Motivic cohomology is a cohomology theory for schemes which in many ways plays the rôle of singular cohomology in algebraic geometry. It was first conjectured to exist by Alexander Beilinson and Stephen Lichtenbaum in the mid 1980s, and it was then defined by Vladimir Voevodsky in the mid 1990s.
Motivic cohomology must not be confused with the hypothetical “universal” cohomology theory envisioned by Alexander Grothendieck in the 1960s as the underlying reason for the standard conjectures on algebraic cycles. The former is an absolute cohomology theory with values in abelian groups, while the latter is a geometric cohomology theory with values in the still conjectural abelian category of mixed motives. They are related in that motivic cohomology with rational coefficents should appear as particular Ext-groups in the category of motives, an idea which can now be made precise using the various existing constructions of the derived category of motives.
The motivic cohomology groups of a scheme form a bigraded family of abelian groups . Several competing definitions of these groups exist but they are known to all agree when is smooth over a field. With rational coefficients, the motivic cohomology groups of are the associated graded of the -filtration on the rational algebraic K-theory groups of (at least if is regular). With coefficients in , they are closely related to the étale cohomology of with coefficients in the sheaf of th roots of unity (if is invertible on ) and to the logarithmic de Rham-Witt cohomology of (if equals the characteristic of ).
The motivic cohomology of a sufficiently nice scheme is also related to the algebraic K-theory of via the motivic spectral sequence
which degenerates rationally. The search for this spectral sequence was one of the motivating factor in the development of motivic cohomology. More generally, spectral sequences whose first page consists of motivic cohomology groups exist for any cohomology theory represented by a motivic spectrum; they are analogous to the Atiyah-Hirzebruch spectral sequences in topology.
We give three definitions of motivic cohomology with integral coefficients, in historical order: the first, due to Bloch and later generalized by Levine, only works for smooth schemes over Dedekind domains. The other two, due to Voevodsky, work for arbitrary schemes. All definitions are known to agree for smooth schemes over fields, but the equivalence of any pair of them is an open question for more general schemes. It is generally accepted that the Bloch–Levine definition produces the desired motivic cohomology groups as far as it applies, but there is no consensus beyond that.
Note that in each definition motivic cohomology is absolute: the groups depend only of the scheme and not on any base scheme.
The first and most elementary definition of motivic cohomology groups was Bloch’s definition of higher Chow groups (Bloch), although they were only recognized as such later by Voevodsky.
The algebraic -simplex is the -scheme
Note that is isomorphic to affine -space . There are obvious coface and codegeneracy maps that turn into a cosimplicial -scheme. The graded simplicial abelian group is the subgroup of generated in simplicial degree by the cycles which intersect all faces properly. One then defines the higher Chow groups by
The groups are the ordinary Chow groups of algebraic cycles modulo rational equivalence.
Voevodsky proved that these groups agree with his definition of motivic cohomology under the re-indexing
In the mid 1990s Vladimir Voevodsky gave the first “official” definition of the motivic cohomology of a scheme as the hypercohomology of certain complexes of sheaves on the Zariski site of (an analog of the category of open subsets of a topological space). The complexes , , are called the motivic complexes; the existence of such complexes was predicted as part of the so-called Beilinson dream.
This is MaVoWe, Definition 3.4.
Voevodsky’s definition, for smooth schemes over fields, has been shown to have most properties that Beilinson and Lichtenbaum had demanded of the hypothetical cohomology theory, except that to date it hasn’t been shown that the cohomology groups vanish in negative degree, as they should. This open question is known as the Beilinson vanishing conjecture.
Voevodsky also gave an accompanying definition of an integral version of the derived category of the hypothetical category of mixed motives (see there for the definition) and showed that the motivic cohomology appears as derived hom-complexes in this derived category (see MaVoWe, rop. 14.16 for a precise statement).
