Motivic cohomology

Motivic cohomology is an invariant of

The four versions of motivic homology and cohomology can be defined with coefficients in any abelian group. The theories with different coefficients are related by the universal coefficient theorem, as in topology.

By Bloch,

${\displaystyle E_{2}^{pq}=H^{p}(X,\mathbf {Z} (-q/2))\Rightarrow K_{-p-q}(X).}$

As in topology, the spectral sequence degenerates after tensoring with the rationals.^[5] For arbitrary schemes of finite type over a field (not necessarily smooth), there is an analogous spectral sequence from motivic homology to G-theory (the K-theory of coherent sheaves, rather than vector bundles).

Motivic cohomology provides a rich invariant already for fields. (Note that a field k determines a scheme Spec(k), for which motivic cohomology is defined.) Although motivic cohomology Hⁱ(k, Z(j)) for fields k is far from understood in general, there is a description when i = j:

${\displaystyle K_{j}^{M}(k)\cong H^{j}(k,\mathbf {Z} (j)),}$

where K_j^M(k) is the jth Milnor K-group of k.^[6] Since Milnor K-theory of a field is defined explicitly by generators and relations, this is a useful description of one piece of the motivic cohomology of k.

Let X be a smooth scheme over a field k, and let m be a positive integer which is invertible in k. Then there is a natural homomorphism (the cycle map) from motivic cohomology to étale cohomology:

${\displaystyle H^{i}(X,\mathbf {Z} /m(j))\rightarrow H_{et}^{i}(X,\mathbf {Z} /m(j)),}$

where Z/m(j) on the right means the étale sheaf (μ_m)^⊗j, with μ_m being the mth roots of unity. This generalizes the

A frequent goal in algebraic geometry or number theory is to compute motivic cohomology, whereas étale cohomology is often easier to understand. For example, if the base field k is the complex numbers, then étale cohomology coincides with

is an isomorphism for all j ≥ i and is injective for all j ≥ i − 1.[7]

For any field k and commutative ring R, Voevodsky defined an R-linear

One basic point of the derived category of motives is that the four types of motivic homology and motivic cohomology all arise as sets of morphisms in this category. To describe this, first note that there are Tate motives R(j) in DM(k; R) for all integers j, such that the motive of projective space is a direct sum of Tate motives:

${\displaystyle M(\mathbf {P} _{k}^{n})\cong \oplus _{j=0}^{n}R(j)[2j],}$

where M ↦ M[1] denotes the shift or "translation functor" in the triangulated category DM(k; R). In these terms, motivic cohomology (for example) is given by

${\displaystyle H^{i}(X,R(j))\cong {\text{Hom}}_{DM(k;R)}(M(X),R(j)[i])}$

for every scheme X of finite type over k.

When the coefficients R are the rational numbers, a modern version of a conjecture by

Conversely, a variant of the Beilinson-Soulé conjecture, together with Grothendieck's standard conjectures and Murre's conjectures on Chow motives, would imply the existence of an abelian category MM(k) as the heart of a t-structure on DM(k; Q).^[9] More would be needed in order to identify Ext groups in MM(k) with motivic cohomology.

For k a subfield of the complex numbers, a candidate for the abelian category of mixed motives has been defined by Nori.

Let X be a smooth projective variety over a number field. The Bloch-Kato conjecture on

This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources in this section. Unsourced material may be challenged and removed. (January 2021) (Learn how and when to remove this message)

The first clear sign of a possible generalization from Chow groups to a more general motivic cohomology theory for algebraic varieties was Quillen's definition and development of algebraic K-theory (1973), generalizing the Grothendieck group K₀ of vector bundles. In the early 1980s, Beilinson and Soulé observed that Adams operations gave a splitting of algebraic K-theory tensored with the rationals; the summands are now called motivic cohomology (with rational coefficients). Beilinson and Lichtenbaum made influential conjectures predicting the existence and properties of motivic cohomology. Most but not all of their conjectures have now been proved.

Bloch's definition of higher Chow groups (1986) was the first integral (as opposed to rational) definition of Borel-Moore motivic homology for quasi-projective varieties over a field k (and hence motivic cohomology, in the case of smooth varieties). The definition of higher Chow groups of X is a natural generalization of the definition of Chow groups, involving algebraic cycles on the product of X with affine space which meet a set of hyperplanes (viewed as the faces of a simplex) in the expected dimension.

In the 1990s, Voevodsky (building on his work with Suslin) defined the four types^[11] of motivic homology and motivic cohomology for smooth schemes over a perfect field, along with a triangulated category of motives inside a very robust framework of ${\displaystyle \mathbb {A} ^{1}}$-homotopy theory.^[12] Different constructions were also given by Hanamura and Levine. These three triangulated categories of motives are now known to be equivalent, by work of Levine, Ivorra, and Bondarko.

Voevodsky also defined a motivic cohomology for singular varieties ^[13] and used it in the proof of the Block-Kato conjecture.^[14] This has become known as cdh motivic cohomology, as it sits in an Atiyah–Hirzebruch style spectral sequence calculating homotopy-invariant algebraic K-theory (the cdh-localization of algebraic K-theory), rather than algebraic K-theory itself.