group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
In order to express this kinship of these different cohomological theories, I formulated the notion of “motive” associated to an algebraic variety. By this term I want to suggest that it is the “common motive” (or “common reason”) behind this multitude of cohomological invariants attached to an algebraic variety, or indeed, behind all cohomological invariants that are a priori possible. (Grothendieck, Recoltes et Semailles)
The similarity of the behaviour of various cohomologies of varieties over a field suggests that there is a universal one among them with values in an intermediate abelian category, called the category of motives. The idea is that to every variety $X$ is associated a motive $M(X)$, such that every good cohomology theory factors through the functor $M$. (Here not every motive is supposed to be the image of a single variety.)
One distinguishes a theory of pure motives for smooth projective varieties from a more general theory of mixed motives for arbitrary smooth varieties. So far, pure motives and mixed motives have only been defined conditionally. 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 for the t-structure which would make it a derived category).
Grothendieck’s original realization of this idea is the category of Chow motives, which is a certain abelianization and completion of a category of spans of smooth projective varieties. Later a more homotopy-theoretic version was given, the Voevodsky motives or derived motives, see below, which subsume the Chow motives faithfully. More generally still, there are definitions for noncommutative motives obtained by passing to noncommutative algebraic geometry. Finally, the construction principle of motives can also be adapted to other flavors of geometry. For instance in noncommutative topology the role of the category of noncommutative motives is played by KK-theory.
Constructions of motives often depend 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 cohomology theories lead to the same pieces of information; therefore there is a symmetry related to this, which is of Galois theory 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. 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.
$L$-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).
There is no generally accepted construction of a $\mathbb{Q}$-linear abelian category of mixed motives, and its existence remains conjectural. However, there exist candidate and conditional constructions which are useful in practice.
Note that “the” abelian category of mixed motives depends on choosing a base scheme $S$, and one speaks of motives (or motivic sheaves) over $S$. Traditionally, $S$ is the spectrum of a field, often of characteristic zero.
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. Nori’s construction unconditionally produces a $\mathbb{Q}$-Tannakian category of mixed motives over any subfield of $\mathbb{C}$.
Pierre Deligne gave a definition of a category of mixed motives over number fields as compatible systems of realizations, essentially bundling together all the structure that mixed motives should give rise to. This approach automatically yields a $\mathbb{Q}$-Tannakian category of mixed motives with all the desired realization functors (Betti, $l$-adic, de Rham, and crystalline). See Deligne for details.
Deligne first suggested that it might be easier to define the derived category $DM(S,\mathbb{Q})$ of the hypothetical abelian category of mixed motives. Once this is done, one can in principle recover the abelian category as the heart of a t-structure on $DM(S,\mathbb{Q})$. It is now well-understood what the triangulated category $DM(S,\mathbb{Q})$ is over any base scheme (see below). The hypothetical t-structure on $DM(S,\mathbb{Q})$ whose heart is the abelian category of mixed motives over $S$ is called the motivic t-structure.
Beilinson proved that, over fields of characteristic zero, the existence of the motivic t-structure implies the standard conjectures on algebraic cycles (see Beilinson), and Bondarko proved that it implies the existence of motivic t-structures for more general schemes (see Bondarko).
While the derived category of mixed motives can also be defined with integral rather than rational coefficients, Voevodksy observed that the derived category of integral motives cannot have a motivic t-structure (Voevodsky, Prop. 4.3.8). Thus, the abelian category of motives always refers to motives with rational coefficients.
The derived category of the hypothetical abelian category of mixed motives has been unconditionally defined over any Noetherian scheme. The first definition was proposed by Voevodsky in the mid 1990s. Since then, several other definitions were formulated: one by Morel, one by Ayoub, and one by Cisinski and Déglise. The latter three are equivalent and support a full-fledged formalism of six operations. However, they are only known to be equivalent to Voevodsky’s definition over excellent? and geometrically unibranch? schemes.
