Holmstrom Mixed motives

Mixed motives

The theory of mixed motives is still to a large extent conjectural, although constructions have been suggested by Nori, Voevodsky and others.

Some of the properties one hopes for: Let $k$ be a field. There should exist a rigid tensor category $MM_k$ containing “Tate objects” $\mathbb{Z}(n)$ and a functor $h: Sm_k^{op} \to D^b(MM_k)$ such that one gets a universal Bloch-Ogus theory by setting

for $X \in Sm_k$.

Some further expected properties: The full subcat of semi-simple objects should be equivalent to the category of homological motives, after tensoring with $\mathbb{Q}$. Structure of neutral Tannakian category (after tensoring with $\mathbb{Q}$ or $\mathbb{Q}_{\ell}$). Natural weight filtration on each object. Certain isomorphism with K-groups, arising from an Atiyah-Hirzebruch type spectral seq.

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Mixed motives

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Mixed motives

AG (Algebraic geometry), AAG (Arithmetic algebraic geometry)

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Mixed motives

Mixed

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Mixed motives

The book by André

There are some Springer Lecture notes by Huber. Are these online somewhere?

Jannsen: Mixed motives and algebraic K-theory. LNM 1400.

Beilinson: Height pairings between algebraic cycles. In LNM 1289.

Jannsen: Motivic sheaves and filtrations on Chow groups (Motives vol)

LNM 1604: Mixed motives and their realizations in derived categories

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Mixed motives

For definition of mixed Tate motives over $Spec(\mathbb{Z})$, see Goncharov and Deligne: Groupes fondamenteaux motiviques de Tate mixte. On arxiv.

Madhav Nori talk: On mixed Tate motives and hyperplane arrangements. An Abelian category of MTM arising from hyperplane arrangements will be constructed. It will be compared with the construction of Beilinson, Goncharov, Schectman and Varchenko. The Ext groups in this category will be compared with the K-groups of the base field.

Fakhruddin: Notes of Nori’s lectures on mixed motives (2000)

Ania Otwinowska talk: Motivic sheaves on curves. Joint with Beilinson and Vologodsky

For mixed motivic sheaves, see Beilinson’s original article, and perhaps Arapura with references.

Hanamura: Mixed motives and algebraic cycles (1995)

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Notes from André

Chapter 14: Why mixted motives?

The passing from smooth proj var’s to general var’s is followed by a new phenomenon of nontrivial extensions. The notion of weight no longer appears as a grading, but in the form of a filtration, in general “non-scindee”. The theory of mixed motives should provide us with a setting in which these Ext’s give us various geometric and arithmetic invariants.

We shetch now the main ideas.

14.1 The weight filtration

We have seen earlier the notion of weight, in Hodge theory it is the total grading ass to the Hodge bigrading. In crystalling or $\ell$-adic theory, it is related to “absolute values” (eigenvalues?) of Frob.

The classical cohomology of a smooth projective var is equipped with a weight grading. The part of weight $i$ is just $H^i$. Conjecturally, this comes from a weight grading of the motive.

For a general variety, all we have is an increasing filtration $W_*$. Example of smooth projective curve minus two points, showing that nontrivial extensions appear (I don’t get this example).

Deligne associated to any $k$-variety of dim $\leq 1$, a 1-motive, denoted $h_1(X)$. Brief description of this, and mention of realizations. After tensoring with the rationals, or modifying the def a little, one gets an abelian cat. Description of graded pieces, and $Ext^1(\mathbf{1}, h_1(X)) = J(X)(k) \otimes \mathbb{Q}$.

14.2 From pure to mixed motives

We have seen in the pure case how we study a contravariant functor $H^*$ from $P(k)$ to, say, and abelian tensor cat, equipped with an action of Chow correspondences, respecting sum and product of two var’s.

We want to generalize this to the cat $L(k)$ of smooth $k$-var’s (or maybe even all $k$-var’s). Trying to do this, one realizes that one needs a few more axioms. This includes: Homotopy invariance, Mayer-Vietoris l.e.s. for a covering by two open sets. If one tries to include ALL (i.e. also non-smooth?) varieties, one also needs a version with proper supports $H^*_c$ (contravariant for proper morphisms, dual to $H^*$ in the smooth case), and a shift of focus to homological rather than cohomological formalism (a variant of Borel-Moore), with corresponding dual axioms.

