

Roughly, to define this cat we force homotopy complexes and M-V complexes to be zero, and then we take the pseudo-abelian envelope. Details as follows:

For a full subcat of a triangulated cat, notion of epaisse (saturated???). For such subcats, have triangulated quotient category. If the subcat is also a tensor ideal, then the quotient inherits a tensor structure.

Def: Let $T$ be the smallest saturated subcat of the homotopy cat which contains all M-V and homotopy complexes.

Recursive process for defining this cat. Start with the zero object and the M-V and homotopy complexes. Enlarge by adding all direct factors of cones formed from this cat. Repeat enlargement.

Prop: All Nisnevich type complexes belong to $T$.

We now form the cat $DM_{gm}^{eff}(k)$ from the homotopy cat by quotienting out by $T$ and taking the pseudo-abelian envelope. We have a canonical functor $M: L(k) \to DM_{gm}^{eff}(k)$, $X \mapsto M(X)$, sending $X$ to the image of $[X]$ placed in degree zero. Write $f_*$ for $M(f)$.

Balmer and Schlichting proved that a pseudo-abelian envelope of a triangulated category is also triangulated. Hence $DM_{gm}^{eff}(k)$ is triangulated. It also carries a tensor structure such that $M(X) \otimes M(Y) = M(X \times Y)$. The unit object is $\mathbf{1} = M(Spec(k)) = \mathbb{Z}(0) = \mathbb{Z}$.

Exercise: The category $H^b(AM(k)_{\mathbb{Z}})$ embeds fully faithfully into $DM_{gm}^{eff}(k)$.

Motives with coefficients: For any commutative ring $F$, can replace everywhere the group $c(X,Y)$ of finite correspondences by $c(X,Y) \otimes F$. Get the cat $DM_{gm}^{eff}(k)_F$, and for every morphism of rings $F \to K$, we get a triangulated tensor functor $DM_{gm}^{eff}(k)_F \to DM_{gm}^{eff}(k)_K$.

Reduced motive of a pointed variety: Let $x: Spec(k) \to X$ be a rational point of the $k$-variety $X$. It induces a decomposition $M(X) = \mathbf{1} \oplus \tilde{M}(X)$ in $DM_{gm}^{eff}(k)$.

16.3 Mayer-Vietoris triangles

Prop: For every $c \in c(X \times \mathbf{A}^1, Y)$, the fibres $c_x$ for $x \in \mathbf{A}^1$ induec the same morphism of motives $M(X) \to M(Y)$.

Example: The reduced motive of the projective line is independent of the rational “base” point.

Lemma: Let $W, V, U, X$ form a distinguished square. There exists a distinguished triangle in $DM_{gm}^{eff}(k)_F$ of the form $M(W) \to M(U) \oplus M(V) \to M(X) \to M(W)(1)$ in which the first and second morphisms are the expected difference and sum, respectively.

Proof: Form the cone triangle on the first morphism. Then the “obvious” map from the cone to $M(X)$ is an iso, because its cone can be identified with the Nisnevich complex $N_{X,U,f}$, which belongs to $T$.

Remarks: Long exact sequence from a distinguished square. Distinguished square of reduced motives, for any rational point of $W$.

Example: $\tilde{M}(\mathbf{P}^1) = \tilde{M}(\mathbf{G}_m)[1]$

Corollary: “An affine bundle induces an iso on motives”.

Corollary: The fibration $\mathbf{P}^n - \{ 0 \} \to \mathbf{P}^{n-1}$ induces as iso on motives.

Example: Jouanolou proved that for every quasi-projective $k$-variety $X$, vector bundle $E$ and a torsor $T \to X$ for this VB, such that $T$ is an affine scheme. The argument was generalized by Thomason for schemes admitting an ample family of line bundles, for example a smooth not necessarily quasi-projective variety. We obtain and isomorphism $M(T) \to M(X)$ in $DM_{gm}^{eff}(k)_F$, and conclude that the motive of a smooth $k$-variety is isomorphic to the motive of an affine smooth $k$-variety.

Chapter 17: Twists and motivic cohomology

In this chapter, we give the definition of twists, the category $DM_{gm}(k)$ obtained by formal inversion of the reduced motive of the projective line, as well as the elementary geometric construction of motivic cohomology.

17.1 Twists and $DM_{gm}(k)$

Following the homological point of view, one regards $\tilde{M}(\mathbf{P}^1)$ as a $h_2(\mathbf{P}^1)[2]$ rather than a $h^2(\mathbf{P}^1)[-2]$. Hence we refer to $\tilde{M}(\mathbf{P}^1)[-2] = \tilde{M}(\mathbf{G}_m)[-1]$ as the Tate motive in $DM_{gm}^{eff}(k)_F$ (and not the Lefschetz motive). We also denote it by $\mathbf{1}(1)$ or $F(1)$. Hence we have the canonical decomposition $M(\mathbf{P}^1) = \mathbf{1} \oplus \mathbf{1}(1)[2]$.

For all $r > 0$, we set $F(r) = \mathbf{1}(r) = \mathbf{1}(1)^{\otimes r}$.

Lemman: The permutation isomorphism $\mathbf{1}(r)[i] \oplus \mathbf{1}(s)[j] \cong \mathbf{1}(s)[j] \oplus \mathbf{1}(r)[i]$ equals, in some sense, $(-1)^{ij}$.

Exercise: Exhibit a canonical iso $M( \mathbf{A}^n \backslash \{0\} ) = \mathbf{1} \oplus \mathbf{1}(n)[2n-1]$, using the M-V exact sequence.

Construction of $DM_{gm}(k)_F$: Objects are couples $(M,m)$ where $M \in DM_{gm}^{eff}(k)$ and $m$ is an integer. Morphisms are given by the expected limit formula over Hom groups in $DM_{gm}^{eff}(k)_F$ with increasing twists $r$. This is an $F$-linear, pseudo-abelian, triangulated, rigid tensor cat.

