相关论文: Gorenstein injective dimension and a generalizatio…

Let $(R,\frak m, k)$ be a noetherian local ring. It is well-known that $R$ is regular if and only if the injective dimension of $k$ is finite. In this paper it is shown that $R$ is Gorenstein if and only if the Gorenstein injective…

If $M$ is a nonzero finitely generated module over a commutative Noetherian local ring $R$ such that $M$ has finite injective dimension and finite Gorenstein dimension, then it follows from a result of Holm that $M$ has finite projective…

The quasi-projective dimension and quasi-injective dimension are recently introduced homological invariants that generalize the classical notions of projective dimension and injective dimension, respectively. For a local ring $R$ and…

Let $R \to S$ be a local ring homomorphism and $N$ a finitely generated $S$-module. We prove that if the Gorenstein injective dimension of $N$ over $R$ is finite, then it equals the depth of $R$.

Let $R$ be a commutative Noetherian ring of Krull dimension $d$ admitting a dualizing complex $D$ and let $\frak a$ be any ideal of $R$, we prove that $\Gamma_{\frak a}(G)$ is Gorenstein injective for any Gorenstein injective $R$-module…

It is proved that if one of the finite modules M and N, over a local ring R, has reducible complexity and has finite Gorenstein dimension then the depth formula holds, provided TorR_i(M,N) = 0 for i>>0. We also study the vanishing of…

Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological…

Following our previous work about quasi-projective dimension, in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective…

For a finitely generated module $ M $ over a commutative Noetherian ring $R$, we settle the Auslander-Reiten conjecture when at least one of ${\rm Hom}_R(M,R)$ and ${\rm Hom}_R(M,M)$ has finite injective dimension. A number of new…

Let $R$ be a commutative Noetherian local ring. We prove a variety of new formulae for modules of finite quasi-projective or finite quasi-injective dimension. These include the Derived Depth Formula, itself an extension of Auslander famous…

In this paper, we study Gorenstein injective, projective, and flat modules over a Noetherian ring $R$. For an $R$-module $M$, we denote by ${\rm Gpd}_RM$ and ${\rm Gfd}_R M$ the Gorenstein projective and flat dimensions of $M$,…

We prove that a commutative noetherian ring $R$ is Gorenstein of dimension at most $d$ if $d+1$ is an upper bound on the G-levels of perfect $R$-complexes. For $R$ local, we prove a formula for levels, with respect to injective or…

Given a homomorphism of commutative noetherian rings R --> S and an S-module N, it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is…

A generalization of Grothendieck's non-vanishing theorem is proved for a module which is finite over a local homomorphism. It is also proved that the Gorenstein injective dimension of such a module, if finite, is bounded below by its Krull…

We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective…

A central problem in the theory of Gorenstein dimensions over commutative noetherian rings is to find resolution-free characterizations of the modules for which these invariants are finite. Over local rings, this problem was recently solved…

In this paper, it is proved that a commutative noetherian local ring admitting a finitely generated module of finite projective and injective dimensions with respect to a semidualizing module is Gorenstein. This result recovers a celebrated…

Let $(R,\mathfrak{m},k)$ be a commutative Noetherian local ring. It is well-known that if $M$ is a finitely generated $R$-module of finite quasi-injective dimension, then $\operatorname{qid}_RM = \operatorname{depth} R$. In this paper, we…

The aim of this article is to consider the spectral sequences induced by tensor-hom adjunction, and provide a number of new results. Let $R$ be a commutative Noetherian local ring of dimension $d$. In the 1st part, it is proved that $R$ is…

For a reduced Noetherian ring $R$ of characteristic $p > 0$, in this paper we discuss an extension of $R$ called its perfect closure $R^\infty$. This extension contains all $p^e$-th roots of elements of $R$, and is usually non-Noetherian.…