Finitistic Weak Dimension of Commutative Arithmetical Rings

TL;DR

This paper establishes that the finitistic weak dimension of any commutative arithmetical ring is at most 2, with specific values depending on local properties such as being IF or semicoherent.

Contribution

It provides a precise classification of the finitistic weak dimension for all commutative arithmetical rings based on their local properties.

Findings

Finitistic weak dimension of commutative arithmetical rings is ≤ 2.

Dimension is 0 if the ring is locally IF.

Dimension is 1 if the ring is locally semicoherent but not IF.

Abstract

It is proven that each commutative arithmetical ring $R$ has a finitistic weak dimension $\leq 2$. More precisely, this dimension is 0 if $R$ is locally IF, 1 if $R$ is locally semicoherent and not IF, and 2 in the other cases.