# Alexander Schmidt: Tame cohomology of schemes and adic spaces

**Time: **
Thu 2022-10-06 13.15 - 14.15

**Location: **
KTH, 3418

**Participating: **
Alexander Schmidt, Universität Heidelberg

Abstract: Étale cohomology with non-invertible coefficients has some unpleasant properties, e.g., it is not \(\mathbb{A}^1\)-homotopy invariant and for constructible coefficients the expected finiteness properties do not hold. To remedy this, we introduce the `tame site'. Tame cohomology coincides with étale cohomology for invertible coefficients but is better behaved in the general case. The associated higher tame homotopy groups hopefully have a better behaviour than the higher étale homotopy groups, which vanish for affine schemes in positive characteristic by a result of Achinger.

There are two approaches to the construction of a tame site. The first one considers the discretely ringed adic space Spa(*X*,*S*) associated with a scheme *X* over a base scheme *S*. In the category of adic spaces, it is natural to define tame morphisms by a tameness condition on residue field extensions. The resulting “adic tame site” Spa(*X*,*S*)_{t} has good local properties and it seems promising to develop technical machinery such as base change theorems and cohomological purity in this setting. The second approach works with étale morphisms of schemes and imposes a tameness condition on coverings. This “algebraic tame site” fits nicely into the framework of motivic cohomology theory as it sits in between the étale and the Nisnevich site.

In this talk we will explain the basic properties of tame cohomology and a comparison theorem between the tame cohomology of an *S*-scheme *X* and the tame cohomology of the associated adic space Spa(*X*,*S*) in the case of pure characteristic.

This is joint work with K. Hübner.