On the motivic spectral sequence. (English) Zbl 1390.19006

A mayor result in the study of \(K\)-theory of algebraic varieties was the discovery of an Atiyah-Hirzabruch type of spectral sequence that relates motivic cohomology groups of an algebraic variety and the algebraic \(K\)-theory groups the variety. There have been at least three ways of constructing these spectral sequences: the motivic spectral sequence (MSS) by S. Bloch and S. Lichtenbaum [“A spectral sequence for motivic cohomology”, Preprint, https://pdfs.semanticscholar.org/7588/efb1d2c7b7c153c73a1506ccc934a85b3f0f.pdf] (MSS-BL), the MSS by D. R. Grayson [\(K\)-Theory 6, No. 2, 97–111 (1992; Zbl 0776.19001)] (MSS-G) and the MSS by V. Voevodsky [Contemp. Math. 293, 371–379 (2002; Zbl 1009.19003); in: Motives, polylogarithms and Hodge theory. Part I: Motives and polylogarithms. Papers from the International Press conference, Irvine, CA, USA, June 1998. Somerville, MA: International Press. 3–34 (2002; Zbl 1047.14012)] (MSS-V), each of these constructions have their own technique. In [in: Number theory, algebra, and algebraic geometry. Collected papers dedicated to the 80th birthday of Academician Igor’ Rostislavovich Shafarevich. Transl. from the Russian. Moskva: Maik Nauka/Interperiodika. 202–237 (2003; Zbl 1084.14025); translation from Tr. Mat. Inst. Im. V. A. Steklova 241, 218–253 (2003)], A. Suslin conjectured that these spectral sequences are isomorphic. Grayson [loc. cit.] proved that this is the case for MSS-BL and MSS-G. The main result of this that MSS-G is isomorphic to MSS-V. The technique consists of developing the theory of bispectra for the category of presheaves of symmetric spectra, attached to this one has the Grayson tower of bispectra and its motivic spectral sequence. On the other hand, one has the slice tower of the bispectrum \(KGL\), next by comparing the towers of bispectra, the authors obtain the appropriate isomorphism of spectral sequences in for perfect fields.


19E08 \(K\)-theory of schemes
55T99 Spectral sequences in algebraic topology
Full Text: DOI arXiv


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