A finiteness result for the compactly supported cohomology of rigid analytic varieties, II
Annales de l'Institut Fourier, Volume 57 (2007) no. 3, p. 973-1017
Let h:XY 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 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 -adic cohomology of a separated scheme of finite type over k and its associated adic space.
Soit h:XY un morphisme séparé d’espaces adiques de type fini sur un corps non archimédien k avec Y affinoïde et de dimension 1. Soit L un sous-ensemble constructible localement fermé dans X et soit g:(X,L)Y le morphisme d’espaces pseudo-adiques induit de h. Soit A un anneau noethérien de torsion première à la caractéristique résiduelle de k et soit F un faisceau de A-modules localement constant de type fini sur (X,L) e ´t . Il y a une classe naturelle 𝒞(Y) des faisceaux de A-modules sur Y e ´t engendrée par des faisceaux de A-modules constructibles et des faisceaux de A-modules Zariski-constructibles. Nous montrons que le faisceau image directe à support propre R n g ! F est génériquement constructible, et si h est localement algébrique, R n g ! F est un élément de 𝒞(Y). En conséquence, on obtient un théorème de comparaison entre cohomologie -adique d’un schéma séparé de type fini sur k et de l’espace adique associé.
DOI : https://doi.org/10.5802/aif.2283
Classification:  14G22,  14F20
Keywords: Rigid analytic spaces, adic spaces, compactly supported cohomology
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {3},
     year = {2007},
     pages = {973-1017},
     doi = {10.5802/aif.2283},
     zbl = {1146.14015},
     mrnumber = {2336836},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/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/item/AIF_2007__57_3_973_0/

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