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arXiv · 2011.09880

\'Etale cohomology of algebraizable rigid analytic varieties via nearby cycles over general bases

Hiroki Kato
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Abstract

We prove a finiteness theorem and a comparison theorem in the theory of \'etale cohomology of rigid analytic varieties. By a result of Huber, for a quasi-compact separated morphism of rigid analytic varieties with target being of dimension $\le1$, the compactly supported higher direct image preserves quasi-constructibility. Though the analogous statement for morphisms with higher dimensional target fails in general, we prove that, in the algebraizable case, it holds after replacing the target with a modification. We deduce it from a known finiteness result in the theory of nearby cycles over general bases and a new comparison result, which gives an identification of the compactly supported higher direct image sheaves, up to modification of the target, in terms of nearby cycles over general bases.

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Hiroki Kato. 2020-11-19. \'Etale cohomology of algebraizable rigid analytic varieties via nearby cycles over general bases. https://arxiv.org/abs/2011.09880

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