In this paper, we introduce and investigate the notions of ξ-strongly copure projective objects in a triangulated category. This extends Asadollahi’s notion of ξ-Gorenstein projective objects. Then we study the ξ-strongly copure projective dimension and investigate the existence of ξ-strongly copure projective precover.

An R-module M is called epi-retractable if every submodule of MR is a homomorphic image of M. It is shown that if R is a right perfect ring, then every projective slightly compressible module MR is epi-retractable. If R is a Noetherian ring, then every epi-retractable right R-module has direct sum of uniform submodules. If endomorphism ring of a module MR is von-Neumann regular, then M is semi-...

In this paper we use "ring changed'' Gorenstein homologicaldimensions to define Cohen-Macaulay injective, projective and flatdimensions. For doing this we use the amalgamated duplication of thebase ring with semi-dualizing ideals. Among other results, we prove that finiteness of these new dimensions characterizes Cohen-Macaulay rings with dualizing ideals.

Max Noether’s Theorem asserts that if ω is the dualizing sheaf of a nonsingular nonhyperelliptic projective curve then the natural morphisms SymH(ω) → H(ω) are surjective for all n ≥ 1. This is true for Gorenstein nonhyperelliptic curves as well. We prove this remains true for nearly Gorenstein curves and for all integral nonhyperelliptic curves whose non-Gorenstein points are unibranch. The re...

A central problem in liaison theory is to decide whether every arithmetically Cohen-Macaulay subscheme of projective n-space can be linked by a finite number of arithmetically Gorenstein schemes to a complete intersection. We show that this can be indeed achieved if the given scheme is also generically Gorenstein and we allow the links to take place in an (n + 1)-dimensional projective space. F...

known as the left big finitistic projective dimension of A, is finite. Here pdM denotes the projective dimension of M . Unfortunately, this number is not known to be finite even if A is a finite dimensional algebra over a field, where, indeed, its finiteness is a celebrated conjecture. On the other hand, for such an algebra, finite flat certainly implies finite projective dimension, simply beca...

We extend the theory of generalized divisors so as to work on any scheme X satisfying the condition S2 of Serre. We define a generalized notion of Gorenstein biliaison for schemes in projective space. With this we give a new proof of the theorem of Gaeta, that standard determinantal schemes are in the Gorenstein biliaison class of a complete intersection.

We investigate the Cohomological Crepant Resolution Conjecture for reduced Gorenstein weighted projective spaces. Using toric methods, we prove this conjecture in some new cases. As an intermediate step, we show that weighted projective spaces are toric Deligne-Mumford stacks. We also describe a combinatorial model for the orbifold cohomology of weighted projective spaces.

For Gorenstein quotient spaces C d =G, a direct generalization of the classical McKay correspondence in dimensions d 4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstei...

We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves over a semi-separated noetherian scheme, generalising complete flat resolutions over a ring. By studying these complexes as objects of the pure derived category of flat sheaves we extend several results about totally acyclic complexes of projective modules to schemes; for example, we prove that a scheme is Goren...