Seminar: Non-archimedean geometry

Amine Koubaa (Universität Frankfurt)

Abstract:
Given a regular scheme $X$ and a normal crossing divisor $D$ one may concider two different cohomology groups.
The first one is the log étale cohomology developed by Illusie, K. Kato and many others: We associate a logarithmic structure $M$ to $X$ and define the log étale site over $(X,M)$.The second one is the tame cohomology developed by Hübner and Schmidt. Here we consider the tame site over the discretely ringed adic space $Spa(X\backslash D,X)$. Tame morphisms are those which are étale and induce at most tamely ramified extension on the valuations.We construct a comparison morphism between these cohomology groups and prove that they are equal for schemes over $\mathbb{F}_p$ and locally constant finite sheaves once we assume resolution of singularities.“