**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

AMS 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.

#### Readership

Graduate students and research mathematicians.