Let $ \pi:X \rightarrow X_{0}$ be a projective morphism of schemes, such that $ X_{0}$ is Noetherian and essentially of finite type over a field $ K$. Let $ i \in \mathbb{N}_{0}$, let $ {\mathcal{F}}$ be a coherent sheaf of $ {\mathcal{O}}_{X}$-modules and let $ {\mathcal{L}}$ be an ample invertible sheaf over $ X$. Let $ Z_{0} \subseteq X_{0}$ be a closed set. We show that the depth of the hig...

The principle “Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra” was given by Henrik Holm. There is a remarkable body of evidence supporting this claim. Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So ...

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.

Motivated by their impact on homological algebra, the change of rings results have been the subject of several interesting works in Gorenstein homological algebra over Noetherian rings. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, sec...

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 $C$ be a semidualizing module‎. ‎We first investigate the properties of‎ ‎finitely generated $G_C$-projective modules‎. ‎Then‎, ‎relative to $C$‎, ‎we introduce and study the rings over which‎ ‎every submodule of a projective (flat) module is $G_C$-projective (flat)‎, ‎which we call $C$-Gorenstein (semi)hereditary rings‎. ‎It is proved that every $C$-Gorenstein hereditary ring is both cohe...

It is a well-known result of Auslander and Reiten that contravariant finiteness the class P ∞ fin (of finitely generated modules finite projective dimension) over an Artin algebra sufficient condition for validity finitistic dimension conjectures. Motivated by fact dimensions can alternatively be computed Gorenstein dimension, we examine in this work counterpart Auslander–Reiten condition, name...

We investigate the relationship between level of a bounded complex over commutative ring with respect to class Gorenstein projective modules and other invariants or ring, such as dimension, Krull dimension. The results build upon work done by J. D. Christensen [7], H. Altmann et al. [1], Avramov [4] for levels finitely generated modules.

We will introduce the notion of Gorenstein category as the convenient setup for doing Gorenstein Homological Algebra in categories of sheaves, or in general in categories without enough projective objects. We will illustrate this notion by showing that the category of Qcoh(X) of quasi-coherent sheaves on a locally Gorenstein projective scheme fits into this setup. Then we will focus on the cate...