# zbMATH — the first resource for mathematics

On the Grayson spectral sequence. (English. Russian original) Zbl 1084.14025
Proc. Steklov Inst. Math. 241, 202-237 (2003); translation from Tr. Mat. Inst. Im. V. A. Steklova 241, 218-253 (2003).
D. Grayson [K-Theory 9, No. 2, 139–172 (1995; Zbl 0826.19003)], provided a motivic analogue of the Atiyah-Hirzebruch spectral sequence starting from Grayson’s motivic cohomology groups and converging to $$K$$-groups $$H^{p-q}(X,Z^{\text{Gr}}(-q)) \Longrightarrow K_{-p-q}(X)$$.
The main result of the paper under review says that the canonical homomorphism of complexes of sheaves $$Z^{\text{Gr}}(n) \to Z(n)$$ is a quasi-isomorphism, where $$Z^{\text{Gr}}(n)$$ are motivic complexes considered by Grayson and $$Z(n)$$ are motivic complexes constructed by Voevodsky. As an immediate consequence one obtains the motivic analogue of the Atiyah-Hirzebruch spectral sequence starting from motivic cohomology introduced by Voevodsky.
The approach offered by the author completely avoids the use of the preprint by S. Bloch and S. Lichtenbaum [A spectral sequence for motivic cohomology, http://www.math.vivc.edu/k-theory/0062] where the mentioned spectral sequence was constructed in the case when $$X=\text{Spec}(k)$$ is a spectrum of a field $$k$$. Instead, it involves the properties of Grayson’s complexes $$Z^{\text{Gr}}(n)$$. The most important of them are a) the cohomology sheaves of $$Z^{\text{Gr}}(n)$$ are homotopy ivariant pretheories, b) the complex $$Z^{\text{Gr}}(n)$$ is defined by a rationally contractible presheaf, c) Grayson’s motivic cohomology satisfy the cohomology purity.
For the entire collection see [Zbl 1059.11002].

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 55T99 Spectral sequences in algebraic topology
##### Keywords:
motivic Atiyah-Hirzebruch spectral sequence