Topology on cohomology of 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.