Astérisque 2004; 257 pp; softcover Number: 293 ISBN10: 2856291546 ISBN13: 9782856291542 List Price: US$82 Individual Members: US$73.80 Order Code: AST/293
 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 quasicoherent \(\mathcal {O_X}\otimes _{\mathbb F_q}{\Lambda }\)modules equipped with a "unit" Frobenius structure. The authors show that the two corresponding derived categories are antiequivalent as triangulated categories, and that this antiequivalence 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. Readership Graduate students and research mathematicians interested in algebra and algebraic geometry. Table of Contents  General introduction
 Introduction to §§112: \(\mathcal{O}_{F,X}\)modules
 Notation and conventions
 \(\mathcal {O}_{F^r}^\Lambda\)modules
 Pullbacks of \(\mathcal {O}_{F^r}^\Lambda\)modules
 Pushforwards of \(\mathcal {O}_{F^r}^\Lambda\)modules
 Relations between \(f_+\) and \(f^!\)
 Unit \(\mathcal {O}_{F^r}^\Lambda\)modules
 Locally finitely generated unit \(\mathcal {O}_{F^r}^\Lambda\)modules
 \(\mathcal {O}_{F^r}^\Lambda\)modules on the étale site
 \(\Lambda\)sheaves on the étale site
 The functor Sol\(_{ét}\)
 The functor M\(_{ét}\)
 The RiemannHilbert correspondence for unit \(\mathcal {O}_{F,X}\)modules
 \(L\)Functions for unit \(F^r\)modules
 Introduction to §§1317: \(\mathcal {D}_{F,X}\)modules
 \(\mathcal {D}_{F,X}^{(u)}\)modules
 Direct and inverse images for \(\mathcal {D}_{F,X}^{(u)}\)modules
 Unit \(\mathcal {D}_{F,X}\)modules
 The RiemannHilbert correspondence for unit \(\mathcal {D}_{F,X}\)modules
 An equivalence of derived categories
 Appendix A: Duality and the Cartier operator
 Appendix B: Homological algebra
 Bibliography
