Automorphism Groups of Quasi-galois Closed Arithmetic Schemes

Abstract

Assume that $X$ and $Y$ are arithmetic schemes, i.e., integral schemes of finite types over $Spec(\mathbb{Z})$. Then $X$ is said to be quasi-galois closed over $Y$ if $X$ has a unique conjugate over $Y$ in some certain algebraically closed field, where the conjugate of $X$ over $Y$ is defined in an evident manner. Now suppose that $\varphi :X\to Y$ is a surjective morphism of finite type such that $X$ is quasi-galois closed over $Y$. In this paper the main theorem says that the function field $k(X)$ is canonically a Galois extension of $k(Y)$ and the automorphism group $Aut(X/Y)$ is isomorphic to the Galois group $Gal(k(X)/k(Y))$; in particular, $\varphi$ must be affine. Moreover, let $\dim X=\dim Y$. Then $X$ is a pseudo-galois cover of $Y$ in the sense of Suslin-Voevodsky.

Sharpened the main result; Added three references; Removed minor typos. 29 pages

Cited by in corpus (1)

• A short note on quasi-galois closed and pseudo-galois covers