Note: Sometimes, we use the term “Tate motive” to refer to (finite sums of) (shifted) powers of $\mathbf{1}(1)$.

17.2 Motivic cohomology

Let $X \in L(k), i \in \mathbb{Z}, r \in \mathbb{N}$.

Def: Let $H^i(X, F(r) ) = DM_{gm}^{eff}(k)_F(M(X), \mathbf{1}(r)[i])$. This is a bigraded $F$-module. In particular, for every $r$, we obtain a functor $H^*(-, F(r) ) : L(k)^{op} \to VecGr_F$.

We will see that motivic cohomology vanishes for $i > dim (X) +r$. Conjecturally, the same is true when $i 0$, at least with rational coefficients.

Lemma: Motivic cohomology is homotopy invariant.

Lemma: Any distinguished Nixnevich square in $L(k)$ induces, for any $i$, a long exact sequence

We have an obvious definition of exterior product. Taking $X=Y$ in this def, we get a notion of cup product by composing the exterior product by the morphism induced by the diagonal of $X$. Note that these products are “additive” in both degrees.

Lemma: The functor $\oplus_r H^*(X, F(r) )$ is compatible with the tensor structure. In particular, interversion of factors behaves like $(-1)^{ij}$. For any $X$, the cup-product gives motivic cohomology the structure of a bigraded commutative $F$-algebra.

17.3 The first Chern class of a line bundle, and the projective bundle formula

We define a natural homomorphism $c_1: Pic(X) \to H^2(X, F(1) )$ which is an iso for $F = \mathbb{Z}$. Both sides give isomorphisms when applied to an affine torsor, so it suffices to give a definition for affine $X$. In this case any line bundle on $X$ is generated by its global sections, so can expressed as the pullback of $O(1)$ on projective $n$-space. We use the previously encountered finite correspondence from $\mathbf{P}^n to \mathbf{P}^1$. This induces a morphisms of motives, hence a cohomology class $u$ in $H^2(\mathbf{P}^n, F(1))$, which can be pulled back to a class in the cohomology of $X$.

Using powers of $u$ used in the proof, one can construct an isomorphism $M(\mathbf{P}^n) \to \oplus_0^n \mathbf{1}(r)[2r]$.

Using similar methods, one also obtains the following.

Prop: For every vector bundle $E$ of rank $n$ on $X \in L(k)$, there is an (explicitly constructed) isomorphism $M(P(E)) \to \oplus_0^{n-1} M(X)(r)[2r]$.

Chapter 18: Fundamental properties of $DM_{gm}(k)$

We shall overview the main properties of $DM_{gm}(k)$, of `DM_{gm}^{eff}(k), and of motivic cohomology. The proofs require sheaf-theoretic tools, to be sketched in later chapters.

We omit treatment of motivic cohomology with finite coeffs, notably Steenrod operations.

In the chapter, $k$ is always a perfect field.

18.1 Blow-ups and Gysin triangle

Let $X \in L(k)$, and $Z \subset X$ a closed smooth subvariety purely of codimension $c$. Write $\tilde{X}$ for the blow-up of $X$ and $E$ for the exceptional divisor.

Thm: In $DM_{gm}^{eff}(k)$, there is a canonical distinguished triangle

and a canonical isomorphism $M(\tilde{X}) = M(X) \oplus_{r=1}^{c-1} M(Z)(r)[2r]$. There is also a canonical distinguished “Gysin” triangle

Corollary: If $char(k)=0$, then the triangulated cat $DM_{gm}^{eff}(k)$ is “generated” by direct factors of objects $M(X)$ with $X$ smooth projective. In every characteristic, this is true at least for $DM_{gm}^{eff}(k)_{\mathbb{Q}}$.

18.2 Simplifiability of twists

Thm: For every $M, N \in DM_{gm}^{eff}(k)$, the canonical homomorphism

is an isomorphism. A fortiori, the canonical functor $DM_{gm}^{eff}(k) \to DM_{gm}(k)$ is fully faithful.

18.3 Links to Chow motives

Thm: For every commutative ring $F$, there is a commutative square of categories and monoidal functors invovning $P(k)$, $L(k)$, $CHM(k)_F^{op}$ and $DM_{gm}(k)_F$. The bottom functor $R$ between the two cats of motives is fully faithful, sends $\mathbf{1}(-1)$ to $\mathbf{1}(1)[2]$, and sends effective motives to effectives.

18.4 Duality

Thm: If $char(k)=0$, then $DM_{gm}(k)$ is a rigid tensor cat: there exists an autoduality $\vee$ (a contavariant autofunctor) such that for any object $N$, the functor $\otimes N^{\vee}$ left adjoint to $\otimes N$, and the functor $N^{\vee} \otimes$ is right adjoint to $N \otimes$. In arbitrary characteristics, this is true at least for $DM_{gm}(k)_{\mathbb{Q}}$. If $X$ is smooth projective of dimension $d$, then $M(X)^{\vee} = M(X)(-d)[-2d]$, the duality pairing being induced by the diagonal embedding.

Remarks:

For $X \in L(k)$ purely of dimension $d$, one can define the motive with compact support $M^c(X)$ as $M(X)^{\vee}(d)[2d]$. In char zero, Voevodsky gives a direct definition, and then verifies this equality.

More generally, to any variety (not nec smooth!), Voevodsky associates objects $M(X)$ and $M^c(X)$ of $DM_{gm}(k)$, and proves, in char zero, “expected properties”. He defines motivic cohomology with compact support as

and also the motivic Borel-Moore homology

These are functorial (covariant, contravariant, resp.) with respect to proper morphisms between $k$-varieties, and also functorial in the other variance (but with a shift) wrt flat equidimensional morphisms.

We have the “usual” localization sequence: If $Z$ is closed in $X$ with open complement $U$, we have the long exact sequences

and

Furthermore, $H_i^{BM}(X, \mathbb{Z}(r) ) = H^{2d-i}(X, \mathbb{Z}(d-r) )$ if $X$ is smooth of dimension $d$, and $H^i_c(X, \mathbb{Z}(r) ) = H^i(X, \mathbb{Z}(r) )$ if $X$ is proper.

