Homological dimensions, the Gorenstein property, and special cases of some conjectures

TL;DR

This paper develops criteria for finiteness of homological dimensions of modules over Noetherian rings, explores their implications in local Cohen-Macaulay rings, and applies these results to characterize Gorenstein rings and address longstanding conjectures.

Contribution

It introduces new criteria for homological dimension finiteness, investigates their effects in Cohen-Macaulay rings, and applies findings to characterize Gorenstein rings and resolve conjectures.

Findings

Criteria for finiteness of projective and injective dimensions

Characterizations of Gorenstein rings via anticanonical modules

Confirmation of Vasconcelos's 1985 conjecture for almost Cohen-Macaulay differentials

Abstract

Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we investigate the impact of the finiteness of certain homological dimensions of $M$ if $R$ is local, mainly when $R$ is Cohen-Macaulay and with a partial focus on duals. Along the way, we produce various freeness criteria for modules. Finally, we give applications, including characterizations of when $R$ is Gorenstein (and other ring-theoretic properties as well, sometimes in the prime characteristic setting), particularly by means of its anticanonical module, and in addition we address special cases of some long-standing conjectures; for instance, we confirm the 1985 conjecture of Vasconcelos on normal modules in case the module of differentials is almost Cohen-Macaulay.