isomorphic in the derived category of X to Gm[−1]. Here Gm (the multiplicative group) denotes the sheaf of invertible regular functions, and the shift [−1] means that this sheaf is viewed as a complex in degree 1. 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. Relations to other cohomology theories 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: 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: 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: where Z/m(j) on the right means the étale sheaf (μm)⊗j, with μm 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)