Group actions of prime order on local normal rings
type: Item
instance of: scholarly article
title: Group actions of prime order on local normal rings
zbMATH Open document ID: 1273.14094
author: Werner Lütkebohmert
author: Franz J. Király
published in: Algebra \& Number Theory
publication date: 2013-05-28T00:00:00Z
full work available at URL: p03.xhtml
review text: Let \(B\) be regular Noetherian normal local ring and let \(G\) be a group of local automorphisms of \(B\). Denote by \(A\) the subring of elements of \(B\) invariant under \(G\), and by \(m_B\) the maximal ideal of \(B\). We get an induced action on the vector space \(T:=m_B/m_B^2\), and \(\sigma \in G\) is called a \textit{pseudoreflection} if its rank on \(T\) is less than or equal to 1. Let \(k_A\), \(k_B\) denote the residue fields of \(A\) and \(B\) respectively. A classical result of \textit{J. P. Serre} [Colloque d'algèbre (Paris, 1967), Secrétariat mathématique, Paris, Exp. 8, 11 p. (1968; Zbl 0200.00002)] states that, provided the characteristic \(p\) of \(k_B\) does not divide the order of \(G\), then \(A\) is regular if and only if \(G\) is generated by pseudoreflections. Now assume \(G\) is divisible by \(p\). In that modular case, Serre [loc. cit.] proved the ``only if'' part of the above, but the ``if'' is known to fail in general. In this article the authors concentrate on the case where \(G\) is a cyclic group of prime order \(p\), and prove a version of the ``if'' part of the above. More precisely, they prove that if the augmentation ideal \(I_B\) of \(B\) is principal, then \(A\) is regular, and furthermore, that under the additional assumption that \(k_A \cong k_B\), \(I_B\) is principal if and only if the generator of \(G\) acts as a pseudoreflection. In the proof, the authors give an explicit description of \(B\) as an \(A\)-module, in terms of an element generating \(I_B\). If \(A\) is regular, then the result of Serre tells is us that \(G\) is generated by pseudoreflections, but it is not known whether \(I_B\) must be principal. The authors conjecture that this is the case when \(G\) is cyclic of prime order, and prove this when \(p=2\) or \(3\). Although the main result of this paper is invariant-theoretic, the authors main interest in the result is an application to algebraic geometry: they wish to study the relationship between the regular…
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 13A50
zbMATH DE Number: 6167113
zbMATH Keywords: algebraic geometry
zbMATH Keywords: commutative algebra
zbMATH Keywords: group actions
reviewed by: Jon Elmer
MaRDI profile type: Publication
OpenAlex ID: W2027113482
DOI: 10.2140/ANT.2013.7.63
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description: scientific article; zbMATH DE number 6167113