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

TL;DR

This paper studies the structure of restriction and corestriction maps in etale cohomology for certain morphisms of schemes, especially quadratic Galois coverings, and explores applications to the Brauer group comparison.

Contribution

It constructs new maps linking kernels and cokernels of restriction and corestriction in etale cohomology for specific morphisms, providing insights into their algebraic structure.

Findings

Identifies kernels and cokernels of key cohomological maps for quadratic Galois coverings.

Establishes relations between cohomological invariants of schemes and their covers.

Provides applications to comparing Brauer groups of schemes and their quadratic Galois covers.

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.