Change-of-Rings Theorems for the Small Finitistic Dimension

TL;DR

This paper explores the small finitistic dimension of commutative rings through finitistic flat homological algebra, establishing change-of-rings theorems and local bounds.

Contribution

It introduces the concept of $FT$-flat dimension, proving change-of-rings results and characterizations related to the small finitistic dimension.

Findings

Established change-of-rings results for $FT$-flat dimension.

Characterized small finitistic dimension via $FT$-flat dimension.

Derived local upper bounds for the small finitistic dimension.

Abstract

In this paper, we study the small finitistic dimension of a commutative ring from the viewpoint of finitistic flat homological algebra. Using the class $FPR(R)$ of modules admitting finite projective resolutions, we investigate the finitistic flat ($FT$-flat) dimension and establish several of its basic properties. We prove change-of-rings results for the $FT$-flat dimension, including quotient and polynomial extension results, as well as localization inequalities. As applications, we obtain characterizations of the small finitistic dimension in terms of $FT$-flat dimension, derive quotient and polynomial extension theorems for the small finitistic dimension, and establish local upper bounds in terms of the small finitistic dimensions of localizations.