The algebra of observables in noncommutative deformation theory

We consider the algebra $\mathcal O(\mathsf M)$ of observables and the (formally) versal morphism $\eta: A \to \mathcal O(\mathsf M)$ defined by the noncommutative deformation functor $\mathsf{Def}_{\mathsf M}$ of a family $\mathsf M = \{ M_1, \dots, M_r \}$ of right modules over an associative $k$-algebra $A$. By the Generalized Burnside Theorem, due to Laudal, $\eta$ is an isomorphism when $A$ is finite dimensional, $\mathsf M$ is the family of simple $A$-modules, and $k$ is an algebraically closed field. The purpose of this paper is twofold: First, we prove a form of the Generalized Burnside Theorem that is more general, where there is no assumption on the field $k$. Secondly, we prove that the $\mathcal O$-construction is a closure operation when $A$ is any finitely generated $k$-algebra and $\mathsf M$ is any family of finite dimensional $A$-modules, in the sense that $\eta_B: B \to \mathcal O^B(\mathsf M)$ is an isomorphism when $B = \mathcal O(\mathsf M)$ and $\mathsf M$ is considered as a family of $B$-modules.Comment: 9 page

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

Let $k$ be an algebraically closed field of characteristic 0, let $R$ be a commutative $k$-algebra, and let $M$ be a torsion free $R$-module of rank one with a connection $\nabla$. We consider the Lie-Rinehart cohomology with values in $End_{R}(M)$ with its induced connection, and give an interpretation of this cohomology in terms of the integrable connections on $M$. When $R$ is an isolated singularity of dimension $d\geq2$, we relate the Lie-Rinehart cohomology to the topological cohomology of the link of the singularity, and when $R$ is a quasi-homogenous hypersurface of dimension two, we give a complete computation of the cohomology.Comment: 13 page

Connections on modules over quasi-homogeneous plane curves

Let k be an algebraically closed field of characteristic 0, and let $A = k[x,y]/(f)$ be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e. a graded A-linear homomorphism $\nabla: \operatorname{Der}_k(A) \to \operatorname{End}_k(M)$ that satisfy the derivation property and preserves the Lie product. In particular, a torsion free module N over the complete local ring $B = \hat A$ admits a natural integrable connection if A is a simple curve singularity, or if A is irreducible and N is a gradable module.Comment: AMS-LaTeX, 12 pages, minor changes. To appear in Comm. Algebr