THE SMALL FINITISTIC DIMENSIONS OF COMMUTATIVE RINGS

Abstract

Let $R$ be a commutative ring with identity. The small finitistic dimension $\text{fPD}(R)$ of $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we characterize a ring $R$ with $\text{fPD}(R)\le n$ using finitely generated semiregular ideals, tilting modules, cotilting modules of cofinite type and vaguely associated prime ideals. As an application, we obtain that if $R$ is a Noetherian ring, then $\text{fPD}(R)=\text{sup}\{\text{grade}(\mathfrak{𝔪},R)|\mathfrak{𝔪}\in \text{Max}(R)\}$ where $\text{grade}(\mathfrak{𝔪},R)$ is the grade of $\mathfrak{𝔪}$ on $R$. We also show that a ring $R$ satisfies $\text{fPD}(R)\le 1$ if and only if $R$ is a $\text{DW}$ ring. As applications, we show that the small finitistic dimensions of strong Prüfer rings and $\text{LPVD}$s are at most one. Moreover, for any given $n\in \mathbb{N}$, we obtain a total ring of quotients $R$ satisfying $\text{fPD}(R)=n$.