On motivic cohomology with \(\mathbb{Z}/l\)-coefficients
type: Item
instance of: scholarly article
title: On motivic cohomology with \(\mathbb{Z}/l\)-coefficients
zbMATH Open document ID: 1236.14026
author: Vladimir Voevodsky
published in: Annals of Mathematics. Second Series
publication date: 2011-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/0805.4430
review text: This paper is a fundamental one which provides the final step to the proof of a long standing conjecture, the Bloch-Kato conjecture, which was announced several years ago. Theorem 1. Let \(k\) be a field of characteristic \(\neq l\), then the norm residue homomorphism \[ K^M_n(k)/l\to H^n_{et}(k, \mu^{\otimes n}_l) \] is an isomorphism for all \(n\). This result is a generalization of the particular case \(l=2\), whose proof appeared in [the author, Publ. Math., Inst. Hautes Étud. Sci. 98, 59--104 (2003; Zbl 1057.14028)]. Here one of the main new results is a theorem which relates two types of cohomological operations in motivic cohomology: one is defined in terms of symmetric power functors in the categories of relative Tate motives and another one in terms of the reduced power operation. Let \(R\) be a ring such that all primes but \(l\) are invertible in \(R\), e.g \(R= \mathbb{Z}_{(l)}\) or \(R= \mathbb{Z}/l\). Using several results about the structure of the symmetric powers \(S^i(M)\), for \(i<l\), when \(M\) is a Tate motive of the form \[ R(p)[2q]\to M\to R(p)[2q+ 1] \] one defines a cohomological operation \[ \Phi_{l-1}: H^{2q+ 1,p}(+,R)\to H^{2q+l,pl}(-, R). \] Then the author proves the following Theorem 2. For any \(n\geq 0\) there exists \(c\in(\mathbb{Z}/l)^*\) such that for any \(\alpha\in\widetilde H^{2n+1,n}({\mathcal X},\mathbb{Z}/l)\) one has \[ \Phi_{l-1}(\alpha)= c\beta P^n(\alpha), \] where \(\beta\) is the Bockstein homomorphism, \(P^n\) is the motivic reduced power operation and \(\widetilde H^{2n+1,n}({\mathcal X},\mathbb{Z}/l)\) are the motivic cohomology groups on pointed simplicial schemes. The proof of the above theorem has a long history: a proof of it appeared in the first version of this paper in 2003 and was based on a lemma, the validity of which is under serious doubt. Then in 2001 C. Weibel suggested another approach of the same result. In the present paper the author uses a modified version of Weibel's approach to prove a new lem…
reviewed by: Claudio Pedrini
Mathematics Subject Classification ID: 14F42
zbMATH DE Number: 5960705
zbMATH Keywords: Beilinson-Lichtenbaum conjecture
zbMATH Keywords: motivic cohomology operations
zbMATH Keywords: Rost motive
MaRDI profile type: Publication
OpenAlex ID: W1969256763
Wikidata QID: Q29013814
arXiv ID: 0805.4430
cites work: The \(K\)-theory of fields in characteristic \(p\)
cites work: The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky
cites work: Massey Higher Products
cites work: Lois de groupes et analyseurs
cites work: The additivity of traces in triangulated categories
cites work: The Steenrod algebra and its dual
cites work: Norm varieties
cites work: Q4513080
cites work: Reduced power operations in motivic cohomology
cites work: Motivic cohomology with \(\mathbb Z/2\)-coefficients
cites work: Simplicial radditive functors
cites work: Motives over simplicial schemes
cites work: Motivic Eilenberg-MacLane spaces
cites work: The norm residue isomorphism theorem
DOI: 10.4007/ANNALS.2011.174.1.11
description: scientific article