cohomology

# Contents

## Idea

### Informal idea

The similarity of the behaviour of various cohomologies of varieties over a field suggest that there is a universal one among them with values in an intermediate abelian category, called the category of motives. Thus we should have a variety $X$, which maps to its motive $M\left(X\right)$, and good cohomologies would factor through that projection. Of course, not every motive is the image of a single variety. There is supposedly also a version with further filtrations (need to be more specific), the mixed motive?s.

So far there are realizations of pure motives, and not of the mixed motives. However there are several equivalent definitions of a triangulated tensor category which has all conjectured structural properties of the derived category of mixed motives (except t-structure which would manifestly make it a derived category); hence it is denoted $D\left(\mathrm{ℳℳ}\right)$.

Constructions of motives depend much on whether we work in prime characteristics or in characteristic zero. Part of the formalism involves more general schemes than varieties.

Another crucial idea leading to motives is that the various cohomologies lead to the same pieces of information; therefore there is a symmetry related to this, which is of Galois nature. For example, over the complex numbers one can compare the Betti cohomology? and de Rham cohomology “realizations”. Thus one has a motivic Galois group, and as usually with representations one has a tensor category structure which is also rigid. Thus one has in fact an abelian tensor category? of motives. The Tannakian reconstruction plays a major role; for pure motives we have neutral Tannakian categories, and for mixed motives we have mixed Tannakian categories. Functions on the torsor of the isomorphism between “realizations” correspond to the matrices of periods in Hodge theory.

The category of motives is roughly something like an abelianization and derivation of the category of schemes: a motive is sometimes realized as a complex of sheaves on a category whose objects are schemes, but whose morphism are certain correspondences between schemes (much like a groupoidification of the category of schemes).

$L$-functions (and $\zeta$-functions in particular) of varieties are also invariants of their motives. The Langlands program indirectly involves motives; in particular its essential part can be expressed as a general modularity conjecture relating $L$-functions to automorphic functions. Most of the deep properties of elliptic curves are of motivic nature, and in particular a major step of the proof of Fermat's last theorem by Wiles and Taylor can be interpreted as a proof of a special case of the modularity conjecture (for elliptic curves).

### Idea of the precise abstract definition of derived motives

We indicate the simple idea of the formal general abstract definition of derived mixed motives, as originally indicated in Deligne’s transcript of Voevodsky’s lectures and then elaborated on by (Ayoub). We follow the introduction of (Cisinski-Déglise).

On some site $𝒞$ of (Nisnevich) schemes (or maybe some other kinds of spaces), consider functors (or maybe rather (∞,1)-functors)

$D:𝒞\to \mathrm{Stab}\infty \mathrm{Cat}$D \colon \mathcal{C} \to Stab\infty Cat

with values in closed monoidal stable (∞,1)-categories (or some presentation theoreof, such as suitably enhanced triangulated categories) such that the following conditions hold (see Cisinski-Déglise, around p. x):

1. for every morphism $f:X\to Y$ in $𝒞$ it assigns a (inverse image $⊣$ direct image) adjunction $\left({f}^{*}⊣{f}_{*}\right):D\left(X\right)\to D\left(Y\right)$ which extends to a six operations formalism $\left({f}_{!}⊣{f}^{!}\right):D\left(X\right)\to D\left(Y\right)$ if $f$ is a separated morphism of finite type;

2. for every closed immersion $i:Z↪X$ with open immersion complement $j:\left(X-Z\right)↪X$ the unit/counits form a (co)fiber sequence

${j}_{!}{j}^{!}\to 1\to {i}_{*}{i}^{*}$j_! j^! \to 1 \to i_* i^*
3. the functor is homotopy-invariant in the sense that for any $X\in 𝒟$, with ${𝔸}^{1}$-the line object (the affine line), the inverse image $D\left(p{\right)}^{*}$ along the projection $p:{𝔸}^{1}×X\to X$ is a full and faithful (∞,1)-functor (??? check ??? we want to say that the composite of taking the product with ${𝔸}^{1}$ and then forgetting the map to ${𝔸}^{1}$ is an equivalence).

