Generalized étale cohomology theories
type: Item
instance of: scholarly article
title: Generalized étale cohomology theories
zbMATH Open document ID: 0868.19003
author: J. F. Jardine
published in: Progress in Mathematics
publication date: 1997-02-25T00:00:00Z
review text: A generalized étale cohomology theory is a representable cohomology theory for presheaves of spectra on an étale site of an algebraic variety. These cohomology theories simultaneously generalize the homotopy-theoretic cohomologies of algebraic topology and the algebraic theories (for example: étale and crystalline) of Grothendieck. Consequently this volume, in developing the techniques of the subject, introduces the reader to the stable homotopy category of simplicial presheaves. This is an extremely delicate development, obstructed by the need for coherent constructions involving very ``large'' objects such as limits of Čech constructions involving presheaves of spectra. The development of an adequate theory, particularly in respect of its applications to algebraic \(K\)-theory, was held up by difficulties with smash-products of spectra and with transfer constructions. This book provides the user with the first complete account which is sensitive enough to be compatible with the sort of closed model category necessary in \(K\)-theory applications [i.e., the closed model structure of \textit{A. K. Bousfield} and \textit{E. M. Friedlander}, Lect. Notes Math. 658, 80-130 (1978; Zbl 0405.55021)]. As an application of the techniques the author gives proofs of the descent theorems of \textit{R. W. Thomason} and \textit{Y. A. Nisnevich}. In particular, this implies the celebrated result of \textit{R. W. Thomason}, ``Algebraic \(K\)-theory and étale cohomology'', Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 437-552 (1985; Zbl 0596.14012)] which identifies \(\text{mod }p\) \(K\)-theory, after being inflicted with Bott periodicity in the manner introduced by the reviewer, with \(\text{mod }p\) étale \(K\)-theory. The book concludes with a discussion of the Lichtenbaum-Quillen conjecture (an approximation to Thomason's theorem without Bott periodicity). The recent proof of this conjecture, by \textit{V. Voevodsky}, when \(p=2\) for fields of characteristic zero makes this volume …
reviewed by: Victor P. Snaith
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 18-02
Mathematics Subject Classification ID: 55-02
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 18F20
zbMATH DE Number: 981539
zbMATH Keywords: étale site of algebraic variety
zbMATH Keywords: generalized étale cohomology
zbMATH Keywords: presheaves of spectra
zbMATH Keywords: closed model category
zbMATH Keywords: Lichtenbaum-Quillen conjecture
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 981539