One-tilting classes and modules over commutative rings

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.

arXiv comment19 pages, 1 figure

References in corpus (6)

• Tilting, cotilting, and spectra of commutative noetherian rings
• Equivalences induced by infinitely generated tilting modules
• Smashing localizations of rings of weak global dimension at most one
• Cotilting modules over commutative noetherian rings
• Tilting modules and universal localization
• Direct limits in the heart of a t-structure: the case of a torsion pair

Cited by in corpus (10)

• Matlis category equivalences for a ring epimorphism
• Closure properties of $\underrightarrow{\lim }\mathcal{C}$
• Flat ring epimorphisms of countable type
• Tilting classes over commutative rings
• Zariski locality of quasi-coherent sheaves associated with tilting
• Flat commutative ring epimorphisms of almost Krull dimension zero
• Flat Mittag-Leffler modules, and their relative and restricted versions
• Enveloping Classes over Commutative Rings
• Telescope conjecture via homological residue fields with applications to schemes
• On the abundance of silting modules