On the other hand, Voevodsky’s definition is the only one among these four which also makes sense with integral coefficients rather than rational coefficients. Recently, Spitzweck proposed a definition of the category of integral motives over general base schemes which also supports a formalism of six operations. It is known to agree with Voevodsky’s definition for fields of characteristic zero. Rationally, however, it agrees with the Morel/Ayoub/Cisinski-Déglise definition over any base scheme.
Associated to a Noetherian scheme $S$ there is an additive category $SmCor_S$ of “finite” correspondences of schemes, whose
objects are smooth schemes of finite type over $S$;
morphisms $SmCor_S(X,Y)$ are the 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 $X$.
See at pure motive for more (see also MaVoWe, Appendix 1A). Associating to a morphism of schemes its graph defines a faithful functor $Sm/S\hookrightarrow SmCor_S$.
An (∞,1)-presheaf with transfers on the category $Sm/S$ of smooth schemes of finite type is an (∞,1)-presheaf on $SmCor_S$ which transforms finite sums into finite (∞,1)-products (and hence take values in connective chain complexes).
The (∞,1)-category $DM^{eff}_{\geq 0}(S)$ is a certain reflexive localization of the (∞,1)-category of presheaves with transfers: it consists of those presheaves with transfers whose underlying presheaves on $Sm/S$ are (∞,1)-sheaves for the Nisnevich topology and are A1-homotopy invariant.
The Tate motive $\mathbb{Z}(1)[2]$ is the image of the pointed scheme $(\mathbb{P}^1,\infty)$ in $DM^{eff}_{\geq 0}(S)$.
The stable (∞,1)-category of (integral) motives $DM(S)$ is the stabilization of $DM^{eff}_{\geq 0}(S)$ at the Tate motive $\mathbb{Z}(1)[2]$.
The (∞,1)-category $DM(S)$ is a stable, symmetric monoidal, and locally presentable (∞,1)-category which is enriched in chain complexes. It is slightly larger than the category $DM^-(S)$ defined in MaVoWe, p. 110 which is not cocomplete.
Voevodsky’s cancellation theorem states that the canonical functor $DM^{eff}_{\geq 0}(S)\to DM(S)$ is fully faithful if $S$ is a perfect field.
Let $\epsilon : \mathbb{G}_m\wedge \mathbb{G}_m\to \mathbb{G}_m\wedge \mathbb{G}_m$ be the transposition. In the stable motivic homotopy category $SH(S)$ this becomes an endomorphism of the motivic sphere spectrum $S^0$ such that $\epsilon^2=1$. Rationally (or even away from 2), we obtain a pair of idempotent elements
which induce a splitting $SH(S)_{\mathbb{Q}}\simeq SH(S)_{\mathbb{Q}_+}\times SH(S)_{\mathbb{Q}_-}$.
$SH(S)_{\mathbb{Q}_+}$ is the stable $(\infty,1)$-category of Morel motives.
In other words, a Morel motive is a rational stable motivic homotopy type on which $\epsilon$ acts as $-1$.
The Hopf element $\eta\in \pi_{1,1}(S^0)$ is the stabilization of the algebraic Hopf fibration $\mathbb{A}^2-0\to\mathbb{P}^1$ over $S$. Morel motives can also be characterized as those rational stable motivic homotopy types that are acted on trivially by the Hopf element.
We have $\epsilon=-1$ if and only if $-1$ is a sum of squares in all the residue fields of $S$, in which case $SH(S)_{\mathbb{Q}}= SH(S)_{\mathbb{Q}_+}$. Thus, the other summand $SH(S)_{\mathbb{Q}_-}$ only appears over formally real fields. It is called the category of Witt motives.
According to Ayoub, the stable $(\infty,1)$-category of motives over a scheme $S$ can be constructed in the same way as the stable motivic homotopy category $SH(S)$, with two variations:
The resulting (∞,1)-category is denoted $DA^{\mathrm{et}}(S,\mathbb{Q})$. Its objects are thus $\mathbb{P}^1$-spectra of $\mathbb{A}^1$-invariant étale (∞,1)-sheaves with values in connective rational chain complexes.
This definition is due to Cisinski and Déglise. The rationalization of the homotopy invariant algebraic K-theory spectrum $KGL\in SH(S)$ splits as a direct sum
for some $E_\infty$ rational motivic ring spectrum $H_B\in SH(S)$.
