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 ∇ :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.Let k be an algebraically closed field of characteristic 0, and let A be a commutative k-algebra. For any A-module M, we consider connections on M, i.e. A-linear homomorphisms ∇ : Der k (A) → End k (M) such that ∇ D (am) = a∇ D (m) + D(a)m for all D ∈ Der k (A), a ∈ A and m ∈ M. A connection is integrable if it is a Lie algebra homomorphism. The present paper is devoted to the following question: Under what conditions on A and M is it possible to find a connection on M?

Lie–Rinehart cohomology and integrable connections on modules of rank one

“…Given a graded torsion free R-module M of rank one on a monomial curve R, it does not necessarily exist a connection on M, see Section 5.2 in Eriksen and Gustavsen [6]. However, if there exists a connection, it is integrable and unique up to analytic isomorphism by Theorem 2.4 and the proposition above.…”

“…Let g be a Lie-Rinehart algebra and assume that there is a g-connection ∇ on M, see Eriksen and Gustavsen [6] for definitions and basic properties. Then there is an induced g-connection ∇ on (1) The induced g-connection ∇ :…”

“…One may consider the category of modules with g-connections, see Section 1 in Eriksen and Gustavsen [6]. Proof.…”

Exact categories, big Cohen-Macaulay modules and finite representation type

“…Also, from the work of Reiten-Van den Bergh [49] the Auslander-Reiten quivers are known in dimension 2. For Cohen-Macaulay orders of higher dimension, only scattered results are known [7,21,39,62]. Enomoto [20] gives an Auslander-correspondence for such orders.…”

Computing obstructions for existence of connections on modules

“…These results led us to conjecture that for any simple hypersurface singularity of dimension d = 3, the only MCM modules that admit connections are the free modules. Using different techniques, we proved a more general result in Eriksen and Gustavsen (2006a): An MCM module over a simple hypersurface singularity of dimension d ≥ 3 admits a connection if and only if it is free.…”