Remark: It is possible to reformulate the functor $R$ and have instead a covariant functor from Chow motives to Voevodsky motives, by applying a duality on one of the sides.

18.5 Comparison theorems

Theorem: For $X \in L(k)$, we have $H^S_i(X) = DM_{gm}(k)(\mathbb{Z}[i], M(X) )$.

Bloch introduced higher Chow groups, for smooth varieties. Definition of these.

Thm: For any smooth variety, $CH^r(X,i) \cong H^{2r-i}(X, \mathbb{Z}(r) )$

Thm: For motivic cohomology of the base field, comparison with Milnor K-theory.

Chapter 19: Complexes of motivic sheaves

In order to prove the theorems in the preceding chapter, one needs some more sophisticated machinery than what we covered in Chapters 15-17. What we do is embedding $DM_{gm}^{eff}(k)$ into a category of “complexes of motivic sheaves”, and then reinterpreting motivic cohomology in terms of this category. The basic idea is to replace $M(X)$ by the Suslin complex $\underline{C}^*(X)$, viewed as a complex of Nisnevich sheaves.

19.1 Presheaves with transfer and homotopy invariance

Def: A presheaf with transfers is an additive contravariant functor from $cL(k)$ to abelian groups. These form an abelian category.

Example: Every $X \in cL(k)$ represents a PST, denoted $\mathbb{Z}_{tr}(X)$ (sometimes $L(X)$). If $X$ is pointed, have decomposition.

Example: Every abelian group $k$-scheme. defines a PST.

Let $F$ be a PST. We define $\underline{C}_*(F)$ as the chain complex associated to the simplicial presheaf $(Y \in L(k), n) \mapsto F(Y \times \Delta^n)$, and also the Suslin complex $\underline{C}^*(F)$ by usual rule.

Def: Homotopy invariant PST.

Example: Let $F$ be a PST. Then $\underline{H}(\underline{C}^*(F) )$ is a homotopy-invariant presheaf (with transfers? I think).

Example: If a group scheme like above is a torus, an abelian or semiabelian variety, or a “reseau etale”, then it defines a homotopy invariant PST.

19.2 Nisnevich topology and transfers

Def: Nisnevich topology on $L(k)$. Note: Local rings are henselian $k$-algebras.

Def: The cat $Nis_{tr}(k)$ of Nisnevich sheaves with transfers.

The two examples in the very beginning of the chapter (representable PST and group schemes) are actually Nisnevich sheaves.

Yoneda implies $F(X) = Nis_{tr}(k)(\mathbb{Z}_{tr}(X), F)$ for any $X \in L(k)$ and any $F \in Nis_{tr}(k)$.

“The Nisnevich topology (like the etale topology) is adapted to multivalued morphisms”.

Lemma: For a Nisnevich covering $(f_j: Y_j \to X)$, with $Y = \coprod Y_j$, the complex

is exact in $Nis(k)$ (should this be $Nis_{tr}(k)$?).

Prop: Let $F$ be a PST. Then the sheafification $F_{Nis} \in Nis(k)$ is canonically equipped with transfers. The sheafification is left adjoint to the inclusion $Nis_{tr}(k) \to PST(k)$.

Corollary: $Nis_{tr}(k)$ is an abelian cat with enough injectives.

In other respects the Nisnevich topology is close to the Zariski topology. The Nisnevich cohomological dimension (just like the Zariski one) of $X$ is equal to $dim(X)$. Actually, under homotopy invariance, we have the following result.

Thm: Let $F$ be a homotopy invariant PST. Then the associated NST $F_{Nis}$ is also homotopy invariant. As a presheaf on $L(k)$, in coincides with $F_{Zar}$. Furthermore, the presheaves $H^i_{Nis}(X, F_{Nis})$ on $L(k)$ are canonically equipped with transfers, and they are homotopy invariant, and they conincide with $H^i_{Zar}(X, F_{Zar})$.

Proof uses that for a homotopy invariant Zariski sheaf with transfers, every restriction homomorphism is injective.

We consider the derived category $D^{-}(Nis_{tr}(k))$ (complexes bounded to the right). We define $DM^{-}(k)$ (alternative notation $DM^{eff}_{-}(k)$) to be the full subcat of this derived cat formed of objects with homotopy invariant cohomology sheaves. It is an additive, pseudoabelian category. Its objects are sometimes referred to as complexes of motivic sheaves, or motivic complexes. They are analogous, in some sense, to complexes of abelian groups in algebraic topology.

Corollary: For every complex $F^*$ in $DM^{-}(k)$, and every $X \in L(k)$, Zariski hypercohomology agrees with Nisnevich hypercohomology, and these groups vanish when $F^*$ is concentrated in degrees $i - dim(X)$.

Corollary: For every $F \in Nis_{tr}(k)$, the cohomology sheaves $H^i_{Nis}(\underline{C}^*(F))$ are in $DM^{-}(k)$.

More generally, $\underline{C}^*$ defines a functor $D^{-}(Nis_{tr}(k)) \to DM^{-}(k)$.

Prop: The functor $\underline{C}^*$ is left adjoint to the embedding in the other direction, and gives an equivalence between $DM_{-}(k)$ and the localisation of $D^{-}(Nis_{tr}(k))$ at the saturated (epaisse) subcat generated by complexes $F^*$ such that $\underline{C}^*(F^*)$ is acyclic.

Proposition: For every $F^* \in D^{-}(Nis_{tr}(k))$ and every $X \in L(k)$, we have \mathbf{H}^i_{Nis}(X, F^) = DM^{-}(k)(\underline{C}^(X), F^[i])`.

