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...

We show that any quasi-Gorenstein deformation of a 3-dimensional Buchsbaum local ring with I-invariant 1 admits maximal Cohen-Macaulay module, provided it is quotient Gorenstein ring. Such class rings includes two instances unique factorization domains constructed by Marcel-Schenzel and Imtiaz-Schenzel, respectively. Apart from this result, motivated the small conjecture in prime characteristic...

We exhibit the elementary but somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals. It leads to the construction of a bountiful set of Cohen–Macaulay Rees algebras.

Unprojection theory analyzes and constructs complicated commutative rings in terms of simpler ones. Our main result is that, on the algebraic level of Stanley-Reisner rings, stellar subdivisions of Gorenstein* simplicial complexes correspond to unprojections of type Kustin-Miller. As an application of our methods we study the minimal resolution of Stanley– Reisner rings associated to stacked po...

In this paper, we introduce and study the notion of linkage modules by reflexive homomorphisms. This unifies generalizes several known concepts enables us to theory over Cohen–Macaulay rings rather than more restrictive Gorenstein rings. It is shown that results for are still true in general case module We also colinkage establish an adjoint equivalence between linked colinked modules.

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.

It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t ±1 ]-module associated to R. This module, denoted J(R), is the free Z[t ±1 ]-module on the isomorphism classes of finitely generated R-modules modulo relations reminiscent of those defining the Grothendieck group of R. The ...

Existence of proper Gorenstein projective resolutions and Tate cohomology is proved over rings with a dualizing complex. The proofs are based on Bousfield Localization which is originally a method from algebraic topology.

This paper is based on a talk given by the author in October, 1997 at a conference at Columbia University in celebration of Hyman Bass’s 65th birthday. The paper details some of the history of Gorenstein rings and their uses.