TOPOLOGY ON COHOMOLOGY OF LOCAL FIELDS

Abstract

Arithmetic duality theorems over a local field$k$are delicate to prove if$\text{char}\,k>0$. 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 ‘Čech topologies’, defined using Čech 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 when$n=1$, 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 Čech topologies. With this as the key input, we prove that the Čech and the classifying stack topologies actually agree. The expected properties of the Čech topologies follow, and these properties streamline a number of arithmetic duality proofs given elsewhere.