Suslin and Voevodsky placed the focus on another structure: functorality of $H^*$ and $H_*$ for “multivalued morphisms”, i.e. finite correspondences. Inspiration drawn from the Dold-Thom theorem (1956) which says that $H_n(X, \mathbb{Z}) = \pi_n(\mathbb{Z}[X])$. Here $\mathbb{Z}[X]$ means the group completion (I guess) of the commutative monoid $\coprod_{n \geq 0} S^n(X)$ (the latter equipped with the law of concatenation). Since the symmetric product still makes sense in the cat of quasi-projective varieties, one can define this monoid, and using algebraic standard simplices, one can make sense of the above homotopy groups. This defines Suslin homology, with functoriality for finite correspondences. Suslin homology played a “role moteur” in Voevodsky’s work.

Returning to the initial aim, we define a mixed Weil cohomology to be a functor $H^*: L(k) \to A$ (where $A$ is a certain abelian tensor cat), equipped with functoriality for finite correspondences between smooth $k$-varieties, satisfying homotopy invariance, Kunneth formula, and the Mayer-Vietoris l.e.s. Every mixed Weil cohomology defines a Weil cohomology by restriction to smooth projectives, and every classical Weil cohomology extends to a mixed Weil cohomology. (Note: This notion of mixed Weil cohomology probably differs from Déglise-Cisinski).

One can ask about the universal target for such a theory. This hypothetical target is the abelian cat $MM(k)$ of mixed motives over $k$. The yoga tells us that the semisimple objects should be pure motives wrt numerical equivalence.

Deligne suggested in the 80s that it might be easier to construct $D^b(MM(k))$. From this point of view, the mixed Weil cohomologies (or rather the corresponding homology theories) should come from certain types of “triangulated fiber functors” $H:D \to A$ from a triangulated tensor cat $D$, equipped with a functor $M: L(k) \to D$. In some sense, the triangulated cat of motives should be the universal such $D$. The axioms that this cat should satisfy are obtained by dualizing the axioms for a mixed Weil cohomology. I.e. action (=functoriality?) for finite correspondences, homotopy invariance, Kunneth formula $M(X) \otimes M(Y) = M(X \times Y)$, and a reinterpreted M-V property: the existence of an exact triangle of the form $M(U \cap V) \to M(U) \oplus M(V) \to M(X) \to M(U \cap V)[1]$.

The above program was carried out by Voevodsky, to be described here. Brief mention of Hanamura, Levine, and higher Chow groups.

14.3 The idea of motivic cohomology

The mixed motives philosophy has led to a development of vast theories of “mixed coefficients” over an arbitrary base, say smooth over a field for simplicity. Examples: mixed $\ell$-adic coefficients, perverse sheaves, Hodge modules, p-adic syntomic coeffs. These coeffs are/should be stable under Grothendieck’s six operations, under some restrictions. These theories produce “absolute cohomology” thoeries $H^*_{abs}(X, *)$, distinct from “geometric” cohomology theories of mixed Weil type, but related to the latter through a spectral sequence

where $Ext$ is taken in the respective categories of mixed coefficients.

Beinlinson, in “Height pairings…”, proposed a vast program, postulating the existence of “motivic coefficients”, verifying a formalism analogous to the formalism of mixed cohomological coefficients. For every $X \in L(k)$, the motivic coeffs should be objects in the derived category of an abelian tensor category $MM(X)$ of certain Zariski sheaves on $X$, the so called motivic sheaves. In particular, writing $p: X \to Spec(k)$ for the structure morphism, one should have an object $h(X) : = R p_* \underline{\mathbb{Z}} \in D(MM(Spec(k)))$ and objects $R^i p_* \underline{\mathbb{Z}} \in MM(k) := MM(Spec(k))$. One should also have objects $\mathbf{1}(r) \in MM(k)$, with $\mathbf{1}(-1) = h^2(\mathbf{P}^1)$.

These “motivic sheaves” should be equipped with a weight filtration (at least after tensoring with the rationals), with semisimple graded pieces. If $X$ is projective, then $h(X)$ is identified with the corresponding numerical pure motive. The motivic coefficients should be related with various types of mixed coefficients through various exact functors, called realizations.

