nLab Tate's acyclicity theorem

Statement

Let $X = Spec_{an}(A)$ be an affinoid space with affinoid algebra $A$.

Then for every finite cover of $X$ by affinoid domains, the corresponding Cech cohomology with coefficients in the structure sheaf $A$, or with coefficients in any finite Banach module over $A$, is concentrated in degree 0.

(Berkovich 09, fact 2.2.6)

References

• S. Bosch, U. Güntzer, Reinhold Remmert, section 8.2 of Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, 1984 (pdf)

• Vladimir Berkovich, Non-archimedean analytic spaces, lectures at the Advanced School on $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

• pdf

• Doosung Park, Affinoid domains, lecture notes (pdf)