19.3 The embedding theorem

The composition of $\mathbb{Z}_{tr}: H^b(cL(k)) \to D^{-}(Nis_{tr}(k))$ with $\underline{C}^*$ factors through $DM^{eff}_{gm}(k)$, and the induced functor $\iota: DM^{eff}_{gm}(k) \to DM^{-}(k)$ is fully faithful. Furthermore $\iota(M(X))$ is the Suslin complex $\underline{C}^*(X)$.

Corollary: Let $X,Y \in L(k)$, and $i$ a natural number. We have

Remarks: The cats $Nis_{tr}(k)$ and $D^{-}(Nis_{tr}(k))$ admit a natural tensor structure. There is a tensor structure on $DM^{-}(k)$ which is compatible with $\underline{C}^*$ but not with the inclusion into $D^{-}(Nis_{tr}(k))$. With this structure, the functor $\iota$ is a tensor functor. The functor $\iota$ does not extend to the cat $DM_{gm}(k)$ of noneffective motives. However, there exists an indirect method for inverting $\iota(\mathbf{1}(1))$ in $DM^{-}(k)$, using spectra. Déglise showed that the cat obtained by this method is equivalent to the cat of Rost’s cycle modules.

19.4 New description of motivic cohomology

Def of the motivic complexes $\underline{\mathbb{Z}}(r)$. First, def of $\tilde{\mathbb{Z}}_{tr}(X_1 \wedge \ldots \wedge X_k)$ as a quotient (but also a direct factor!) of $\mathbb{Z}_{tr}(X_1 \times \ldots \times X_k)$ in $Nis_{tr}(k)$. Then we define

viewed as an object of $DM^{-}(k)$ concentrated in degrees $\leq r$.

Remark: There is a canonical decomposition of $\mathbb{Z}_{tr}(\mathbf{G}_m^n)$ into a direct sum of $\tilde{\mathbb{Z}}_{tr}$ terms (over various index subsets of $1, \ldots r$). This can be used to prove that $\underline{\mathbb{Z}}(r) \cong \underline{\mathbb{Z}}(1)^{\otimes r}$.

Lemma: We have $\iota(\mathbb{Z}(r)) = \underline{\mathbb{Z}}(r)$.

Theorem: For every $X \in L(k)$, we have $H^i(X, \mathbb{Z}(r)) = \mathbf{H}^i_{Zar}(X, \underline{\mathbb{Z}}(r) )$, and this group is zero if $i > r + dim(X)$.

Proof: Clear from previous stuff.

Theorem: $\underline{\mathbb{Z}}(1) = \mathbf{G}_m[-1]$. In other words, $\iota(\tilde{M}(\mathbf{G}_m)) = \mathbf{G}_m$.

Corollary: $H^1(X, \mathbb{Z}(1)) = \mathcal{O}_X^*$ and $H^2(X, \mathbb{Z}(1)) = Pic(X)$.

Remark: The embedding theorem together with some stuff from the section on the first Chern class gives a canonical isomorphism $\underline{\mathbb{Z}}(r) = \underline{C}^*(\mathbb{Z}_{tr}(\mathbf{P}^r) / \mathbb{Z}_{tr}(\mathbf{P}^{r-1}))[-2r]$.

Remark: Let $z_{equi}(X,0)$ be the Nisnevich sheaf which to $Y \in L(k)$ assigns the groups of cycles on $Y \times X$ which are quasi-finite over $Y$. Friedlander-Suslin proved that

Moreover, identifying $\Delta^r$ with an open of $\mathbf{P}^r \ \mathbf{P}^{r-1}$, one obtains a morphism $\mathbb{Z}_{tr}(\mathbf{P}^r) / \mathbb{Z}_{tr}(\mathbf{P}^{r-1}) \to z_{equi}(\Delta^r,0)[-2r]$ in $Nis_{tr}(k)$, and Voevodsky shows that after applying $\underline{C}^*$ one gets a quasi-iso. Hence the link between motivic cohomology and higher Chow groups.

Chapter 20: 1-motives and mixed Tate motives

In this chapter we briefly cover two important classes of examples of mixed motives.

20.1 One-motives

Deligne defines a 1-motive over $k$ as the data consisting of (1) A lattice $\Lambda$ equipped with a Galois action, (2) A torus $T$, an abelian variety $A$, and an extension $G$ of $A$ by $T$, (3) A Galois equivariant morphism $\Lambda \to G(\bar{k})$.

On can consider a $-motive as a complex of abelian sheaves for the fppf topology, using the number (3) above, placed in degrees$[-1,0]`.

The 1-motives form an additive cat, denoted $1-Mot(k)$. Tensoring it by the rationals gives an abelian $\mathbb{Q}$-linear cat. Objects in this are called 1-isomotives, or 1-motives up to isogeny. This cat contains the cat $AM(k)_{\mathbb{Q}}$ of Artin motives, identified with motives $\Lambda \to G$ with $G=0$.

Since each of the two components of a 1-motive defines a Nisnevich sheaf with transfers, we have a canonical functor $1-Mot(k) \to DM^{-}(k)$. After tensoring with the rationals and passing to the associated simple complex, one gets a functor $D^b(1-Mot(k)_{\mathbb{Q}}) \to DM^{-}_{\mathbb{Q}}$.

Prop: This functor is fully faithful, and its image can be identified with the full saturated subcategory of $DM_{gm}^{eff}(k)_{\mathbb{Q}}$ generated by motives $MM(X)$ with $X$ smooth and one-dimensional over $k$.

Proof: See Orgogozo (2004).

There is also a direct link (without tensoring with the rationals) between the Suslin complex and the (homological) 1-motive of $X$. See Lichtenbaum: Suslin homology and Deligne 1-motives.

If $k = \mathbb{C}$, then $1-Mot(k)$ is canonically equivalent to a certain cat of mixed Hodge structures. Ref to Ramachandran and Barbieri-Viale, Rosenschon, Saito.