4. some further technical conditions …

The idea here is that one wants to think of $D\left(X\right)$ as a subcategory of the derived category $𝒟\left(X\right)$ of quasicoherent sheaves on those that serve as certain nice coefficients for abelian sheaf cohomology (for instance in that they exhibit nice gluing over open immersions, give homotopy-invariant cohomology groups etc.)

Then (with the list of such axioms suitably completed) we want to say that:

Idea of Definition.

1. The initial object – to be denoted $\mathrm{SH}$ – in the evident (∞,2)-category of such functors $D$ is the motive (or the “motive functor” or the like).

2. For every $X\in 𝒞$ the exceptional direct image ${p}_{#}\left(1\right)\in \mathrm{SH}\left(\mathrm{Spec}k\right)$ of the tensor unit under the projection $X\to \mathrm{Spec}k$ to the base space is the motive of $X$.

Idea of one central theorem. The assignment that sends $X\in 𝒞$ to the stable A1-homotopy theory over $X$ is such an initial object.

(Warning: apparently the full initiality has not been shown yet, but it has been shown that there is a morphism from this $\mathrm{DM}$ to any other $D$ which seems to be essentially unique in the right sense).

## Constructions

### Voevodsky motives

Associated to a Noetherian scheme $S$ there is a category ${\mathrm{Cor}}_{S}$ of “finite” correspondences of schemes, whose

• objects are schemes of finite type over $S$;

• morphisms ${\mathrm{Cor}}_{S}\left(X,Y\right)$ form an abelian group of cycles on the fiber product $X{×}_{S}Y$ that are “universally integral relative to $X$” and each of whose components are finite and and surjective over $X$.

Details are in appendix 1A of MaVoWe.

The triangulated category of motives over a field $k$ is…

…defined for instance in lecture 14, def 14.1 of MaVoWe.

It is a localization of the derived category of (bounded) complexes of sheaves on this category of correspondences, ${\mathrm{Cor}}_{S}$.

### Motivic cohomology

The derived hom-sets in the category of motives, at least between special objects, compute what is called motivic cohomology.

### Nori motives

Madhav Nori? has an approach to the theory of motives based on a peculiar kind of Tannakian reconstruction, the so called Nori's Tannakian theorem.

### Extensions

Correspondences are interesting in noncommutative geometry of the operator algebra flavour. For example, KK-groups are in fact themselves sort of correspondences; Connes and Skandalis had an early reference very much paralleling some ideas from the algebraic world. More recently, motives in the operator algebraic setup have been approached by Connes, Marcolli and others.

In derived noncommutative algebraic geometry based on ${A}_{\infty }$-categories, Kontsevich proposed a theory of noncommutative motives. There is now already a more general setup (than Kontsevich’s) due Cisinski and Tabuada (see Refs.).

In birational geometry, Bruno Kahn defined the appropriate version. In rigid analytic geometry, ${A}^{1}$-homotopy theory is replaced by ${B}^{1}$-homotopy theory and the appropriate analogue of the Voevodsky’s category of mixed motives has been constructed; the construction follows the same basic pattern.

## References

### General

A brief exposition is in

Voevodsky’s formalization of motives was sketched in

and worked out in detail in

• Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I), Astérisque, vol. 314, Soc. Math. France, 2007.

Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II), Astérisque, vol. 315, Soc. Math. France, 2007.

A summary of the axioms and of the main theorems is in the introduction of

A modern introduction to Voevodsky’s theory is

An outline of the big picture can be found in the introduction to

A formal discussion of motives can be found in lecture 14 of

There is also

Some recent generalizations of the theory, using derivators and similar techniques, are in

Some other aspects

• M.V. Bondarko, Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), arxiv/0704.4003
• Yuri Manin, Motives and quantum cohomology, talk at Colloque Grothendieck, video

### Relation to Hodge theory

Explicit discussion of the relation to Hodge theory is in

### Relation to KK-theory / bivariant K-theory

Relation of motivic cohomology to bivariant algebraic K-theory (see also at KK-theory) is discussed in