Details

Autor(en) / Beteiligte

• Cubides Kovacsics, Pablo
• Edmundo, Mário J.
• Ye, Jinhe

Titel

Cohomology of algebraic varieties over non-archimedean fields

Ist Teil von

• Forum of mathematics. Sigma, 2022-01, Vol.10, Article e94

Ort / Verlag

Cambridge, UK: Cambridge University Press

Erscheinungsjahr

2022

Quelle

EZB Electronic Journals Library

Beschreibungen/Notizen

• We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean 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 $\Gamma _{\infty }$ , where $\Gamma $ 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 $\Gamma _{\infty }$ . In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the singular cohomology of the analytification of algebraic varieties concerning finiteness and invariance.