Dual of Bass numbers and dualizing modules

TL;DR

This paper characterizes dualizing modules over Noetherian rings using relative homological dimensions, Bass numbers, and local cohomology, extending previous results and providing explicit structures for minimal flat resolutions.

Contribution

It introduces new criteria for a semidualizing module to be dualizing, generalizes existing theorems, and explicitly describes minimal flat resolutions of local cohomology modules.

Findings

C is dualizing iff a Cohen-Macaulay module of type 1 with finite G_C-dimension exists.

C is dualizing iff certain Bass number invariants vanish for C-injective modules.

Explicit minimal flat resolution of local cohomology modules in Cohen-Macaulay rings.

Abstract

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using relative homological dimensions with respect to $C$, we impose various conditions on $C$ to be dualizing. First, we show that $C$ is dualizing if and only if there exists a Cohen-Macaulay $R$-module of type 1 and of finite G$ _C $-dimension. This result extends Takahashi \cite[Theorem 2.3]{T} as well as Christensen \cite[Proposition 8.4]{C}. Next, as a generalization of Xu \cite[Theorem 3.2]{X2}, we show that $C$ is dualizing if and only if for an $R$-module $M$, the necessary and sufficient condition for $M$ to be $C$-injective is that $ \pi_i(\fp , M) = 0 $ for all $ \fp \in \Spec(R) $ and all $ i \neq \h(\fp) $, where $ \pi_i $ is the invariant dual to the Bass numbers defined by E.Enochs and J.Xu \cite{EX}. We use the later result to give an explicit structure of the minimal flat resolution of $ \H_{\fm}^d(R) $, where $ (R, \fm) $ is a $ d $-dimensional Cohen-Macaulay local ring possessing a canonical module. As an application, we compute the torsion product of these local cohomology modules.