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

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:224

Article

Gonzalez-Aviles, Cristian D.¹ 

[1] Univ La Serena, Dept Matemat, La Serena, Chile

关键词: Restriction map;    Norm map;    Quadratic Galois cover;    Relative ideal class group;    Tate-Shafarevich group;    Relative Brauer group;   

DOI  :  10.1016/j.jpaa.2019.05.022

来源: Elsevier

【 摘 要 】

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 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 )circle beta(r )= beta(r )circle alpha(r )= n. 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. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free