Topology on cohomology of local fields

TL;DR

This paper introduces a new topology on the cohomology groups of local fields using classifying stacks, simplifying the proof of properties needed for arithmetic duality theorems in positive characteristic.

Contribution

It proposes a novel approach to topologize cohomology groups via classifying stacks, replacing ch topologies and establishing their equivalence.

Findings

Classifying stack topologies satisfy expected properties of ch topologies.

The new topology simplifies proofs of arithmetic duality theorems.

The approach confirms the compatibility of topologies for cohomology groups over local fields.

Abstract

Arithmetic duality theorems over a local field $k$ are delicate to prove if $\mathrm{char} k > 0$. In this case, the proofs often exploit topologies carried by the cohomology groups $H^n(k, G)$ for commutative finite type $k$-group schemes $G$. These "\v{C}ech topologies", defined using \v{C}ech cohomology, are impractical due to the lack of proofs of their basic properties, such as continuity of connecting maps in long exact sequences. We propose another way to topologize $H^n(k, G)$: in the key case $n = 1$, identify $H^1(k, G)$ with the set of isomorphism classes of objects of the groupoid of $k$-points of the classifying stack $\mathbf{B} G$ and invoke Moret-Bailly's general method of topologizing $k$-points of locally of finite type $k$-algebraic stacks. Geometric arguments prove that these "classifying stack topologies" enjoy the properties expected from the \v{C}ech topologies. With this as the key input, we prove that the \v{C}ech and the classifying stack topologies actually agree. The expected properties of the \v{C}ech topologies follow, which streamlines a number of arithmetic duality proofs given elsewhere.