The stable $(\infty,1)$-category of Beilinson motives is the $(\infty,1)$-category of modules over $H_B$. Equivalently, it is the full subcategory of $SH(S)_{\mathbb{Q}}$ consisting of $H_B$-local objects.
Cisinski and Déglise have shown that $H_B$ is exactly the $+$-summand $S^0_{\mathbb{Q}_+}$ of the rational motivic sphere spectrum, and hence that a Beilinson motive is the same thing as a Morel motive. They have also shown that Beilison/Morel motives are equivalent to Ayoub motives. Finally, they have shown that Beilinson motives are equivalent to rational Voevodsky motives $DM(S,\mathbb{Q})$ when $S$ is excellent? and geometrically unibranch?. Over such schemes, all four definitions of the derived category of mixed motives are therefore equivalent.
One idea to define a category of integral motives with a formalism of six operations is to first define an $E_\infty$ motivic ring spectrum $M_{\mathbb{Z}}\in SH(Spec \mathbb{Z})$. If $f: S\to Spec \mathbb{Z}$ is any scheme, we obtain an $E_\infty$-algebra $M_S = f^\ast(M_{\mathbb{Z}})$ in $SH(S)$. The categories of modules over $M_S$ for varying $S$ then inherit a complete formalism of six operations from $SH$.
Spitweck defined such an $E_\infty$-algebra $M_{\mathbb{Z}}$ such that
The stable $(\infty,1)$-category of $M_S$-modules is thus a well-behaved candidate for a derived category of integral motives, but it is only known to agree with Voevodsky’s definition when $S$ is a field of characteristic zero (by Rondigs-Ostvaer, Theorem 5.5).
The definition of Voevodsky motives can be found in
and the definition of Ayoub motives in
For the definition of Beilinson and Morel motives, the equivalences of the various definitions, and the formalism of six operations, see
The fact that $DM(k)$ is equivalent to the category of $H(\mathbb{Z})$-modules if $\mathrm{char}(k)=0$ is proved in
Spitzweck’s definition of a motivic cohomology spectrum over $Spec \mathbb{Z}$ is in
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.
Motivic structures show up in quantum field theory, for instance
The pull-push quantization in Gromov-Witten theory is naturally understood as a “motivic quantization” in terms of Chow motives of Deligne-Mumford stacks (Behrend-Manin 95).
See at KK-theory in the section As an analog of motives in noncommutative topology.
pure motive, mixed motive, Voevodsky motive, Nori motive, Chow motive
See also at KK-theory – Relation to motives.
A brief exposition is in
A review is also in chapter I of
Lectures include
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
An outline of the big picture can be found in the introduction to
Some pretty useful comments on motives are at this MathOverflow thead:
See also the blog post
A formal discussion of motives can be found in lecture 14 of
There is also
James S. Milne, Motives – Grothendieck’s Dream
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)
Marc Levine, Mixed motives, Math. Surveys and Monographs 57, Amer. Math. Soc. 1998, free pdf
For a noncommutative analogue to the theory of motives, see noncommutative motives.
Some other aspects
Explicit discussion of the relation to Hodge theory is in
Relation of motivic cohomology to bivariant algebraic K-theory (see also at KK-theory) is discussed in
Guillermo Cortiñas, Andreas Thom, Bivariant algebraic K-theory. J. Reine Angew. Math. 510 (2007), 71–124. (arXiv:math/0603531)
Grigory Garkusha, Ivan Panin, K-motives of algebraic varieties (arXiv:1108.0375)
Grigory Garkusha, Algebraic Kasparov K-theory. II (arXiv:1206.0178)
See also at KK-theory – Relation to motives.
For a collection of literature see also paragraph 1.5 in
(in the context of noncommutative motives).
See also at motivic multiple zeta values.
That the pull-push quantization in Gromov-Witten theory is naturally understood as a “motivic quantization” in terms of Chow motives of Deligne-Mumford stacks was suggested in
Further investigation of these stacky Chow motives then appears in
For more see at motives in physics.