Homological dimensions of Burch ideals, submodules and quotients

TL;DR

This paper characterizes local rings using homological invariants of Burch ideals, submodules, and quotients, providing new criteria for properties like Gorenstein and Cohen-Macaulay conditions.

Contribution

It introduces novel homological characterizations of local rings based on Burch ideals and submodules, extending previous work and offering practical criteria.

Findings

Rings are Gorenstein iff certain Ext groups vanish for three consecutive degrees.

Rings are Cohen-Macaulay iff CM-dimension of specific modules is finite.

Burch ideals and submodules serve as tools for homological classification of rings.

Abstract

The notion of Burch ideals and Burch submodules were introduced (and studied) by Dao-Kobayashi-Takahashi in 2020 and Dey-Kobayashi in 2022 respectively. The aim of this article is to characterize various local rings in terms of homological invariants of Burch ideals, Burch submodules, or that of the corresponding quotients. Specific applications of our results include the following: Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring. Let $M=I$ be an integrally closed ideal of $R$ such that ${\rm depth}(R/I)=0$, or $M = \mathfrak{m} N \neq 0$ for some submodule $N$ of a finitely generated $R$-module $L$ such that either ${\rm depth}(N)\ge 1$ or $L$ is free. It is shown that: (1) $I$ has maximal projective $($resp., injective$)$ complexity and curvature. (2) $R$ is Gorenstein if and only if ${\rm Ext}_R^n(M,R)=0$ for any three consecutive values of $n \ge \max\{{\rm depth}(R)-1,0\}$. (3) $R$ is CM (Cohen-Macaulay) if and only if CM-$\dim_R(M)$ is finite.