**Astérisque**

Volume: 293;
2004;
257 pp;
Softcover

MSC: Primary 14;
Secondary 13

**Print ISBN: 978-2-85629-154-2
Product Code: AST/293**

List Price: $82.00

Individual Member Price: $73.80

# The Riemann-Hilbert Correspondence for Unit $F$-crystals

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*Matthew Emerton; Mark Kisin*

A publication of the Société Mathématique de France

Let \(\mathbb {F}_q\) denote the finite field of order \(q\) (a power of a prime \(p\)), let \(X\) be a smooth scheme over a field \(k\) containing \(\mathbb {F}_q\), and let \(\Lambda \) be a finite \(\mathbb {F}_q\)-algebra. We study the relationship between constructible \(\Lambda \)-sheaves on the étale site of \(X\), and a certain class of quasi-coherent \(\mathcal {O_X}\otimes _{\mathbb F_q}{\Lambda }\)-modules equipped with a "unit" Frobenius structure. The authors show that the two corresponding derived categories are anti-equivalent as triangulated categories, and that this anti-equivalence is compatible with direct and inverse images, tensor products, and certain other operations.

They also obtain analogous results relating complexes of constructible \(\mathbb {Z}/p^n\mathbb {Z}\)-sheaves on smooth \(W_n(k)\)-schemes, and complexes of Berthelot's arithmetic \(\mathcal {D}\)-modules, equipped with a unit Frobenius.

The volume is suitable for graduate students and researchers interested in algebra and algebraic geometry.

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

## The Riemann-Hilbert Correspondence for Unit $F$-crystals

#### Readership

Graduate students and research mathematicians interested in algebra and algebraic geometry.