The small finitistic dimensions of commutative rings, III

TL;DR

This paper characterizes the small finitistic dimension of commutative rings through Ext vanishing conditions and explores its relation to other homological dimensions and ring classes.

Contribution

It provides a new characterization of the small finitistic dimension in terms of Ext vanishing and connects it to the self-FP-injective dimension of rings.

Findings

fPD(R)  d iff Ext conditions hold for all finitely generated ideals

fPD(R)  FP-Id_R R for any commutative ring R

Applications to (weak) (n,d)-rings, DW-rings, and Prufer-type rings

Abstract

The small finitistic dimension fPD$(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we show that a commutative ring $R$ has fPD$(R)\leq d$ if and only if for any finitely generated ideal $I$ of $R$, if $Ext_R^i(R/I,R)=0$ for each $i=0,\dots,d$, then $Ext_R^i(R/I,R)=0$ for all $i\geq 0.$ As applications, we obtain that, for any commutative ring $R$, fPD$(R)\leq \mbox{FP-}Id_RR$, the self-FP-injective dimension of $R$. We also give some applications of these results to (weak) $(n,d)$-rings, DW-rings and rings of Prufer type.