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Let f: S' --> S be a finite and faithfully flat morphism of locallynoetherian schemes of constant rank n and let G be a smooth, commutative andquasi-projective S-group scheme with connected fibers. For every r>0, letRes_{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 andcorestriction 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 andcokernel of \beta_{r} with the kernel and cokernel of another map CokerCores_{G}^{(r-1)}---> KerRes_{G}^{(r+1)}. We then discuss several applications,for example to the problem of comparing the (cohomological) Brauer group of ascheme S to that of a quadratic Galois cover S' of S.

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