We study certain Schur functors which preserve singularity categories of rings and we apply them to study the singularity category of triangular matrix rings. In particular, combining these results with Buchweitz–Happel’s theorem, we can describe singularity categories of certain non-Gorenstein rings via the stable category of maximal Cohen–Macaulay modules. Three concrete examples of finitedim...

Let R be a local ring and C semidualizing module of R. We investigate the behavior certain classes generalized Cohen-Macaulay R-modules under Foxby equivalence between Auslander Bass with respect to C. In particular, we show that are invariant this if M is finitely generated R-module in class such C\otimes_RM surjective Buchsbaum, then also Buchsbaum.

We extend a reciprocity theorem of Stanley about enumeration of integer points in polyhedral cones when one exchanges strict and weak inequalities. The proof highlights the roles played by Cohen–Macaulayness and canonical modules. The extension raises the issue of whether a Cohen–Macaulay complex of dimension d embedded piecewise-linearly in R is necessarily a d-ball. This is observed to be tru...

Recently, Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslander and Bridger and the Cohen–Macaulay approximation theorem due to Auslander and Buchweitz. In this paper we generalize these results to much more general case over non-commutative rings. As an application, w...

R.G. and Perez proved that under certain conditions the test ideal of a module closure agrees with trace closure. We use this fact to compute ideals various rings respect closures coming from their indecomposable maximal Cohen–Macaulay modules. also give an easier way hypersurface ring in three variables particular type matrix factorization.

This paper shows that Cohen-Macaulay algebras can be algebraically approximated in such a way their Cohen-Macaulayness and minimal Betti numbers are preserved. is achieved by showing finitely generated modules over power series rings manner preserves diagrams of initial exponents numbers. These results also applied to obtain an approximation result for flat homomorphisms from algebras.

We extend a reciprocity theorem of Stanley about enumeration of integer points in polyhedral cones when one exchanges strict and weak inequalities. The proof highlights the roles played by Cohen–Macaulayness and canonical modules. The extension raises the issue of whether a Cohen–Macaulay complex of dimension d embedded piecewise-linearly in Rd is necessarily a d-ball. This is observed to be tr...

Abstract Let be a regular local ring and non‐zero element of . A theorem due to Knörrer states that there are finitely many isomorphism classes maximal Cohen–Macaulay (CM) ‐modules if only the same is true for double branched cover , is, hypersurface which defined by in We consider an analogue this statement case instead In particular, we show hypersurface, refer as ‐fold has finite CM represen...

We consider the notion of a connection on a module over a commutative ring, and recall the obstruction calculus for such connections. This obstruction calculus is defined using Hochschild cohomology. However, in order to compute with Gröbner 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. F...

In 1966 [1], Auslander introduced a class of finitely generated modules having a certain complete resolution by projective modules. Then using these modules, he defined the G-dimension (G ostensibly for Gorenstein) of finitely generated modules. It seems appropriate then to call the modules of G-dimension 0 the Gorenstein projective modules. In [4], Gorenstein projective modules (whether finite...