One-tilting classes and modules over commutative rings

TL;DR

This paper classifies 1-tilting classes and modules over commutative rings, establishing a correspondence with Gabriel topologies and Thomason subsets, and generalizes classical tilting modules with applications to localization.

Contribution

It provides a comprehensive classification of 1-tilting classes and modules over arbitrary commutative rings, linking them to spectral and topological structures.

Findings

Classified all 1-tilting classes over commutative rings.

Established a correspondence with faithful Gabriel topologies and Thomason subsets.

Generalized classical Fuchs and Salce tilting modules.

Abstract

We classify 1-tilting classes over an arbitrary commutative ring. As a consequence, we classify all resolving subcategories of finitely presented modules of projective dimension at most 1. Both these collections are in 1-1 correspondence with faithful Gabriel topologies of finite type, or equivalently, with Thomason subsets of the spectrum avoiding a set of primes associated in a specific way to the ring. We also provide a generalization of the classical Fuchs and Salce tilting modules, and classify the equivalence classes of all 1-tilting modules. Finally we characterize the cases when tilting modules arise from perfect localizations.