nLab
motive

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 $M(X)$, 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 $D(\mathrm{ℳℳ})$.

Constructions of motives depend much on whether we work in prime characteristic?s 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).

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 ${\mathrm{Cor}}_S(X,Y)$ form an abelian group of cycles on the fiber product $X{\times }_{S}Y$ 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.

See prop. 14.16, p. 114 of MaVoWe.

Nori motives

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

Related entries

• pure motive

• mixed motive?

• Voevodsky motive

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 $A_{\mathrm{\infty }}$-categories, Kontsevich proposed a theory of noncommutative motive?s. 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

A modern introduction to Voevodsky’s theory is

• A. Beilinson, V. Vologodsky, A DG guide to Voevodsky’s motives, math.AG/0604004

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

• Dmitri Kaledin?, Motivic structures in non-commutative geometry, arxiv/1003.3210

Some pretty useful comments on motives are at this MathOverflow thead:

• What’s the “Yoga of Motives?”

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

• Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lectures in motivic cohomology (web pdf)

There is also

• James S. Milne, Motives – Grothendieck’s Dream

• Mihnyong Kim, Classical Motives: Motivic $L$-functions

• Bruno Kahn, pdf slides on pure motives

• Florence Lecomte, Nathalie Wach, Réalisations des complexes motiviques de Voevodsky (arxiv:0911.5611)

• Marc Levine, Smooth motives, arxiv:0807.2265

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

• D-C. Cisinski, F. Déglise, Triangulated categories of mixed motives, arxiv/0912.2110

• D-C. Cisinski, G. Tabuada, Symmetric monoidal structure on Non-commutative motives, arxiv/1001.0228