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.

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

References in corpus (5)

• On the existence of geometric models for function fields in several variables
• On the arithmetic fundamental groups
• Affine Structures on a Ringed Space and Schemes
• On the etale fundamental groups of arithmetic schemes
• Nonhomeomorphic conjugates of connected Shimura varieties

Cited by in corpus (12)

• On the existence of geometric models for function fields in several variables
• On the arithmetic fundamental groups
• On the étale fundamental groups of arithmetic schemes, revised
• On the algebraic fundamental groups
• On the etale fundamental groups of arithmetic schemes
• Notes on the quasi-galois closed schemes
• On the section conjecture of Grothendieck
• On the transcendental Galois extensions
• Notes on the section conjecture of Grothendieck
• On the arithmetic Galois covers of higher relative dimensions
• Algebraic Anabelian Functors
• A short note on quasi-galois closed and pseudo-galois covers