From the point of view of the motivic homotopy theory of Morel and Voevodsky, one would like the motivic cohomology of to be representable in the stable motivic homotopy category over . Voevodsky gave a definition of motivic cohomology in this setting as the bigraded cohomology theory represented by the motivic Eilenberg–Mac Lane spectrum .
The motivic spectrum is built out of motivic Eilenberg–Mac Lane spaces . Below we only discuss the definition of these spaces over a field . The definition in general is essentially the same, but it relies on the notion of finite correspondence over more general bases which is technical.
To define motivic Eilenberg-Mac Lane spaces, a first guess might be to apply the general definition of an Eilenberg-Mac Lane object in the Nisnevich (∞,1)-topos and then take its -localization. While this is an interesting construction, these spaces can only be assembled into an -spectrum and we want a -spectrum. This is not easy: Voevodsky states in his ICM-talk article (on p. 596) that every morphism is trivial in the -homotopy category.
Instead one applies a recipe which, when applied to the usual topological spheres produces the (topological) Eilenberg-Mac Lane spaces, to the algebro-geometric sphere :
The Dold-Thom theorem says that in topology the reduced singular homology of a space can be produced as
where is the free strictly commutative monoid on and denotes group completion. Inserting the topological n-sphere yields that is an Eilenberg-Mac Lane space.
The symmetric powers also make sense for quasi-projective -schemes, and they can be formally extended to pointed presheaves on such schemes. If is a pointed presheaf, we have maps (lengthening an -letter word by one, attaching the base point) and the colimit over these maps, followed by group completion, gives a functor . Over a field of characteristic zero, one defines the motivic Eilenberg-Mac Lane spaces by
These assemble to give the motivic Eilenberg-Mac Lane spectrum with bonding maps induced by (i.e. take the extra -point as new coordinate in the bigger -product of s).
This definition does not quite work over fields of positive characteristic. In general one has to take cycles as described by Denis-Charles Cisinski below. Intuitively the points of are finite formal sums of points of X, i.e. zero-cycles, which links this story to the functor described below. In characteristic zero both coincide. The link to higher Chow groups however only becomes apparent in the cycle description.
To keep things simple, let us assume we work over a perfect field . The easiest part of motivic cohomology which we can get is the Picard group (i.e. the Chow group in degree 1). This works essentially like in Top: in the (model) category of simplicial Nisnevich sheaves (over smooth -schemes), the classifying space of the multiplicative group has the -homotopy type of the infinite dimensional projective space.
Moreover, as the Picard group is homotopy invariant for regular schemes (semi-normal is even enough), the fact that reads as
where are the hom-sets in the motivic homotopy category .
There are several models for , one of the smallest being constructed as follows. What is explained above is that is the infinite projective space. is simply the constant sheaf. For higher , here is the following construction due to Voevodsky.
Given a -scheme , denote by the presheaf with transfers associated to , that is the presheaf of abellian groups whose sections over a smooth -scheme are the finite correspondences from to (i.e. the finite linear combinations of cycles in such that is finite and surjective over ). This is a presheaf, where the pullbacks are defined using the pullbacks of cycles (the condition that the ; are finite and surjective over a smooth (hence normal) scheme makes that this is well defined without working up to rational equivalences, and as we consider only pullbacks along maps with and smooth (hence regular) ensures that the multiplicities which will appear from these pullbacks will always be integers). The presheaf is a sheaf for the Nisnevich topology. This construction is functorial in (we will need this functoriality only for closed immersions).
Let (resp. ) be the cartesian product of (resp. ) copies of the projective line. The point at infinity gives a family of maps . Then a model of the Eilenberg-Mac Lane object is the sheaf of sets obtained as the quotient (in the category of Nisnevich sheaves of abelian groups) of by the subsheaf generated by the images of the maps .
The original definition of motivic Eilenberg–Mac Lane spaces and spectra is in
More details are in §4 of
J. F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445–553
A discussion of an equivariant version of motivic cohomology is in
For a discussion of the relation betwen motives and motivic cohomology, see for instance section 0.1.8 of