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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].

14F42 Motivic cohomology; motivic homotopy theory
55T99 Spectral sequences in algebraic topology