BIBLIOS

Document type
Journal articles

Document subtype
Full paper

Title
Cohomology of algebraic varieties over non-archimedean fields

Participants in the publication
Pablo Cubides Kovacsics (Author)
Mário J. Edmundo (Author)
Dep. Matemática
CMAFcIO
Jinhe Ye (Author)

Summary
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 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 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.


Institution
FACULDADE DE CIÊNCIAS DA UNIVERSIDADE DE LISBOA

Where published
Forum of Mathematics, Sigma

Publication Identifiers
ISSN - 2050-5094

Publisher
Cambridge University Press (CUP)


Document Identifiers
DOI - https://doi.org/10.1017/fms.2022.84
URL - http://dx.doi.org/10.1017/fms.2022.84

Rankings
Web Of Science Q1 (2021) - 1.389 - MATHEMATICS - SCIE
SCIMAGO Q1 (2021) - 1096 - Geometry and Topology