20.2 Mixed Tate motives

Let $D$ be a triangulated cat. Notion of “extension of $M$ by $N$” (obvious thing). Let $A$ be a full abelian subcat of $D$. Can ask the following: Consider the full subcat of $D$ of successive extensions of objects in $A$. Is is also abelian? A sufficient condition for this to be true, is that for all $M, N \in A$ and all $i 0$, $D(A, B[i]) = 0$.

A mixed Tate motive over $k$ with rational coeffs is an iterated extension of objects of the form $\mathbf{1}(r)$, $r \in \mathbb{Z}$, in the triangulated $\mathbb{Q}$-linear cat $DM_{gm}(k)_{\mathbb{Q}}$. Notation: $MTM(k)_{\mathbb{Q}}$ for this full subcat.

The condition above, for the full abelian subcat formed of finite sums $\oplus \mathbf{1}(r_n)$ in $DM_{gm}(k)_{\mathbb{Q}}$, turns out to be the Beilinson-Soulé vanishing conjecture: For every $i0$ and every $r \geq 0$, we have $H^i(Spec(k), \mathbb{Q}(r) ) = 0$.

By Borel, this holds for a number field, so in this case the cat of mixed Tate motives is abelian, and actually even Tannakian over $\mathbb{Q}$.

Details on weight filtration and fiber functor on MTM, omitted here.

Still for a number field, the $Ext^i$ vanish for $i \geq 2$.

Proposition: (For a number field) The full (triangulated) subcat $DTM(k)_{\mathbb{Q}}$ of $DM_{gm}(k)_{\mathbb{Q}}$ formed of iterated extensions of objects $\mathbf{1}(r)[i]$ is canonically isomorphic to $D^b(MTM(k)_{\mathbb{Q}})$.

Remark: The fundamental triangles in $DM_{gm}$ allows us to exhibit various $k$-varieties such that $M(X) \in DTM(k)_{\mathbb{Q}}$, for example projective space, “varietes de drapeaux”, and the moduli space of genus zero curves with $n$ marked points, and also its compactification.

Remark: There are other approaches to mixed Tate motives which do not rely on Beilinson-Soulé. See Kriz-May and Bloch-Kriz.

20.3 Kummer motives

Description of the intersection of the two classes of examples.

Chapter 21: Towards the heart of $DM_{gm}(k)$

We look at the problem of passing from a triangulated cat of mixed motives to an abelian cat containing the cat of numerical motives. Also links to Chow motives.

We assume that $k$ is perfect.

21.1 t-structure and heart

Recall $NM(k)_{\mathbb{Q}}$, supposedly the universal Weil cohomology. The numerical motives should be the semisimple objects of $MM(k)_{\mathbb{Q}}$. There should also be a functor $Var(k)^{op} \to MM(k)_{\mathbb{Q}}$ playing the role of the universal mixed Weil cohomology.

In order to complete the yoga for $MM(k)_{\mathbb{Q}}$, there should be: (1) An increasing filtration by weight of $MM(k)_{\mathbb{Q}}$ extending the weight grading of $NM(k)_{\mathbb{Q}}$ (which exists under the Kunneth standard conjecture). The associated graded of any object should be semisimple, hence in $NM(k)_{\mathbb{Q}}$. (2) A tensor structure on $MM(k)_{\mathbb{Q}}$ extending the one on ${}^{\bullet}NM(k)_{\mathbb{Q}}$. The tensor functor $Gr_W$ should be a faithful and exact left quasi-inverse of the inclusion ${}^{\bullet}NM(k)_{\mathbb{Q}} \to MM(k)_{\mathbb{Q}}$. (3) As the universal mixed Weil cohomology, the functor $h$ should satisfy Kunneth, homotopy invariance, and Mayer-Vietoris. There should also be functors $h_c$ and $h^{BM}$ of opposite variance, defined only for proper morphisms, playing the role of cohomology with compact supports and Borel-Moore homology; these should verify the usual localization property.

One reason that we have not been successful in constructing $MM(k)_{\mathbb{Q}}$ from $NM(k)_{\mathbb{Q}}$ might be a question of signs. Recall that we could not lift the modified commutativity constraint on $NM(k)_{\mathbb{Q}}$ to Chow motives. For mixed motives the situation seems to be the opposite.

Observing that the usual cohomology theories come from a functor $R \Gamma$ with values in a triangulated category (in practice, a derived cat) and that the Chow correspondences act on the level of $R \Gamma$, Deligne suggested that one should start by constructing the derived cat of mixed motives, equipped with a functor from $Var(k)^{op}$. This was done by Levine, Hanamura, and Vooevodsky with his $DM_{gm}(k)_{\mathbb{Q}}$. Idea: recover $MM(k)_{\mathbb{Q}}$ as the heart of the triangulated cat, given a suitable “t-structure”.

Note: “Strictement pleine” means: Every object isomorphic to an object in the subcat is actually in the subcat.

Def: Let $D$ be a triangulated cat. A t-structure on $D$ is the data of two (strictement pleines) subcats $D^{\leq 0}$ and $D^{\geq 0}$ such that:

• For every $A \in D^{\leq 0}$ and every $B \in D^{\geq 0}$, we have $D(A[1],B) = 0$,
• We have $D^{\leq 0}[1] \subset D^{\leq 0}$ and $D^{\geq 0} \subset D^{\geq 0}[1]$.
• For every $E \in D$, there exists a distinguished triangle $A \to E \to B[-1] \to A[1]$ such that $A \in D^{\leq 0}$ and $B \in D^{\geq 0}$.

The inclusion of $D^{\leq 0}$ (and $D^{\geq 0}$, resp.) into $D$ then admits a right adjoint $\tau_{\leq 0}$ (and a left adjoint $\tau_{\geq 0}$, resp.) and in the distinguished triangle above we have $A = \tau_{\leq 0}$ and $B = \tau_{\geq}(E[1])$ (up to unique isomorphism).

The heart of the t-structure is the full subcategory $A := D^{\leq 0} \cap D^{\geq 0}$ of $D$.

