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.Comment: 19 pages, 1 figur

Silting Modules over Commutative Rings

“…This extends the results in [12] from the tilting to the silting case, and it is a further piece of evidence for the close relationship between silting modules and localisation theory.…”

“…Here we give a direct proof by providing an explicit construction for a silting module corresponding to a Gabriel filter of finite type. It generalises the construction of a Fuchs-Salce tilting module in [12]. Construction 4.5.…”

“…Then F ′ is closed under direct limits, and thus D = Mod-R/A by [7] or [12,Theorem 4.6]. In particular, A is an idempotent ideal.…”

“…Tilting modules over commutative rings were recently classified in [12]: they correspond bijectively to faithful Gabriel topologies of finite type. In this note we extend this classification by dropping faithfulness.…”

Matlis category equivalences for a ring epimorphism

“…One of the main results of this paper is based on some recent results of Hrbek and Angeleri Hügel-Hrbek [11,1]. We show that whenever u : R −→ U is a homological ring epimorphism and U is an R-module of projective dimension 1, it follows that U is a flat R-module.…”