Generalized étale cohomology theories (Q5961533)

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 compulsory reading for all who want to be au fait with current trends in algebraic \(K\)-theory!