The principal result of Beilinson-Bernstein-Deligne is the following:

Lemma: The heart $A$ is an abelian cat. The functors ${}^{\tau} H^i: D \to A, A \mapsto \tau_{\geq 0} \tau_{\leq 0}(A[i])$ are cohomological functors. If $\cap_{j \in \mathbf{N}} D^{\leq 0}[j] = \cap_{j \in \mathbf{N}} D^{\geq 0}[-j] = \{ 0 \}$, then the system of these functors is a conservative system, and an object $A \in D$ lies in $D^{\leq 0}$ ($D^{\geq 0}$, resp.) iff ${}^{\tau} H^i(A) = 0$ for all $i>0$ ($i0$, resp).

Example: The derived cat $D^b(A)$ (or $D^{+}(A)$, etc) of an abelian cat. Then $D^{\leq 0}$ is formed from the complexes which are acyclic in degrees $>0$. The heart is $A$.

The wished-for “motivic” t-structure on $DM_{gm}(k)_{\mathbb{Q}}$ would give rise to a commutative square with corners $CHM(k)_{\mathbb{Q}}, DM_{gm}(k)_{\mathbb{Q}}, NM(k)_{\mathbb{Q}}, MM(k)_{\mathbb{Q}}$, where the functor on the right is $\sum {}^{\tau} H^i$.

Remark: The first two conditions in the definition of a t-structure, applied to $A = \mathbf{1}$ and $B= \mathbf{1}(r)$, immediately implies the Beilinson-Soulé vanishing conjecture for $k$.

Remark: A weight filtration would also follow, I think? Don’t really understand this. “Every object in $MM(k)_{\mathbb{Q}}$ would be artinian.

A good t-structure should be compatible with the tensor structures. Details on what this means. Even more optimistically, one could hope that there should be a canonical equivalence between triangulated rigid tensor cats $D^b(MM(k)_{\mathbb{Q}}) \cong DM_{gm}(k)_{\mathbb{Q}}$.

Status: For $k$ a number field, a t-structure with the desired properties has been constructed on $DTM(k)_{\mathbb{Q}}$, with heart $MTM(k)_{\mathbb{Q}}$.

Remark: Let $k$ be a finite field. Then, because “Frob scinderait la filtration pare les poids de $MM(k)_{\mathbb{Q}}$”, the cat $MM(k)_{\mathbb{Q}}$ would be semisimple and coincide with ${}^{\bullet} NM(k)_{\mathbb{Q}}$. The diagram above would just say that $CHM(k)_{\mathbb{Q}} = NM(k)_{\mathbb{Q}}$. One also expects a “trivial” description of motivic cohomology of a smooth projective variety over $k$: it should be zero everywhere except the classical Chow groups.

21.2 Motives of smooth affine varieties, and the weak Lefschetz “theorem” for motivic cohomology

Here is a suggestion for the motivic t-structure (ref Voev letter to Beilinson, and Beilinson: Remarks on n-motives and correspondences at the generic point). $D^{\geq 0}$ should be formed of objects $B$ such that for affine smooth $k$-variety $U$ of dimension $d$, and for every $i > d$, $DM_{gm}(k)_{\mathbb{Q}} (B, M^{\vee}(U)[i] ) = 0$, and $D^{\leq 0}$ should be formed of objects $A$ such that for every $B \in D^{\geq 0}$, $DM_{gm}(k)(A[1], B) = 0$. The idea is that such an $U$ should satisfy ${}^{\tau} H^i(M^{\vee}(U) )= 0$ for $i>d$.

Applying this to $B = \mathbf{1}(r-d) \in D^{\geq 0}$, writing $M^c(U) = M^{\vee}(U)(d)[2d]$, one would obtain that the motivic cohomology with compact supports $H^j_c(U, \mathbb{Q}(r) )$ vanishes in all degrees $j d$.

Now let $X$ be smooth projective, and let $Y$ be the inclusion of a smooth hyperplane section. The complement $U = X \backslash Y$ is then affine smooth. Taking into account that $H = H_c$ for smooth projective varieties, and using the long exact sequence for $H_c$ with rational coeffs, one obtains an analogue of the weak Lefschetz thm: $H^i(X, \mathbb{Q}(r) ) \to H^i(Y, \mathbb{Q}(r) )$ is an isomorphism for $i d-1$, and injective for $i = d-1$.

For $i=2$ and $r=1$, one recovers a well-known thm of Lefschetz on the Picard group of a smooth hyperplane section (true even without tensoring with the rationals).

Remark: By a dual argument, one would show that $H^i(U, \mathbb{Q}(r) )$ vanishes for $i > d$. From the Gysin triangle one then obtains that the Gysin morphism $H^{i-2}(Y, \mathbb{Q}(r-1) ) \to H^i(X, \mathbb{Q}(r) )$ is an iso for $i > d$, and surjective for $i=d$.

21.3 Mixed motives and the BBM conjectures

Assume hom = num. A good t-structure would imply the BBM conjecture and the B-S vanishing conjecture. Conversely, these two conjectures would imply a good t-structure. Details, omitted here.

Refs: Beilinson: Height pairings…, Jannsen in Motives vol, ICM talk, and Equivalence relations.

21.4 Nori’s category

Details omitted.

Chapter 22: Mixed realisations and regulators

Recall the enriched realisations of pure motives (Hodge, Tate, de Rham-Betti).

We will sketch the generalisation to the mixed case. Ref Huber, and Levine’s heavy book. These realizations are triangulated tensor functors from $DM_{gm}(k)$ with values in certain cats of complexes (up to qis) bounded to the left. From these one deduces certain homomorphisms, called regulators, from motivic cohomology to various “absolute” cohomologies.

We assume $char(k) = 0$ throughout this chapter.

