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.
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arXiv subjects
Trond S. Gustavsen
Publications and source records attributed to Trond S. Gustavsen.
Computing obstructions for existence of connections on modules
We consider the notion of a connection on a module over a commutative ring, and recall the obstruction calculus for such connections. The obstruction calculus is defined using Hochschild cohomology. However, in order to compute with Grobner bases, we need the conversion to a description using free resolutions. We describe our implementation in Singular 3.0, available as the library conn.lib. Finally, we use the library to verify some known results and to obtain a new theorem for maximal Cohen-Macaulay (MCM) modules on isolated singularities. For a simple hypersurface singularity of dimension one or two, it is known that all MCM modules admit connections. We prove that for a simple threefold hypersurface singularity of type A_n, D_n or E_n, only the free MCM modules admit connections if n is at most 50.
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Connections on modules over singularities of finite CM representation type
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 on A and M is it possible to find a connection on M? We consider 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 over a simple singularity of dimension at most two, any MCM module admits an integrable connection. We prove that over a simple singularity of dimension at least three, an MCM module admits connections 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 do 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 at least two. We show that this condition is sufficient for the canonical module of 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 of A admits an integrable connection if and only if A is Gorenstein.
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