正态方案的艾克斯-格罗登第定理的逆定理

Lázaro O. Rodríguez Díaz
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引用次数: 0

摘要

给定一个正常的不可还原的诺特和简单连接方案的\'etale 内形变,我们证明如果这个内形变是可射的,那么它就是可射的。这个证明是基于刘氏构造的一个由射出\'etale态构成的伽罗瓦盖。如果我们放弃投射性假设,假设内态性是分离的,那么我们就可以证明诱导的域扩展是伽罗瓦的。在仿射空间在特征为零的代数闭域上的\'etale内形变的情况下,坎贝尔定理意味着弹射性假设是多余的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A converse of Ax-Grothendieck theorem for étale endomorphisms of normal schemes
Given an \'etale endomorphism of a normal irreducible Noetherian and simply connected scheme, we prove that if the endomorphism is surjective then it is injective. The proof is based on Liu's construction of a Galois cover out of a surjective \'etale morphism. If we give up of the surjectivity hypothesis and suppose the endomorphism is separated, then we prove that the induced field extension is Galois. In the case of an \'etale endomorphism of the affine space over an algebraically closed field of characteristic zero, Campbell's theorem implies that the assumption of surjectivity is superfluous.
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