Submission date: 22 May 2020

Abstract:

We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field K based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group Γ_∞, where Γ denotes the value group of K. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of Γ_∞. In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide vanishing bounds in each case. Due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, as an application, we recover and extend results on the topological cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

Mathematics Subject Classification: Primary 55N30, 12J25, 03C98, 03C64 Secondary 14G22, 14T05

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Full text arXiv 2005.11229: pdf, ps.