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>0, let Res_{G}^{(r)}: H^{r}(S_{et},G)---> H^{r}(S'_{et},G) and Cores_{G}^{(r)}: H^{r}(S'_{et},G)---> H^{r}(S_{et},G) be, respectively, the restriction and corestriction maps in etale cohomology induced by f. For certain pairs (f, G), we construct maps \alpha_{r}: Ker Cores_{G}^{(r)}---> Coker Res_{G}^{(r)} and \beta_{r}: Coker Res_{G}^{(r)}---> Ker Cores_{G}^{(r)} such that \alpha_{r}o\beta_{r}=\beta_{r}o\alpha_{r}=n. In the simplest nontrivial case, namely when f is a quadratic Galois covering, we identify the kernel and cokernel of \beta_{r} with the kernel and cokernel of another map Coker Cores_{G}^{(r-1)}---> KerRes_{G}^{(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.