Beilinson conjectured the existence of motivic coefficients $\underline{\mathbb{Z}}(r)$ (complexes of Zariski sheaves on $L(k)$) satisfying $\underline{\mathbb{Z}}(0) \cong \underline{\mathbb{Z}}$, and $\underline{\mathbb{Z}}(1) \cong \mathbf{G}_m[-1]$, and $\underline{\mathbb{Z}}(r) \cong \underline{\mathbb{Z}}(1)^{ {}^L \otimes r}$ for $r \geq 2$, and he defines motivic cohomology by setting as `\mathbf{H}^i_{Zar}(X, \underline{\mathbb{Z}}(r) ).

As in algebraic topology, motivic cohomology is related to algebraic K-theory by

degenerating at $E_2 \otimes \mathbb{Q}$.

Comparisons: $\mathbf{H}^{2r}_{Zar}(X, \underline{\mathbb{Z}}(r) ) = CH^r(X)$ and $\mathbf{H}^{r}_{Zar}(Spec(k), \underline{\mathbb{Z}}(r) ) = K^M_r(k)$.

Finally, motivic cohomology should be related to the various absolute cohomologies by so called regulator maps.

A substantial proportion of the above program is already realized, at least for $char(k) = 0$. The triangulated tensor category $DM_{gm}(k)$ is a substitute for $D(MM(k))$. The coefficients $\underline{\mathbb{Z}}(r)$ have been constructed, and we also have the formula $\mathbf{H}^{i}_{Zar}(X, \underline{\mathbb{Z}}(r) ) = Hom_{DM_{gm}(k)}( \mathbf{1}, h(X)(r)[i] )$, where $h(X)$ is the (contravariant) image of $X$ in $DM_{gm}$. Also, comparison thm ans AHSS have been established.

If one would have $MM(k)$ with its associated formalism, it would also give new insights on pure motives. One would have the Beilinson spectral sequence:

For $X$ projective, one would have $h(X)_{\mathbb{Q}} = \oplus h^i(X)_{\mathbb{Q}}[i]$ and the above spectral seq would degenerate at $E_2 \otimes \mathbb{Q}$.

Taking $i=2r$, the abutment coincides with the Chow group, tensored with $\mathbb{Q}$, and the filtration induced on these groups should/would be the same as BBM filtration discussed earlier. The degeneration of the Beilinson spectral sequence and the previous properties actually implies the Bloch-Beilinson conjecture, and the formula

This might be the most natural reason to believe in the B-B conjecture.

Next: Chapter 15, see Voevodsky motives

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Dear Irene,

Thanks for your email. I think that the transition from Grothendieck to Voevodsky is through Beilinson, Lichtenbaum and Bloch. Beilinson formulated the idea of a universal Bloch-Ogus cohomology, whereas Grothendieck’s motives are supposed to furnish a universal Weil cohomology, the difference is two-fold:

1. Weil cohomology is for smooth projective varieties, Bloch-Ogus for arbitrary (or maybe arbitrary smooth) varieties
2. Weil cohomology satisfies a Kuenneth formula on the level of cohomology groups, Bloch-Ogus satisfies a Kuenneth formula in an appropriate derived category of sheaves.

Beilinson and Lichtenbaum conjectured that there should be complexes whose hypercohomology gave the universal theory. Bloch gave a first construction of such complexes with his cycle complexes.

Beilinson later conjectured that the universal theory should occur as Ext groups in an abelian category of mixed motives. This has not been constructed. However, in the earlier 90’s Voevodsky, Hanamura and myself constructed triangulated categories that had many of the properties that one would expect of the derived category of Beilnson’s abelian category of mixed motives.

I think the book of Yves André on mixed motives gives a very nice treatment. You can also look at some articles in the Handbook of K-theory (I wrote one, see also Bruno Kahn’s article). I think there are many open problems mentioned in th Handbook of K-theory.

As far as my notes are concerned, they have not been published yet. I can let you know when this occurs.

I hope this has been helpful, please feel free to write with questions or comments.

