TOPOLOGY ON COHOMOLOGY OF LOCAL FIELDS

Kęstutis Česnavičius

doi:10.1017/fms.2015.18

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.

Highlights

The study of cohomology groups H n(k, G) for commutative finite k-group schemes G is facilitated by Tate local duality: H n(k, G) and H 2−n(k, G D) are Pontryagin duals, (‡)
The aim of this paper is to present a new way to define these topologies
For convenience of a reader not interested in generalities, we summarize our findings in the case of commutative finite type group schemes G over local fields k

Summary

Introduction

Let h : R → R be a continuous homomorphism between Henselian local topological rings that satisfy (α)–(β) and are etale-open, and let X , Y be locally of finite type R-algebraic stacks such that ∆X /R has a separated R-fiber over the closed point of Spec R. In Propositions 2.14–2.16 we record several situations in which morphisms of algebraic stacks (for example, of schemes) induce closed maps on R-points These results will be useful in Section 4; see the proof of Proposition 4.4. Let R be a local topological ring that is Hausdorff and satisfies (α)–(β), and let X be a locally of finite type R-algebraic stack such that ∆X /R has a separated R-fiber over the closed point of Spec R. In case (10), this follows from Proposition 2.15(c) (supplemented by [SP, 04XS] or by [LMB00, 4.2] again)

Topologies on cohomology sets via classifying stacks

Topological properties of maps in cohomology sequences