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Log-isocristaux surconvergents et holonomie

Part of:(Co)homology theory

Published online by Cambridge University Press:  24 September 2009

Daniel Caro
Daniel Caro*
Affiliation:
Laboratoire de Mathématiques Nicolas Oresme, Université de Caen, Campus 2, 14032 Caen Cedex, France (email: daniel.caro@math.unicaen.fr)
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Abstract

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Let đ’± be a complete discrete valuation ring of unequal characteristic with perfect residue field. Let $\X $ be a separated smooth formal đ’±-scheme, đ’” be a normal crossing divisor of $\X $, $\X ^\#:= (\X , \ZZ )$ be the induced formal log-scheme over đ’± and $u: \X ^\# \rightarrow \X $ be the canonical morphism. Let X and Z be the special fibers of $\X $ and đ’”, T be a divisor of X and ℰ be a log-isocrystal on $\X ^\#$ overconvergent along T, that is, a coherent left $\D ^\dag _{\X ^\#} (\hdag T) _{\Q }$-module, locally projective of finite type over $ \O _{\X } (\hdag T) _{\Q }$. We check the relative duality isomorphism: $u_{T,+} (\E ) \riso u_{T,!} (\E (\ZZ ))$. We prove the isomorphism $u_{T,+} (\E ) \riso \D ^\dag _{\X } (\hdag T) _{\Q } \otimes _{\D ^\dag _{\X ^\#} (\hdag T) _{\Q }} \E (\ZZ )$, which implies their holonomicity as $\D ^\dag _{\X } (\hdag T) _{\Q }$-modules. We obtain the canonical morphism ρℰ : uT,+(ℰ)→ℰ(†Z). When ℰ is moreover an isocrystal on $\X $ overconvergent along T, we prove that ρℰ is an isomorphism.

Keywords

arithmetical 𝒟-moduleslog-isocrystalsholonomicity

MSC classification

Secondary:14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials14F30: $p$-adic cohomology, crystalline cohomology

Information

Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 6 , November 2009, pp. 1465 - 1503
Copyright
Copyright © Foundation Compositio Mathematica 2009

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