Cotorsion pairs generated by modules of bounded projective dimension

TL;DR

This paper investigates the properties of cotorsion pairs generated by modules with bounded projective dimension, focusing on their closure properties and conditions for finite type over certain rings.

Contribution

It establishes criteria for cotorsion pairs of finite type generated by modules of projective dimension at most one, linked to the finitistic dimension of the ring.

Findings

Cotorsion pairs generated by modules of bounded projective dimension have specific closure properties.

Finite type of these cotorsion pairs is characterized by the ring's finitistic dimension.

Results apply to classes of rings including semiprime Goldie and Cohen-Macaulay noetherian rings.

Abstract

We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an $\aleph_0$-noetherian ring Q of small finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and Cohen Macaulay noetherian commutative rings.