**Astérisque**

Volume: 363;
2014;
619 pp;
Softcover

MSC: Primary 12; 13; 14; 18; 20;
**Print ISBN: 978-2-85629-790-2
Product Code: AST/363/364**

List Price: $210.00

Individual Member Price: $168.00

# Travaux de Gabber sur L’uniformisation Locale et la Cohomologie Étale des Schémas Quasi-Excellents

Share this page *Edited by *
*Luc Illusie; Yves Laszlo; Fabrice Orgogozo*

with the collaboration of F. Déglise, A. Morau, V. Pilloni, M. Raynaud, J. Riou, B. Stroh, M. Temkin, and W. Zheng

A publication of the Société Mathématique de France

The work of Ofer Gabber presented in this book can be roughly divided into two closely related parts: a geometric one and a cohomological one. The first part contains local uniformization theorems which state that any pair consisting of a quasi-excellent noetherian scheme and a nowhere dense closed subscheme becomes isomorphic, after localization by suitable étale morphisms and alterations, to a pair consisting of a regular scheme and a normal crossings divisor. These are local results, but their proofs have global theorems as corollaries, refining alteration theorems of de Jong for schemes of finite type over a field or a Dedekind ring. Techniques from logarithmic geometry and, concerning the finest results, canonical desingularization in characteristic zero, play a key role in the proofs.

In the second part, the book gives applications, with examples and counterexamples, to abelian finiteness theorems, as well as theorems on cohomological dimension and duality in étale cohomology over quasi-excellent schemes. In particular, Grothendieck's local duality conjecture is proved, and his absolute cohomological purity conjecture is proved by a new method. Non-abelian rigidity and finiteness results are also established in the final exposés.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Table of Contents

# Table of Contents

## Travaux de Gabber sur L'uniformisation Locale et la Cohomologie Etale des Schemas Quasi-Excellents

#### Readership

Graduate students and research mathematicians.