Differential operators on varieties with a quotient subvariety
type: Item
instance of: scholarly article
title: Differential operators on varieties with a quotient subvariety
zbMATH Open document ID: 0844.13016
author: Martin P. Holland
author: Rob C. Cannings
published in: Journal of Algebra
publication date: 1996-08-27T00:00:00Z
review text: The authors consider a commutative ring \(A\), without nilpotents and finitely generated over the algebraically closed field \(k\) of characteristic zero. A finite group \(G\) acts as \(k\)-algebra automorphism of \(A\) and stabilizes a radical ideal \(J\) of \(A\). The action is assumed to be generically without inertia; that is, \(G\) acts non-trivially and for every minimal prime ideal \(P\) of \(A\) \[ \{\sigma \in G : f - f^\sigma \in P, \;\forall f \in A\} = 1. \] Let \(B = A^G + J\). Let \({\mathcal D} (\cdot)\) denote the ring of differential operator functors. The point of the paper is to consider the relation of \({\mathcal D} (B)\) and \({\mathcal D} (A)\), and in particular when differential operators on \(B\) are restrictions of operators on \(A\). For \(A\) regular, the authors obtain the following theorem: Let \({\mathcal I} (J) = \{\theta \in {\mathcal D} (A) : \theta \cdot J \subseteq J\}\). Then \({\mathcal D} (B) = {\mathcal I} (J)^G + J {\mathcal D} (A)\) (and so \({\mathcal D} (B) \subset {\mathcal D} (A))\); and \(J{\mathcal D} (A)\) is the unique minimal non-zero ideal of \({\mathcal D} (B)\), with the quotient isomorphic to \({\mathcal D} (A/J)^G\). The paper also includes a number of results for the case when \(A\) is not regular, and some special cases of regular \(A\) for which more detailed results can be obtained, as well as some examples.
Mathematics Subject Classification ID: 13N10
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 16S32
zbMATH DE Number: 705003
zbMATH Keywords: invariants of finite group
zbMATH Keywords: automorphism
zbMATH Keywords: differential operators
reviewed by: Andy R. Magid
MaRDI profile type: Publication
full work available at URL: jabr.1994.1363
OpenAlex ID: W2070858122
DOI: 10.1006/JABR.1994.1363
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description: scientific article; zbMATH DE number 705003