Generalized local duality, canonical modules, and prescribed bound on projective dimension

TL;DR

This paper advances the theory of generalized local cohomology, extending local duality results to broader rings, and establishes bounds on projective dimension, along with new characterizations of Cohen-Macaulay modules.

Contribution

It introduces a generalized local duality theorem for a wider class of rings and provides new bounds on projective dimension based on cohomological conditions.

Findings

Extended local duality to broader rings

Established upper bounds for projective dimension

Characterized Cohen-Macaulay modules via generalized local cohomology

Abstract

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module.