Related papers: Multiplicities of Indecomposable Injectives

Examples are given to show that the support of a complex of modules over a commutative noetherian ring may not be read off the minimal semi-injective resolution of the complex. The same examples also show that a localization of a…

A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative…

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields…

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 ring. Enochs and Huang [EH] proved that over a Gorenstein ring of Krull dimension $d$, every Gorenstein injective module admits a finite filtration of Gorenstein injective submodules. In this paper, we…

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, we describe the structure of the localization of Ext^{i}_{R}(R/P,M), where P is a prime ideal and M is a module, at certain Ore sets X. We first study the situation for FBN rings, and then consider matters for a large class…

We consider trace ideals in Noetherian rings and focus our attention to one-dimensional analytically irreducible local rings. For such rings we classify those Gorenstein rings which admit only a finite number of trace ideals.

We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring $R$, we prove that the flat dimension of the injective envelope of any (Gorenstein)…

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…

Let $R$ be a commutative Noetherian ring. In this paper, we study those finitely generated $R$-modules whose Cousin complexes provide Gorenstein injective resolutions. We call such a module a G-Gorenstein module. Characterizations of…

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…

Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. This paper proves that M is a dualizing complex for A if and only if the trivial extension A \ltimes M is a Gorenstein Differential Graded Algebra.…

Let $(R,\fm)$ be a local ring and let $C$ be a semidualizing $R$--module. In this paper, we are concerned in $C$--injective and $G_{C}$--injective dimensions of certain local cohomology modules of $R$. Firstly, the injective dimension of…

For a ring $A$, we consider the question whether every bounded above cochain complex of injective $A$-modules which is acyclic is null-homotopic. We show that if $A$ is left and right noetherian and has a dualizing complex, then this…

We describe a general correspondence between injective (resp. projective) recollements of triangulated categories and injective (resp. projective) cotorsion pairs. This provides a model category description of these recollement situations.…

In this note, it is proved that over a commutative noetherian henselian non-Gorenstein local ring there are infinitely many isomorphism classes of indecomposable totally reflexive modules, if there is a nonfree cyclic totally reflexive…

Separative von Neumann regular rings exist in abundance. For example, all regular self-injective rings, unit regular rings, regular rings with a polynomial identity are separative. It remains open whether there exists a non-separative…

It is proven that each indecomposable injective module over a valuation domain $R$ is polyserial if and only if each maximal immediate extension $\widehat{R}$ of $R$ is of finite rank over the completion $\widetilde{R}$ of $R$ in the…

Let $R\subset A$ be a Frobenius extension of rings. We prove that: (1) for any left $A$-module $M$, $_{A}M$ is Gorenstein projective (injective) if and only if the underlying left $R$-module $_{R}M$ is Gorenstein projective (injective). (2)…