We consider the notion of a connection on a module over a commutative ring, and deduce an obstruction calculus for determining if there exist such connections. The obstruction calculus is defined using Hochschild cohomology. However, in order to compute with Gröbner bases, we need the non-trivial conversion to a description using free resolutions. We describe our implementation in Singular 3.0,...

In 1982 Richard P. Stanley conjectured that any finitely generated Zn-graded module M over a finitely generated Nn-graded K-algebra R can be decomposed as a direct sum M = ⊕t i=1 νi Si of finitely many free modules νi Si which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least...

We study rings of integral modular forms for congruence subgroups as modules over the ring full group. In many cases these are free or decompose at least into well-understood pieces. apply this to characterize which Cohen--Macaulay and prove finite generation results. These theorems based on decomposition results about vector bundles compactified moduli stack elliptic curves.

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small possible. In this paper we call such far-flung Gorenstein rings. investigate in relation with endomorphism algebras maximal ideals and numerical semigroup show that solution Rohrbach problem additive number theory provides an upper bound for multiplicity Reflexive modules over are also studied.