Homological dimensions of Burch ideals, submodules and quotients

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.