A finiteness result for the compactly supported cohomology of rigid analytic varieties, II

Roland Huber[1]

  • [1] Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Gaussstr. 20 42097 Wuppertal (Germany)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 3, page 973-1017
  • ISSN: 0373-0956

Abstract

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Let h : X → Y be a separated morphism of adic spaces of finite type over a non-archimedean field k with Y affinoid and of dimension ≤ 1 , let L be a locally closed constructible subset of X and let g : ( X , L ) → Y be the morphism of pseudo-adic spaces induced by h . Let A be a noetherian torsion ring with torsion prime to the characteristic of the residue field of the valuation ring of k and let F be a constant A -module of finite type on ( X , L ) e ´ t . There is a natural class 𝒞 ( Y ) of A -modules on Y e ´ t generated by the constructible A -modules and the Zariski-constructible A -modules. We show that, for every n ∈ ℕ 0 , the higher direct image sheaf with proper support R n g ! F is generically constructible, and if h is locally algebraic, R n g ! F is an element of 𝒞 ( Y ) . As an application we obtain a comparison isomorphism for the ℓ -adic cohomology of a separated scheme of finite type over k and its associated adic space.

How to cite

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Huber, Roland. "A finiteness result for the compactly supported cohomology of rigid analytic varieties, II." Annales de l’institut Fourier 57.3 (2007): 973-1017. <http://eudml.org/doc/10248>.

@article{Huber2007,
abstract = {Let $h:X \rightarrow Y$ be a separated morphism of adic spaces of finite type over a non-archimedean field $k$ with $Y$ affinoid and of dimension $\le 1$, let $L$ be a locally closed constructible subset of $X$ and let $g:(X,L) \rightarrow Y$ be the morphism of pseudo-adic spaces induced by $h$. Let $A$ be a noetherian torsion ring with torsion prime to the characteristic of the residue field of the valuation ring of $k$ and let $F$ be a constant $A$-module of finite type on $(X,L)_\{\rm \acute\{e\}t\} $. There is a natural class $\mathscr\{C\} (Y)$ of $A$-modules on $Y_\{\rm \acute\{e\}t\}$ generated by the constructible $A$-modules and the Zariski-constructible $A$-modules. We show that, for every $n \in \mathbb\{N\} _0$, the higher direct image sheaf with proper support $R^ng_!F$ is generically constructible, and if $h$ is locally algebraic, $R^ng_!F$ is an element of $\mathscr\{C\} (Y)$. As an application we obtain a comparison isomorphism for the $\ell $-adic cohomology of a separated scheme of finite type over $k$ and its associated adic space.},
affiliation = {Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Gaussstr. 20 42097 Wuppertal (Germany)},
author = {Huber, Roland},
journal = {Annales de l’institut Fourier},
keywords = {Rigid analytic spaces; adic spaces; compactly supported cohomology; rigid analytic spaces},
language = {eng},
number = {3},
pages = {973-1017},
publisher = {Association des Annales de l’institut Fourier},
title = {A finiteness result for the compactly supported cohomology of rigid analytic varieties, II},
url = {http://eudml.org/doc/10248},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Huber, Roland
TI - A finiteness result for the compactly supported cohomology of rigid analytic varieties, II
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 3
SP - 973
EP - 1017
AB - Let $h:X \rightarrow Y$ be a separated morphism of adic spaces of finite type over a non-archimedean field $k$ with $Y$ affinoid and of dimension $\le 1$, let $L$ be a locally closed constructible subset of $X$ and let $g:(X,L) \rightarrow Y$ be the morphism of pseudo-adic spaces induced by $h$. Let $A$ be a noetherian torsion ring with torsion prime to the characteristic of the residue field of the valuation ring of $k$ and let $F$ be a constant $A$-module of finite type on $(X,L)_{\rm \acute{e}t} $. There is a natural class $\mathscr{C} (Y)$ of $A$-modules on $Y_{\rm \acute{e}t}$ generated by the constructible $A$-modules and the Zariski-constructible $A$-modules. We show that, for every $n \in \mathbb{N} _0$, the higher direct image sheaf with proper support $R^ng_!F$ is generically constructible, and if $h$ is locally algebraic, $R^ng_!F$ is an element of $\mathscr{C} (Y)$. As an application we obtain a comparison isomorphism for the $\ell $-adic cohomology of a separated scheme of finite type over $k$ and its associated adic space.
LA - eng
KW - Rigid analytic spaces; adic spaces; compactly supported cohomology; rigid analytic spaces
UR - http://eudml.org/doc/10248
ER -

References

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