Reduced power operations in motivic cohomology
type: Item
instance of: scholarly article
title: Reduced power operations in motivic cohomology
zbMATH Open document ID: 1057.14027
published in: Publications Mathématiques
publication date: 2004-10-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0107109
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104196
review text: Cohomology operations are very familiar from algebraic topology, where they form a basic toolkit of great power and applicability. Their theory has been extensively developed by N. Steenrod, J. Milnor, and others some decades ago [\textit{N. E. Steenrod}, Cohomology. Ann. Math. Stud. 50 (1962; Zbl 0102.38104)], later on also in the framework of \(K\)-theory. In the paper under review, the author develops a theory of cohomology operations in the motivic cohomology of smooth simplicial schemes over a perfect field \(k\) with coefficients in \(\mathbb{Z}/\ell\), where \(\ell\) is a prime number different from the characteristic of the field \(k\). In this context, he constructs the so-called reduced power operations in motivic cohomology, establishes the motivic analogues of the classical Cartan formulas and the Adem relations, describes explicitely the subalgebra generated by these reduced power operations \(P^i\) in the algebra of all (bistable) operations in the motivic cohomology, characterizes an analogue of the Bockstein operator and analyzes the multiplication by the motivic cohomology classes of \(\text{Spec}(k)\). This includes a thorough comparison of the usual topological Steenrod algebra and the motivic Steenrod algebra, the construction of the Thom isomorphism in motivic cohomology, the description of Euler classes of vector bundles in this framework, the analysis of the dual of the motivic Steenrod algebra, and the interrelation between motivic cohomology operations and characteristic classes. This highly original and pioneering program is carried out in the fourteen sections of the present paper, with a wealth of ingenious new ideas and novel, extremely subtle and fundamental constructions. Whereas the introduction of the motivic reduced power operations via the so-called total power operation follows the classical approach in topology (à la Steenrod-Epstein [loc. cit.]), the author's construction of the motivic total power operation is completely new an…
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 19E15
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 55S05
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14F45
zbMATH DE Number: 2111676
zbMATH Keywords: cohomology operations
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: vector bundles
zbMATH Keywords: characteristic classes
author: Vladimir Voevodsky
reviewed by: Werner Kleinert
MaRDI profile type: Publication
OpenAlex ID: W2045667896
Wikidata QID: Q29540580
arXiv ID: math/0107109
cites work: Q4309177
cites work: Q5485722
cites work: The Steenrod algebra and its dual
cites work: Algebraic \(K\)-theory and quadratic forms. With an appendix by J. Tate
cites work: \(\mathbb{A}^1\)-homotopy theory of schemes
cites work: Q4952444
cites work: Q4513080
cites work: Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic.
cites work: Motivic cohomology with \(\mathbb Z/2\)-coefficients
cites work: Cycles, Transfers, and Motivic Homology Theories. (AM-143)
DOI: 10.1007/S10240-003-0009-Z
description: scientific article