22.1 de Rham-Betti, Hodge, and Tate realizations

Principle: We start with a Tannakian cat $A$ over a field $K$ of char zero, equipped with Tate twists $\otimes K(r)_A$, and a functor $C_A: L(k)^{op} \to C^{\geq 0}(A)$ satisfying the Kunneth formula (up to qis, see Huber). Under some technical hyps (homotopy, Mayer-V, descent for Galois coverings and proper morphisms), Huber proves that $C_A$ extends to a triangulated tensor functor $DM_{gm}(k)_K^{op} \to D^{+}(A)$ sending $\mathbf{1}(r)$ to $K(-r)$. Composing with the duality of the LHS, on gets a covariant functor $R_A$, sending $\mathbf{1}(r)$ to $K(r)$.

Simple examples:

• If $k \subset \mathbb{C}$, take $A = Vec_{\mathbb{Q}}$, the functor sending $X$ to the cochain complex attached to $X(\mathbb{C})$. Get Betti realization.
• Taking $A = Vec_k$, the functor $X \mapsto \Gamma(X, God(\Omega^*_{X/k}) )$. Get de Rham realization.

Examples of enriched realizations:

• Period (de Rham-Betti) realizations
• For $k \subset \mathbb{C}$: Real Hodge realization $DM_{gm}(k)_{\mathbb{R}} \to D^{+}(MHS_{\mathbb{R}})$ and other finer variants of this (e.g. with rational coeffs, with a real Frob)
• Tate realization
• (Syntomic: See Besser: Syntomic regulators and p-adic integration I: rigid syntomic regulators)

Also other variants.

Conjecture: For any of these cases, $R_A$ actually takes values in $D^b(A)$, and is a conservative functor.

Note that a mixed realization allows us to recover the pure realization by composing $R_A$ on both sides with the obvious things.

22.2 Regulators

The calculation of $R_A$ on morphisms involves in particular (and can be reduced to?) the case of motivic cohomology $H^i(X, K(r) )$. Composing $R_A$ with the functor $R^{+}A(K, -): D^{+}(A) \to D^{+}(Vec_K)$, we obtain, for every $X \in L(k)$ (or $Var(k)$), and every integer $r$, objects

and the cohomology groups $H^i_A(X, r)$ of this object are sometimes called absolute cohomology groups. There is a spectral sequence

We consider the Hodge and the Tate cases.

Hodge: The spectral sequence puts the absolute Hodge cohomology groups in an exact sequence (write subscript $\mathbb{R}$ rather than $MHS_{\mathbb{R}}$)

Note that for any real MHS $V$, we can compute $Ext^i_{\mathbb{R}}({\mathbb{R}}, V)$ through the representation $W_0 V \oplus F^0(W_0 V)_{\mathbb{C}} \to (W_0 V)_{\mathbb{C}}$ of $R^{+}MHS_{\mathbb{R}}({\mathbb{R}}, - )$ given by the difference of the two inclusions.

Tate: For $Rep_{\ell} Gal(\bar{k} / k)$, the absolute cohomology is given by Jannsen’s continuous $\ell$-adic cohomology. Here we get an exact sequence

On the level of cohomology, we get regulators $H^i(X, \mathbb{R}(r) ) \to H^i_{MHS-\mathbb{R}}(X, \mathbb{R}(r) )$ and $H^i(X, \mathbb{Q}_{\ell}(r)) \to H^i_{cont}(X, \mathbb{Q}_{\ell}(r) )$ induced by the above realizations.

Example: Explicit description of regulators in the case $X = Spec(k)$, $r=1$.

22.3 Expected properties of the realizations of $MM(k)$

In this section, we assume the formalism of mixed motives, i.e. a good t-structure on $DM_{gm}(k)_{\mathbb{Q}}$.

Conjecture: Every mixed realization $R_A: DM_{gm}(k)_{K} \to D^{+}(A)$ respects the t-structure.

Actually, this conjecture together with conservativity determines the t-structure on the first cat. Furthermore, $R_A$ would induce an exact conservative faithful tensor functor $MM(k)_K \to A$, extending the corresponding realization on numerical motives.

Also, to every mixed realization (simple, with coeffs in a field $K$), there should be an associated $K$-group, the absolute mixed motivic Galois group (an extension of the pure motivic Galois group ($K$-group) by a pro-unipotent group.

The fullness conjectures (Hodge, Tate) should also extend to the mixed case.

We have seen in 21.3 how the filtration on $CH^r(X)_{\mathbb{Q}}$ should be the filtration induced from the spectral sequence

with $p+q=2r$. Applying the $\ell$-adic realization, this is sent to the “Hochschild-Serre type” spectral sequence

connecting geometric and continuous etale cohomology.

If the $\ell$-adic regulator $CH^r(X)_{\mathbb{Q}} \to H^{2r}_{cont}(X, {\mathbb{Q}}_{\ell}(r))$ is injective, one could define the BBM filtration, via this $\ell$-adic route, and maybe prove the BBM conjecture. A very optimistic conjecture states that the injectivity holds whenever $k$ is finitely generated over its prime field. It would follow that BBM filtration always is zero after the level equal to the Kronecker dimension of $k$. In particular, over a number field, the filtration would only have two levels. In particular, $AJ_X$ would be injective for $X$ defined over $k$ and $k$-rational zero cycles.

From the above conjectures, including the conservativity of the $\ell$-adic realization, Beilinson derived a “curious” finiteness statement for a Chow group of a generic point times itself.

Remark: On the transcendental part of the Chow motive of a surface.

22.4 Values of L-functions, periods, regulators

One can define the L-function of a mixed motive over a number field $k$ by the same cohomological formalism as in the pure case. Conjecturally one gets a Dirichlet series with rational coeffs indep of $\ell$, with meromorphic continuation.

We consider here the case of $k = {\mathbb{Q}}$ and principal values at integer points.

