Trond Stølen Gustavsen

CONNECTIONS ON MODULES OVER SINGULARITIES OF FINITE CM REPRESENTATION TYPE

Abstract Let A be a commutative k -algebra, where k is an algebraically closed field of characteristic 0 , and let M be an A -module. We consider the following question: Under what conditions is it possible to find a connection ∇ : Der k ( A ) → End k ( M ) on M ? We consider the maximal Cohen–Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assume that the singularities have finite CM representation type. It is known that any MCM module over a simple singularity of dimension d ≤ 2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d ≥ 3 admits a connection if and only if it is free. Among singularities of finite CM representation type, we find examples of curves with MCM modules that do not admit connections, and threefolds with non-free MCM modules that admit connections. Let A be a singularity not necessarily of finite CM representation type, and consider the condition that A is a Gorenstein curve or a Q -Gorenstein singularity of dimension d ≥ 2 . We show that this condition is sufficient for the canonical module ω A to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if A is a monomial curve singularity, then the canonical module ω A admits an integrable connection if and only if A is Gorenstein.

Equivariant Lie-Rinehart cohomology

In this paper, we study Lie-Rinehart cohomology for quotients of singularities by finite groups, and interpret these cohomology groups in terms of integrable connection on modules.

Connections on Modules over Singularities of Finite and Tame CM Representation Type

Let R be the local ring of a singular point of a complex analytic space, and let M be an R-module. Under what conditions on R and M is it possible to find a connection on M? To approach this question, we consider maximal Cohen–Macaulay (MCM) modules over CM algebras that are isolated singularities, and review an obstruction theory implemented in the computer algebra system Singular. We report on results, with emphasis on singularities of finite and tame CM representation type.