Best regards,

Marc Levine

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Brief memo notes from Huber’s introduction

Description of “dream world”: Consider the category of varieties over a field (say $\mathbb{Q}$). For any variety $X$, there is an abelian category $MM(X)$ of motivic sheaves (“sheaves of motives”). There is also a triangulated category $D_{MM(X)}$ equipped with a t-structure with $MM(X)$ as its heart. This might or might not be the derived category of $MM(X)$.

There is a formalism of six Grothendieck functors. That is a tensor product on $D_{MM(X)}$, a notion of duals, and for a morphism $f:X \to X'$ of varieties, four functors $Rf^*$, $Rf^{!}$, $Rf_*$, $Rf_{!}$ between the triangulated categories. Furthermore, any object $M$ of $D_{MM(X)}$ has a “weight filtration”, such that the graded parts $Gr_n^W(M)$ are “pure of weight $n$”. Various properties and compatibilities between these structures.

Motivic sheaves would be the universal theory satisfying this formalism. The category $MM = MM(Spec(k))$ is called the category of mixed motives. The subcategory of objects whose weight filtrations split should agree with Grothendieck’s category of pure motives.

This world also contains two special CTs: geometric and absolute motivic cohomology. By geometric cohomology we mean the system of objects

in $Ob(MM)$. Here $s$ is the structure morphism of $X$, and $1_X = s^*1$, where $1$ signifies (I think) the constant sheaf on $Spec(k)$ (i.e. $\mathbb{Q}$??). This higher direct image is usually called the i-motive of $X$. Morally it is like a sheaf, not like a group.

By absolute cohomology, we mean the system of abelian groups:

Here $\Gamma(X, -) = Hom_{MM(X)}(1_X, -)$ is the global sections functor. A conjecture of Beilinson says that these groups should agree with the Adams eigenspaces of algebraic K-theory.

From the formal properties above, one easily shows:

This relation is usually expressed in terms of the Leray spectral sequence:

The above formalism, except for universality, is fulfilled in two important settings: l-adic and Hodge (what about crystalline???).

There is a theory of l-adic sheaves, with the perverse t-structure. Usual $\ell$-adic cohomology has a Galois action, and should be viewed as the geometric cohomology theory, but with $\mathbb{Q}$ above replaced by $\mathbb{Q}_{\ell}$. The absolute cohomology in this setting is Jannsen’s continuous étale cohomology.

Saito’s Hodge modules. The category of “Hodge sheaves” on $Spec(k)$ is the category of mixed Hodge structures. The geometric cohomology is de Rham cohomology, equipped with a Hodge structure. The absolute cohomology is Deligne cohomology (absolute Hodge cohomology).

One route towards the construction of the category $MM$ is via realizations. Jannsen constructed the category of “mixed realizations”, an approximation to $MM$, and also the corresponding geometric cohomology theory. In this book, we construct the absolute cohomology. This absolute cohomology satisfies the Bloch-Ogus axioms (in particular, cohomology with compact supports, homology, cup and cap product). Also Chern classes from K-theory.

Remark on relation with Beilinson’s conjectures. Scholl’s reformulation in terms of extensions of mixed motives. Get motivic extensions from cycles homologically equivalent to zero, via the Abel-Jacobi map.

An object of the category $D_{MR}$ that we construct, consists of a tuple of complexes, corresponding to étale, singular and de Rham cohomology. We work with filtered complexes in order to get weight filtrations. For a smooth variety, get weight filtration from a smooth compactification and the Leray spectral sequence. For the singular case, follow Deligne and use that “a singular variety behaves for cohomological question as a certain smooth simplicial variety does”. So: we get a functor on the category of smooth simplicial varieties. Sheaf theory methods vs simplicial techniques.

Note: the realization functor $R_{MR}$ can be extended to a functor on the category of Chow motives.

Remark: Get Chern classes into singular, de Rham, l-adic, absolute Hodge and continuous étale cohomology.

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Mixed motives

See Toen AIM talk on Homotopy types of algebraic varieties. Towards the end he talks about the motivic schematic homotopy theory, which should be closely related to mixed motives.

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Mixed motives

A category of abelian mixed motives has not yet been constructed, but there are suggestions, by Nori and maybe also by others. For the formalism of mixed motives, see various articles in the Motives volumes.

