Homology with the theme of Matlis

TL;DR

This paper extends Matlis' homological properties from 1-dimensional to higher-dimensional cases, exploring co-torsion, splitting criteria, projective dimensions, and co-Hopfian modules, with applications to non-noetherian contexts.

Contribution

It generalizes key homological properties of Matlis to higher ranks and introduces new classes of modules, broadening the scope of Matlis' theory.

Findings

Computed the projective dimension of the completion $\, ext{ extasciitilde} R$

Presented non-noetherian versions of Grothendieck's localization problem

Constructed a new class of co-Hopfian modules

Abstract

Matlis proved a lot of homological properties of the fraction field of an integral domain. In this paper, we simplify and extend some of them from 1-dimensional (resp. rank one) cases to the higher dimensional (resp. finite rank) cases. For example, we study the weakly co-torsion property of Ext$(-,\sim)$, and use it to present splitting criteria. These are equipped with several applications. For instance, we compute the projective dimension of $\widehat{R}$ and present some non-noetherian versions of Grothendieck's localization problem. We construct a new class of co-Hopfian modules and extend Matlis' decomposability problem to higher ranks. In particular, this paper deals with the basic properties of Matlis' quadric $(Q,Q/R, \widehat{R},\overset{\sim}R).$