Comparison of Kummer logarithmic topologies with classical topologies

TL;DR

This paper compares Kummer flat and etale cohomology with classical cohomology for various group schemes, focusing on algebraic tori, and includes specific computations for certain base log schemes.

Contribution

It provides a detailed comparison between Kummer and classical topologies, highlighting differences and computations for algebraic tori and special base schemes.

Findings

Kummer flat cohomology differs from classical flat cohomology in specific cases.

Explicit computations for algebraic tori in Kummer flat topology.

Insights into the behavior of cohomology with logarithmic multiplicative groups.

Abstract

We compare the Kummer flat (resp. Kummer etale) cohomology with the flat (resp. etale) cohomology with coefficients in smooth commutative group schemes, finite flat group schemes and the logarithmic multiplicative group of Kato. We will be particularly interested in the case of algebraic tori in the Kummer flat topology. We also make some computations for certain special cases of the base log scheme.