Let be a separated morphism of adic spaces of finite type over a non-archimedean field with affinoid and of dimension , let be a locally closed constructible subset of and let be the morphism of pseudo-adic spaces induced by . Let be a noetherian torsion ring with torsion prime to the characteristic of the residue field of the valuation ring of and let be a constant -module of finite type on . There is a natural class of -modules on generated by the constructible -modules and the Zariski-constructible -modules. We show that, for every , the higher direct image sheaf with proper support is generically constructible, and if is locally algebraic, is an element of . As an application we obtain a comparison isomorphism for the -adic cohomology of a separated scheme of finite type over and its associated adic space.
Soit un morphisme séparé d’espaces adiques de type fini sur un corps non archimédien avec affinoïde et de dimension . Soit un sous-ensemble constructible localement fermé dans et soit le morphisme d’espaces pseudo-adiques induit de . Soit un anneau noethérien de torsion première à la caractéristique résiduelle de et soit un faisceau de -modules localement constant de type fini sur . Il y a une classe naturelle des faisceaux de -modules sur engendrée par des faisceaux de -modules constructibles et des faisceaux de -modules Zariski-constructibles. Nous montrons que le faisceau image directe à support propre est génériquement constructible, et si est localement algébrique, est un élément de . En conséquence, on obtient un théorème de comparaison entre cohomologie -adique d’un schéma séparé de type fini sur et de l’espace adique associé.
Keywords: Rigid analytic spaces, adic spaces, compactly supported cohomology
Mot clés : espace analytique rigide, espace adique, cohomologie à support compact
Huber, Roland 1
@article{AIF_2007__57_3_973_0, author = {Huber, Roland}, title = {A finiteness result for the compactly supported cohomology of rigid analytic varieties, {II}}, journal = {Annales de l'Institut Fourier}, pages = {973--1017}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {3}, year = {2007}, doi = {10.5802/aif.2283}, mrnumber = {2336836}, zbl = {1146.14015}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2283/} }
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 SP - 973 EP - 1017 VL - 57 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2283/ DO - 10.5802/aif.2283 LA - en ID - AIF_2007__57_3_973_0 ER -
%0 Journal Article %A Huber, Roland %T A finiteness result for the compactly supported cohomology of rigid analytic varieties, II %J Annales de l'Institut Fourier %D 2007 %P 973-1017 %V 57 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2283/ %R 10.5802/aif.2283 %G en %F AIF_2007__57_3_973_0
Huber, Roland. A finiteness result for the compactly supported cohomology of rigid analytic varieties, II. Annales de l'Institut Fourier, Volume 57 (2007) no. 3, pp. 973-1017. doi : 10.5802/aif.2283. https://aif.centre-mersenne.org/articles/10.5802/aif.2283/
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