Cotorsion pairs and Tor-pairs over commutative noetherian rings

TL;DR

This paper classifies hereditary cotorsion pairs over commutative noetherian rings using functions on the spectrum, linking them to Tor-pairs via duality, and describes their structure through local depth conditions.

Contribution

It provides a complete classification of hereditary cotorsion pairs cogenerated by pure-injective modules with finite injective dimension over such rings, using spectrum-based functions.

Findings

Classification of hereditary cotorsion pairs via spectrum functions

Explicit duality between cotorsion pairs and Tor-pairs

Description of local depth conditions for cotorsion pairs

Abstract

For a commutative noetherian ring $R$, we classify all the hereditary cotorsion pairs cogenerated by pure-injective modules of finite injective dimension. The classification is done in terms of integer-valued functions on the spectrum of the ring. Each such function gives rise to a system of local depth conditions which describes the left-hand class in the corresponding cotorsion pair. Furthermore, we show that these cotorsion pairs correspond by explicit duality to hereditary Tor-pairs generated by modules of finite flat dimension.