arXiv subjects

Shrikant Shekhar

Publications and source records attributed to Shrikant Shekhar.

2 recordsLinked to original sources

arXiv2026

A Study of Good and Bad Artinian Gorenstein local Rings

We say that a local ring $R$ is good, in the sense of Roos, if all finitely generated $R$-modules have rational Poincaré series that share a common denominator; otherwise, $R$ is said to be bad. An important class of good rings is the class of generalized Golod rings. In this paper, we show that connected sums of Artinian Gorenstein generalized Golod rings are good. We provide a criterion for decomposing Artinian Gorenstein local rings as connected sums. As a key application, we prove that a Gorenstein local ring $R$ with maximal ideal $\mathfrak{m}$ is good under either of the following conditions: (1) the multiplicity of $R$ is at most $12$ and its $h$-vector is different from $(1, 5, 5, 1)$, (2) $\mathfrak{m}^4$ = 0 and $\mathfrak{m}^2$ is generated by at most four elements. The above result records partial progress towards resolving a question posed by L.~Avramov. We also present examples of bad Artinian Gorenstein local rings of any multiplicity greater than or equal to $18$. In all these cases, the results establishing that the rings are good are obtained by showing that the rings are generalized Golod rings.

math.AC↗

arXiv2025

On Tor-vanishing of local rings

Let $R$ be a local ring with residue field $k$ and $M$, $N$ be finitely generated modules over $R$. It is well known that $Tor^R_i(M, N) = 0$ for $i \gg 0$ if $pd_R(M) < \infty$ or $pd_R(N) < \infty$. The ring $R$ is said to satisfy the Tor-vanishing property if the converse holds, that is, $Tor^R_i(M, N) = 0$ for $i \gg 0$ implies $pd_R(M) < \infty$ or $pd_R(N) < \infty$. Interest in the Tor-vanishing property stems from the fact that Cohen-Macaulay local rings satisfying this property also satisfy the Auslander-Reiten conjecture. In this article, we study a variant of this property. If $R$ is a generalized Golod ring, we prove that $Tor^R_i(M, N) = 0$ for $i \gg 0$ implies $\{curv_R M, curv_R N \} \cap \{0, 1\} \neq \emptyset$. A key intermediate step in our proof is to show that $curv_R M \in \{0, 1, curv_R k\}$ for any module $M$ over a generalized Golod ring $R$. As an application, we prove that generic Gorenstein local rings, non-trivial connected sums of generalized Golod-Gorenstein rings satisfy the Tor-vanishing property and consequently the Auslander-Reiten conjecture. Our method suggests a uniform approach and recovers many old results on the Tor-vanishing property.

math.AC↗