Let [Formula: see text] be an extriangulated category with a proper class of text]-triangles. We study complete cohomology objects in by applying text]-projective resolutions and text]-injective coresolutions constructed text]. Vanishing detects finite dimension dimension. As consequence, we obtain some criteria for the validity Wakamatsu tilting conjecture give necessary sufficient condition v...

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.

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...

In this article we investigate the relations between Gorenstein projective dimensions of [Formula: see text]-modules and their socles for text]-minimal Auslander–Gorenstein algebras text]. First give a description projective-injective in terms socles. Then prove that text]-module text] has dimension at most if only its socle is cogenerated by text]-module. Furthermore, show can be characterised...

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...

We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify special cases of a question of Takahashi and White. Mathematics Subject Classification (2000). 13C05, 13D05, 13H10.

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra A $A$ with radical J $J$ will be said to short provided 3 = 0 $J^3 0$ . As case, we show: If a has an indecomposable non-projective Gorenstein-projective module M $M$ , then either is self-injective (so that all modules are Gorenstein-projective) and then, of cou...

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.