Because of bad reduction a motive and its semi-simplification may have different L-functions. However, this will not happen provided that the weight filtration on $H_{\ell}(M)$ splits, as a representation of the inertia group at $p$, for every pair $(p, \ell)$ of distinct prime numbers. Following Scholl, we call such a motive a “mixed motive over $\mathbb{Z}$”. Note that every object in $NM({\mathbb{Q}})_{\mathbb{Q}}$ satisfies this condition. Write $MM({\mathbb{Z}})_{\mathbb{Q}}$ for the full subcat.

Remark: The only Kummer motive in this cat is $\mathbf{1} \oplus \mathbf{1}(1)$.

Remark: There exists a sketch of a “finer” theory of motives over $\mathbb{Z}$, or in some other ring of integers, for which $Ext^2(\mathbf{1}, \mathbf{1}(1) )$ is not zero, but equal to the first Arakelov Chow group of $Spec \mathbb{Z}$. See Deninger, Nart, Scholl, and Jannsen ICM.

Let $r$ be an integer. The canonical Betti-De Rham comparison isomorphism gives rise to a homomorphism

and we say that $r$ is a critical value for $L(M,s)$ if this homomorphism is bijective. In this case, write $c^{+}(M(r))$ for the determinant of this homomorphism wrt fixed rational bases.

Conjecture (Deligne conjecture, gen by Scholl to mixed case): Let $M$ be a mixed motive over $\mathbb{Z}$. If $r$ is a critical value, then $L(M,r) / c^{+}(M(r)) \in \mathbb{Q}$.

Beilinson conjecture, reformulated by Scholl: Let $M$ be a mixed motive over $\mathbb{Z}$. Then

Furthermore, if $Ext^1_{MM(\mathbb{Z})}(M(r), \mathbf{1}(1) ) = MM(\mathbb{Z}) (M(r), \mathbf{1}(1) ) = Ext^1_{MM(\mathbb{Z})}(\mathbf{1}, M(r) ) = MM(\mathbb{Z}) (\mathbf{1}, M(r) ) = 0$, then $r$ is a critical value for $L(M,s)$, and $L(M,r) \neq 0$. End of conjecture.

Description of a trick of Scholl, which allows us to recover the principal value of an L-function of an $M \in MM(\mathbb{Z})$ at any integer $r$, from the above two conjectures.

It remains to understand $c^{+}(M(r))$. If one of the “crans extremes” of this filtration is nonzero, one sees a real regulator. If these two “crans” is simultaneously nonzero, (typically this happens when $M$ is pure of weight $2r-1$), one also must take into account a height pairing between one “cran” and the dual of the other.

References: Deligne, 2 x Beilinson, Scholl, Nekovar in Motives, and finally Kings (2003) for the Bloch-Kato conj.

Example: The Beilinson-Deligne conjecture is valid for $M = \mathbf{1}$ and all integers $r$. Details omitted.

PART III: PERIODS

Will not take notes right now. H^i_c(X, \mathbb{Z}(r) )

DM_{gm}^{eff}(k)

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Voevodsky motives

See also Mixed motives, Motivic cohomology

Use Friedlander’s Bourbaki article.

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Voevodsky motives

Huber: Realization of Voevodsky’s motives (with a Corrigendum)

Spitzweck: Some constructions for Voevoedsky’s triangulated category of motives

In the letter to Beilinson, Voevodsky formulates axioms for a homology theory. He considers $\partial \Delta^n$ as an (n-1)-dim sphere, write also S for the 1-dim sphere in this sense. Let Sch/k be the cat of separated schemes of finite type over a base k. Then a homological theory is a functor from Sch/k together with a family of natural isos $H(X \times S) \to H(X) \oplus H(X)[1]$. This functor should satisfy some conditions: Morally, homotopy invariance, MV exact triangle, an exact triangle for blowups, and transfer for flat finite morphisms. Get a 2-cat of homological theories over $k$. Examples: Algebraic K-th with rational coeffs, l-adic homology, Hodge homology ass to a complex embedding. Thm: There is an initial object $DM_k^{ft}$ in this cat, which we call the triang cat of eff mixed motives over k. Notion of reduced homological theory, and reduced motive of a scheme. Any motive in the above sense is of the form $\tilde{M}(X)[n]$, where we may assume $X$ affine and $n \leq 0$. Tate object and comparison with K-theory. Bigger cat $DM_k$ which contains the previous as a full triang subcat, but admits a more explicit description rather than just the universal property. Can also be viewed as the closure of the previous, wrt direct sums and inductive limits. Need the h-topology, in particular coverings including surjective blowups, finite surjetive maps, etale coverings. Various filtrations on $DM_k$ (homotopy canonical, geometrical, motivic canonical, weight). The weight filtr should be related to pure numerical motives.

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Voevodsky motives

A fundamental idea in Voevodsky’s thesis and in Homology of schemes I, is the homological cat of a site with interval. This might not be so much emphasized for example in MVW, but it seems very natural, and gives some intuition for the def of DM.

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Voevodsky motives

http://ncatlab.org/nlab/show/Voevodsky+motive

Voevodsky: Homology of schemes I. Has a really nice introduction, describing some intuition for the construction of $DM(S)$, and also about the notion of universal cohomology. See section 4 for construction of DM. Possibly this is improved in later writings and/or by Deglise and Cisinski.

For triangulated cats of motives over simplicial schemes, see Voevoedsky: Motives over simplicial schemes. Any simplicial scheme $X$ defines a complex of presheaves with transfers, and hence we can define motivic cohomology of $X$ as $Hom(M(X), A(q)[p])$, taken in the cat $DM(k)$ or $DM_{-}^{eff}(k)$ I think. The main goal of the paper is to define a tensor triang cat $DM_{-}^{eff}(X)$ such that the motivic cohomology of $X$ can be expressed as Hom from the unit object to $A(q)[p]$ in this cat. It seems like we always work with simplicial schemes over a perfect field.

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Voevodsky motives

arXiv:1102.0579 Some remarks on the integral Hodge realization of Voevodsky’s motives from arXiv Front: math.AG by Vadim Vologodsky 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 for motives with rational coefficients.

nLab page on Voevodsky motives