Topology on cohomology of local fields
Abstract
Arithmetic duality theorems over a local field k are delicate to prove if char k > 0. In this case, the proofs often exploit topologies carried by the cohomology groups Hn(k, G) for commutative finite type k-group schemes G. These "Cech topologies", defined using Cech 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 Hn(k, G): in the key case n = 1, identify H1(k, G) with the set of isomorphism classes of objects of the groupoid of k-points of the classifying stack 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 Cech topologies. With this as the key input, we prove that the Cech and the classifying stack topologies actually agree. The expected properties of the Cech topologies follow, which streamlines a number of arithmetic duality proofs given elsewhere.