Group actions of prime order on local normal rings.

*(English)*Zbl 1273.14094Let \(B\) 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 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 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 and stable models of smooth projective curves over fields of fractions of discrete valuation rings. They briefly explain the consequences of their result and conjecture in the final section.

A classical result of 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 and stable models of smooth projective curves over fields of fractions of discrete valuation rings. They briefly explain the consequences of their result and conjecture in the final section.

Reviewer: Jon Elmer (Aberdeen)