Relation to K-theory By Bloch,
Lichtenbaum,
Friedlander,
Suslin, and Levine, there is a
spectral sequence from motivic cohomology to
algebraic K-theory for every smooth scheme
X over a field, analogous to the
Atiyah-Hirzebruch spectral sequence in topology: :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. 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).
Relation to Milnor K-theory 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: :K_j^M(k) \cong H^j(k, \mathbf{Z}(j)), where
KjM(
k) is the
jth
Milnor K-group of
k. 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.
Map to étale cohomology 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: :H^i(X,\mathbf{Z}/m(j))\rightarrow H^i_{et}(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
cycle map from the Chow ring of a smooth variety to étale cohomology. 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
singular cohomology (with finite coefficients). A powerful result proved by Voevodsky, known as the
Beilinson-Lichtenbaum conjecture, says that many motivic cohomology groups are in fact isomorphic to étale cohomology groups. This is a consequence of the
norm residue isomorphism theorem. Namely, the Beilinson-Lichtenbaum conjecture (Voevodsky's theorem) says that for a smooth scheme
X over a field
k and
m a positive integer invertible in
k, the cycle map :H^i(X,\mathbf{Z}/m(j))\rightarrow H^i_{et}(X,\mathbf{Z}/m(j)) is an isomorphism for all
j ≥
i and is injective for all
j ≥
i − 1.
Relation to motives For any field
k and commutative ring
R, Voevodsky defined an
R-linear
triangulated category called the
derived category of motives over
k with coefficients in
R, DM(
k;
R). Each scheme
X over
k determines two objects in DM called the
motive of
X, M(
X), and the
compactly supported motive of
X, Mc(
X); the two are isomorphic if
X is
proper over
k. 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: :M(\mathbf{P}^n_k)\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 :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
Beilinson predicts that the subcategory of compact objects in DM(k;
Q) is equivalent to the bounded derived category of an
abelian category MM(
k), the category of
mixed motives over
k. In particular, the conjecture would imply that motivic cohomology groups can be identified with
Ext groups in the category of mixed motives. This is far from known. Concretely, Beilinson's conjecture would imply the
Beilinson-Soulé conjecture that
Hi(X,
Q(
j)) is zero for
i < 0, which is known only in a few cases. 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). 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. If a category
MM(
k) with the expected properties exists (notably that the Betti realization functor from
MM(
k) to
Q-vector spaces is
faithful), then it must be equivalent to Nori's category. == Applications to arithmetic geometry ==