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The Riemann-Hilbert Correspondence for Unit $$F$$-crystals
Matthew Emerton and Mark Kisin, Northwestern University, Evanston, IL
A publication of the Société Mathématique de France.
 Astérisque 2004; 257 pp; softcover Number: 293 ISBN-10: 2-85629-154-6 ISBN-13: 978-2-85629-154-2 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 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. Readership Graduate students and research mathematicians interested in algebra and algebraic geometry. Table of Contents General introduction Introduction to §§1-12: $$\mathcal{O}_{F,X}$$-modules Notation and conventions $$\mathcal {O}_{F^r}^\Lambda$$-modules Pull-backs of $$\mathcal {O}_{F^r}^\Lambda$$-modules Push-forwards 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 Riemann-Hilbert correspondence for unit $$\mathcal {O}_{F,X}$$-modules $$L$$-Functions for unit $$F^r$$-modules Introduction to §§13-17: $$\mathcal {D}_{F,X}$$-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 Riemann-Hilbert 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