We develop in this paper a stable theory for projective complexes, by which we mean to consider chain complex of finitely generated modules as an object the factor category homotopy modulo split complexes. As result are able prove that over generically Gorenstein ring is exact if and only its dual exact. This shows dependence total reflexivity conditions ring.

We propose a conjectural framework for computing Gorenstein measures and stringy Hodge numbers in terms of motivic integration over arcs smooth Artin stacks, we verify this the case fantastacks, which are certain toric stacks that provide (non-separated) resolutions singularities varieties. Specifically, let $\mathcal{X}$ be stack admitting good moduli space $\pi: \mathcal{X} \to X$, assume $X$...

For a suitable triangulated category $\mathcal{T}$ with Serre functor $S$ and full precovering subcategory $\mathcal{C}$ closed under summands extensions, an indecomposable object $C$ in is called Ext-projective if Ext$^1(C,\mathcal{C})=0$. Then there no Auslander-Reiten triangle end term $C$. In this paper, we show that if, for such $C$, minimal right almost split morphism $\beta:B\rightarrow ...

In this note, we introduce the notion of Gorenstein algebras. Let R be a commutative Gorenstein ring and A a noetherian R-algebra. We call A a Gorenstein R-algebra if A has Gorenstein dimension zero as an R-module (see [2]), add(D(AA)) = PA, where D = HomR(−, R), and Ap is projective as an Rpmodule for all p ∈ Spec R with dim Rp < dim R. Note that if dim R = ∞ then a Gorenstein R-algebra A is p...

Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of Gorenstein rings has led to the advent of a whole branch of homological algebra, known as Gorenstein homological algebra. This paper solves one of the open pr...

A semi-dualizing module over a commutative noetherian ringA is a finitely generated module C with RHomA(C,C) ≃ A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call C–Gorenstein projective, C–Gorenstein injective, and C–Gorenstein flat dimension, and investigate the properties of these dimensions.

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 two special cases of a question of Takahashi and White.

Let X be an n-dimensional normal projective variety with terminal, Gorenstein, Qfactorial singularities. Let L be an ample line bundle on X . Let t be the nefvalue of ðX ;LÞ. Then we classify ðX ;LÞ, describing the structure of the nefvalue morphism of ðX ;LÞ, when t satisfies n k < t < n k þ 1 and nd 2k 3, kd 4. In the smooth case, we discuss the case n 1⁄4 2k 4, kd 5, as well.

We prove that if a positively-graded ring R is Gorenstein and the associated torsion functor has finite cohomological dimension, then the corresponding noncommutative projective scheme Tails(R) is a Gorenstein category in the sense of [10]. Moreover, under this condition, a (right) recollement relating Gorensteininjective sheaves in Tails(R) and (graded) Gorenstein-injective R-modules is given.

In this paper, we prove that the global Gorenstein projective dimension of a ring R is equal to the global Gorenstein injective dimension of R, and that the global Gorenstein flat dimension of R is smaller than the common value of the terms of this equality.