Specific classes of mixed motives include Mixed Tate motives, 1-motives, Laumon 1-motives.

The triangulated category of mixed motives has been constructed by Voevodsky, see Voevodsky motives. This is supposed to be the derived category of the above-mentioned abelian category. Voevodsky mentions in the introduction to Homology of schemes I, that the abelian category cannot exist when working with integral or finite coeffs (but possibly with rational coefficients), although for finite coeffs it is possible that the triangulated cat of mixed motives is the derived cat of an exact category.

See also Motives, and Motivic cohomology

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Mixed motives

A general idea is to assume one is given some category M of pure motives and then to try to construct a Hopf algebra or a co-Lie algebra H in the category M such that corepresentations of H in M give rise to (and conjecturally are equivalent to) mixed motives whose weight graded pieces lie in M. The focus becomes the study of H and its representations just as in number theory one studies the Galois group and its representations. In the case of motives, H is constructed using algebraic cycles, and cycle classes in some Weil cohomology lead to realizations of the mixed motives.” Quote from MathSciNet review of an article by Bloch

Voevodsky: Nilpotence theorem for cycles algebraically equivalent to zero. Discusses smash nilpotence, the nilpotence conjectures, and various ideas related to mixed motives, theories of motivic type, the standard conjectures, and stuff about algebraic equivalence, for example mixed motives modulo alg equiv.

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Mixed motives

Bloch describes a possible approach to mixed motives in Remarks on elliptic motives

Bloch and Kriz: Mixed Tate motives.

Bruguieres: On a Tannakian result due to Nori (preprint?)

Goncharov: Periods and mixed motives (2002, arXiv)

Nori: Constructible sheaves (2002)

MR1141206 (92m:14012) Saito, Morihiko(J-KYOT-R) Hodge conjecture and mixed motives. I. Complex geometry and Lie theory (Sundance, UT, 1989), 283–303, Proc. Sympos. Pure Math., 53, Amer. Math. Soc., Providence, RI, 1991. Also a follow-up: II.

MR1649192 (99i:19004) Goncharov, Alexander(1-BRN) Volumes of hyperbolic manifolds and mixed Tate motives. J. Amer. Math. Soc. 12 (1999), no. 2, 569–618.

Hanamura: Mixed motives and algebraic cycles I, II, III.

Hanamura: Homological and cohomological motives of an algebraic variety

Masaki Hanamura, The mixed motive of a projective variety (183–193)

Terasoma: Mixed Tate motives and multiple zeta values (Invent., 2002)

arXiv:0911.5611 Réalisations des complexes motiviques de Voevodsky from arXiv Front: math.AG by Florence Lecomte, Nathalie Wach Over a number field k, we construct realizations of Voevodsky motivic complexes, realizations as presented by Fontaine and Perrin-Riou [FPR94]. Our realization functors are defined from the category of motivic complexes constructed by Voevodsky and are obtained as cohomological functors which are, up to some limits, representable. The De Rham realization is represented by the De Rham motivic complex defined in [LW09]. We obtain integral Betti and l-adic realizations. Our realization functors are related by comparison arrows, which become isomorphisms when restricted to the category of geometrical motives. Furthermore, on geometrical motives, the realizations are endowed with Bondarko’s weight filtration [Bo09], the Hodge realization is constructed and all these realizations coincide rationally with those defined by A. Huber [H00].

arXiv:1001.5112 Mixed Motives over $k[t]/{(t^{m+1})}$ from arXiv Front: math.AG by Amalendu Krishna, Jinhyun Park For a perfect field $k$, we construct a triangulated category of mixed motives over $k[t]/{(t^{m+1})}$. The ext groups in this category are given by higher Chow groups, and additive higher Chow groups.

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Mixed motives

[CDATALevines heavy book.]

A better source is probably his recent survey article, published on paper in Handbook of K-theory. See in particular page 28-29 in the online version, or page 456-457 in the paper version.

There is also some notes by Bloch, and the article by Nekovar, and other articles in the Motives volumes.

Deninger: On extensions of mixed motives

For Nori motives, see von Wangenheim: http://front.math.ucdavis.edu/1111.5146

nLab page on Mixed motives