**Astérisque**

Volume: 371;
2015;
239 pp;
Softcover

MSC: Primary 11; 14;
**Print ISBN: 978-2-85629-807-7
Product Code: AST/371**

List Price: $96.00

Individual Member Price: $76.80

# Relative $p$-adic Hodge Theory: Foundations

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*Kiran S. Kedlaya; Ruochuan Liu*

A publication of the Société Mathématique de France

The authors describe a new approach to relative \(p\)-adic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean analytic geometry in the style of both Berkovich and Huber. They give a thorough development of \(\varphi\)-modules over a relative Robba ring associated to a perfect Banach ring of characteristic \(p\), including the relationship between these objects and étale \({\mathbb Z}_p\)-local systems and \({\mathbb Q}_p\)-local systems on the algebraic and analytic spaces associated to the base ring, and the relationship between (pro-)étale cohomology and \(\varphi\)-cohomology. They also make a critical link to mixed characteristic by exhibiting an equivalence of tensor categories between the finite étale algebras over an arbitrary perfect Banach algebra over a nontrivially normed complete field of characteristic \(p\) and the finite étale algebras over a corresponding Banach \({\mathbb Q}_p\)-algebra. This recovers the homeomorphism between the absolute Galois groups of \({\mathbb F}_{p}((\pi))\) and \({\mathbb Q}_{p}(\mu_{p}\infty)\) given by the field of norms construction of Fontaine and Wintenberger, as well as generalizations considered by Andreatta, Brinon, Faltings, Gabber, Ramero, Scholl, and, most recently, Scholze.

Using Huber's formalism of adic spaces and Scholze's formalism of perfectoid spaces, the authors globalize the constructions to give several descriptions of the étale local systems on analytic spaces over \(p\)-adic fields. One of these descriptions uses a relative version of the Fargues–Fontaine curve.

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

## Relative $p$-adic Hodge Theory: Foundations

#### Readership

Graduate students and research mathematicians.