The small finitistic dimensions of commutative rings

TL;DR

This paper characterizes the small finitistic dimension of commutative rings using various algebraic tools and applies these results to specific classes of rings, providing bounds and examples.

Contribution

It introduces new characterizations of the small finitistic dimension using semi-regular ideals, tilting modules, and prime ideals, and applies these to Noetherian, Prüfer, and total quotient rings.

Findings

For Noetherian rings, finitistic dimension equals the supremum of grades of maximal ideals.

Rings with finitistic dimension ≤ 1 are exactly the DW rings.

Examples of total rings of quotients with arbitrary finitistic dimension n are constructed.

Abstract

Let $R$ be a commutative ring with identity. The small finitistic dimension $\fPD(R)$ 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 $\fPD(R)\leq n$ using finitely generated semi-regular ideals, tilting modules, cotilting modules of cofinite type or vaguely associated prime ideals. As an application, we obtain that if $R$ is a Noetherian ring, then $\fPD(R)= \sup\{\grade(\m,R)|\m\in \Max(R)\}$ where $\grade(\m,R)$ is the grade of $\m$ on $R$ . We also show that a ring $R$ satisfies $\fPD(R)\leq 1$ if and only if $R$ is a $\DW$ ring. As applications, we show that the small finitistic dimensions of strong \Prufer\ rings and $\LPVD$s are at most one. Moreover, for any given $n\in \mathbb{N}$, we obtain examples of total rings of quotients $R$ with $\fPD(R)=n$.