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 H n (k, G) for commutative finite type k-group schemes G. These 'Čech topologies', defined usingČech 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 BG 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 theČech topologies. With this as the key input, we prove that theČech and the classifying stack topologies actually agree. The expected properties of theČech topologies follow, and these properties streamline a number of arithmetic duality proofs given elsewhere.

p-Selmer growth in extensions of degree p

“…The crucial topological input is closedness and discreteness of the image of a certain global‐to‐local pullback map. The analysis of this map in § 2 rests in part on the results of and leads to several improvements to the literature on arithmetic duality in positive characteristic, notably to [, § 4] and . To be able to simultaneously prove the growth of class groups mentioned in Remark 1.5, in § 3 we present a general framework for Selmer groups that extends the framework of Selmer structures of Mazur and Rubin to arbitrary global fields (in positive characteristic the unramified subgroups that play the decisive role in Selmer structures tend to be too small).…”

“…As mentioned in § 1.9, topology carried by cohomology groups of local fields will play an important role in treating all at once. This topology is always taken to be the one defined in [, 3.1–3.2]. To avoid cluttering the proofs with repetitive citations, in § 1.11 we gather the main topological properties that we will need.…”

“…Fix a place of , a commutative finite ‐group scheme , and an . By [, 3.5 (c), 3.6–3.8], is a locally compact Hausdorff abelian topological group that is discrete if . By [, 3.5 (b)], if is étale (in particular, if ), then is also discrete.…”

“…By [, 3.5 (c), 3.6–3.8], is a locally compact Hausdorff abelian topological group that is discrete if . By [, 3.5 (b)], if is étale (in particular, if ), then is also discrete. By [, 3.10], if and is a commutative finite flat ‐model of , then the pullback map identifies with a compact open subgroup of (if , then by [, 3.4]).…”

“…By [, 3.5 (b)], if is étale (in particular, if ), then is also discrete. By [, 3.10], if and is a commutative finite flat ‐model of , then the pullback map identifies with a compact open subgroup of (if , then by [, 3.4]). By [, 4.2], if fits into an exact sequence of commutative finite ‐group schemes, then the maps in the resulting cohomology sequence are continuous.…”