One-tilting classes and modules over commutative rings

Abstrakt

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.

Klíčová slova

• Commutative ring
• Tilting module
• Cotilting module
• Zariski spectrum
• Gabriel topology