Related papers: A counterexample to a conjecture of Ding

We generalize a theorem of Ding relating the generalized Loewy length $\text{g}\ell\ell(R)$ and index of a one-dimensional Cohen-Macaulay local ring $(R,\mathfrak{m},k)$. Ding proved that if $R$ is Gorenstein, the associated graded ring is…

Let $R=k[|t^a,t^b,t^c|]$ be a complete intersection numerical semigroup ring over an infinite field $k$, where $a,b,c\in\BN$. The generalized Loewy length, which is Auslander's index in this case, is computed in terms of the minimal…

A complete local ring of embedding codepth 3 has a minimal free resolution of length 3 over a regular local ring. Such resolutions carry a differential graded algebra structure, based on which one can classify local rings of embedding…

An example is constructed of a local ring and a module of finite type and finite projective dimension over that ring such that the module is not rigid. This shows that the rigidity conjecture is false.

Let $(A,\mathfrak{m})$ be a local Gorenstein local ring and let $M$ be an $A$ module of finite length and finite projective dimension. We prove that the Lowey length of $M$ is greater than or equal to order of $A$. This generalizes a result…

Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d \geq 1$. Suppose there exists be a non-zero $A$ module $M$ of finite length and finite projective dimension such that $\ell\ell(M)$, the Lowey length of $M$, is equal to…

In this paper, we compare the Gorenstein homological dimension of a ring $R$ and of its trivial ring extension by an module $E$.

In this paper, we show that the Gorenstein global dimension of trivial ring extensions is often infinite. Also we study the transfer of Gorenstein properties between a ring and its trivial ring extensions. We conclude with an example…

In this paper, we establish, as a generalization of a result on the classical homological dimensions of commutative rings, an upper bound on the Gorenstein global dimension of commutative rings using the global cotorsion dimension of rings.…

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…

In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings…

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…

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…

It is well-known that for a large class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we exploit this…

We answer a question of Celikbas, Dao, and Takahashi by establishing the following characterization of Gorenstein rings: a commutative noetherian local ring $(R,\mathfrak m)$ is Gorenstein if and only if it admits an integrally closed…

We provide a counterexample to the Lagrangian Poincar\'e recurrence conjecture of Ginzburg and Viterbo in all dimensions $6$ and greater.

Let $(R,\fm,k)$ be a commutative noetherian local ring with dualizing complex $\dua R$, normalized by $\Ext^{\depth(R)}_R(k,\dua R)\cong k$. Partly motivated by a long standing conjecture of Tachikawa on (not necessarily commutative)…

This concerns a conjecture of Lynch on annihilators of local cohomology modules; we present a counterexample that has dimension three.

We expand on two existing characterizations of rings of Gorenstein (weak) global dimension zero and give two new characterizations of rings of finite Gorenstein (weak) global dimension. We also include the answer to a question of Y.~Xiang…

Over a commutative Noetherian ring, we show that the Auslander-Reiten conjecture holds true for the class of (finitely generated) modules whose dual has finite complete intersection dimension. We provide another result that validates the…