\(G\)-endomorphisms of affine \(G\)-varieties which induce automorphisms of the invariant subrings of the coordinate rings
type: Item
instance of: scholarly article
title: \(G\)-endomorphisms of affine \(G\)-varieties which induce automorphisms of the invariant subrings of the coordinate rings
zbMATH Open document ID: 1112.14069
author: K. Masuda
published in: Journal of Algebra
publication date: 2007-01-26T00:00:00Z
review text: Let \(G\) be an algebraic group defined over \(\mathbb{C},\) and let \(X\) be an affine \(G\)-variety, \(X=\operatorname{Spec}R.\) A \(G\)-equivariant endomorphism, or \(G\)-endomorphism, \(\varphi\) of \(X\) induces a \(G\)-endomorphism \(f\) of \(R\) as well as an endomorphism \(\bar{f}\) of the invariant subalgebra \(R^{G}.\) It is known that if \(\varphi\) is an automorphism then so is \(\bar{f}.\) This paper is concerned with the converse. Suppose \(\bar{f}\) is an endomorphism of \(R^{G}\) arising from a \(G\)-endomorphism \(\varphi\) on \(X\). If \(\bar{f}\) is an automorphism, under what conditions is \(\varphi\) an automorphism? Suppose that \(G\) is unipotent and \(X\) is factorial. Then it is shown that if \(\bar{f}\) is an automorphism, then so it \(\varphi.\) This is accomplished by proving it in the case where \(G=G_{a},\) the additive group, where the \(G\)-action can be described in terms of a derivation on \(R\). The result for unipotent \(G\) follows by considering a chain of closed normal subgroups of \(G\) whose quotients are all isomorphic to \(G_{a}\). Now suppose that \(G\) is reductive. Then \(R^{G}\) is finitely generated and \(X//G=\operatorname{Spec}R^{G}.\) We will denote the induced endomorphism on \(X//G\) by \(\overline{\varphi}.\) Let \(\pi\) be the quotient map \(X\rightarrow X//G.\) Suppose \(\overline{\varphi}\) is an automorphism and there exists a closed set \(V\subset X//G\) with codim\(_{X}\pi^{-1}\left( V\right) \geq2.\) If each fiber of \(\pi\) consists of a single closed orbit, then \(\varphi\) is a \(G\) -automorphism. Essential to this result is a theorem by \textit{S. Kaliman} [Proc. Am. Math. Soc. 133, No. 4, 975--977 (2005; Zbl 1060.14002)] which states that if the restriction \(\overline{\varphi}\) of an endomorphism \(\varphi\) of an affine algebraic variety to an open set whose complement has codimension at least 2 is injective, then \(\phi\) is an automorphism. Furthermore, suppose \(X\) is normal. Then, using the n…
reviewed by: Alan Koch
Mathematics Subject Classification ID: 14R20
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 5118806
zbMATH Keywords: \(G\)-endomorphism
zbMATH Keywords: affine \(G\)-variety
zbMATH Keywords: algebraic quotient
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.09.004
OpenAlex ID: W1976918217
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DOI: 10.1016/J.JALGEBRA.2006.09.004
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description: scientific article; zbMATH DE number 5118806