Cokernels of restriction maps and subgroups of norm one, with applications to quadratic Galois coverings

Abstract

Let f:S′→S be a finite and faithfully flat morphism of locally noetherian schemes of constant rank n and let G be a smooth, commutative and quasi-projective S-group scheme with connected fibers. For every r≥1, let ResG(r):Hr(Se´t,G)→Hr(Se´t′,G) and CoresG(r):Hr(Se´t′,G)→Hr(Se´t,G) be, respectively, the restriction and corestriction maps in étale cohomology induced by f. For certain pairs (f,G), we construct maps αr:KerCoresG(r)→CokerResG(r) and βr:CokerResG(r)→KerCoresG(r) such that αr∘βr=βr∘αr=n. In the simplest nontrivial case, namely when f is a quadratic Galois covering, we identify the kernel and cokernel of βr with the kernel and cokernel of another map CokerCoresG(r−1)→KerResG(r+1). We then discuss several applications, for example to the problem of comparing the (cohomological) Brauer group of a scheme S to that of a quadratic Galois cover S′ of S.