\(\mathbb A^1\)-algebraic topology over a field
type: Item
instance of: scholarly article
title: \(\mathbb A^1\)-algebraic topology over a field
zbMATH Open document ID: 1263.14003
DOI: 10.1007/978-3-642-29514-0
author: Fabien Morel
published in: Lecture Notes in Mathematics
publication date: 2012-05-14T00:00:00Z
review text: This book deals with \(\mathbb{A}^1\)-homotopy theory over a base field, i.e., with the natural homotopy theory associated to the category of smooth varieties over a field, in which the affine line is imposed to be contractible. It is a natural sequel to to the paper by \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)]. A main technical result contained in this text is the precise nature of the \(\mathbb{A}^1\)-homotopy sheaves of \(\mathbb{A}^1\)-connected pointed spaces. Using these results the author introduces the \(\mathbb{A}^1\)-homology sheaves of a space and then proves several results, which are the analogues of classical theorems in algebraic topology such as the Brouwer degree theorem and Hurewicz theorem. Let \(Sm_k\) be the category of smooth schemes over a field \(k\). A sheaf of groups \({\mathcal G}\) on \(Sm_k\) in the Nisnevich topology is said to be strongly \(\mathbb{A}^1\)-invariant if for any \(X\in Sm_k\) the map \[ H^i_{Nis}(X,{\mathcal G})\to H^i_{Nis}(X\times \mathbb{A}^1,{\mathcal G}) \] induced by the projection \(\mathbb{A}^1\times X\to X\) is a bijection , for \(i= 0,1\). \({\mathcal G}\) is said to be strictly \(\mathbb{A}^1\)-invariant if the above map is a bijection for all \(i\in\mathbb{N}\). A basic example of strictly \(\mathbb{A}^1\)-invariant sheaves are Voevodsky's \(\mathbb{A}^1\)-invariant sheaves with transfers. The notion of strong \(\mathbb{A}^1\)-invariance given here is new: there are important examples of non-commutative strongly \(\mathbb{A}^1\)-invariant sheaves groups , e.g., \(\pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)\). The computation given in 7.3 of the book \(\pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)\) shows that there is a central extension of sheaves of groups \[ 0\to \mathbb{K}^{MW}_2\to \pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)\to \mathbb{G}_m\to 0, \] where, for a field \(F\), \(K^{MW}_*(F)\) is the Milnor-Witt \(K\)-theory of \(F\) and \(\mathbb{K}^{MW}_…
reviewed by: Claudio Pedrini
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14F42
zbMATH DE Number: 6034616
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-29514-0
OpenAlex ID: W573948285
description: scientific article; zbMATH DE number 6034616