Cohomological splitting, realization, and finiteness

TL;DR

This paper investigates criteria for module splitting and finiteness over local rings using cohomological functors, connecting to realization problems and extending classical results to broader ring classes.

Contribution

It introduces new cohomological criteria for module splitting and finiteness, linking to realization problems and extending known results to non-Cohen-Macaulay rings.

Findings

Recovered results of Jensen using simple methods

Computed projective dimensions of certain injective modules

Extended classical results to broader classes of rings

Abstract

We search for some splitting (resp. finiteness) criteria of a given module $M$ over a local ring $(R,\fm,k)$ in terms of the splitting (resp. finiteness) property of certain cohomological functors evaluated at $M$. In particular, we deal with the cohomological splitting question posted by Vasconcelos. We present a connection from our approach to the realization problem of Nunke. This is equipped with several applications. For instance, we recover some results of Jensen (and others) by applying simple methods. Additional applications, including a computation of the projective dimension of some injective modules, are given. This enables us to extend some results of Matlis (resp. Osofsky) on the projective dimension of $E_R(k)$ (resp. $\mathcal{Q}$) from Cohen-Macaulay rings (resp. regular rings) to non-Cohen-Macaulay (resp. Cohen-Macaulay) rings.