Differential operators on varieties with a quotient subvariety
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.
scientific article; zbMATH DE number 705003
scientific article