[Paper Review] Tilting modules and universal localization

Lidia Angeleri Hügel, Maria Archetti|ArXiv.org|Aug 13, 2009

Algebraic structures and combinatorial models11 references3 citations

TL;DR

This paper establishes a deep connection between tilting modules of projective dimension one and universal localization, proving that every such tilting module arises naturally from universal localization at a set of finitely presented modules of projective dimension one. The key contribution is a characterization showing that tilting modules of the form $S \oplus S/R$ correspond precisely to universal localizations and perfect Gabriel localizations over Artin algebras and Prüfer domains.

ABSTRACT

We show that every tilting module of projective dimension one over a ring R is associated in a natural way to the universal localization (in the sense of Schofield) of R at a set of finitely presented modules of projective dimension one. We then investigate tilting modules arising from universal localization. Furthermore, we discuss the relationship between universal localization and the localization given by a perfect Gabriel topology. Finally, we give some applications to Artin algebras and to Pruefer domains.

Motivation & Objective

• To clarify the structural relationship between tilting modules of projective dimension one and universal localization over arbitrary rings.
• To investigate when tilting modules of the form $S \oplus S/R$ arise from universal localization or perfect Gabriel topologies.
• To establish equivalences between tilting module constructions and localization theories in special classes of rings, particularly Artin algebras and Prüfer domains.
• To recover and generalize Salce's result on tilting modules over Prüfer domains using universal and perfect localizations.
• To characterize when a tilting class arises from a perfect Gabriel topology or universal localization via homological conditions on the associated module sequence.

Proposed method

• Use the perpendicular category construction $\mathcal{X}_{T_1} = \{M \mid \operatorname{Hom}_R(T_1,M) = \operatorname{Ext}^1_R(T_1,M) = 0\}$ to associate a tilting module $T$ with a ring epimorphism $\lambda: R \to S$.
• Apply Gabriel and de la Peña's result to realize $\mathcal{X}_{T_1}$ as the category of modules over a ring $S$, and show $\lambda$ is the universal localization $R \to R_{\mathcal{U}}$ at a set $\mathcal{U}$ of finitely presented modules of projective dimension one.
• Establish equivalence between tilting modules $S \oplus S/R$ and perfect Gabriel localizations $R \to Q_{\mathcal{G}}$ via homological conditions on $T_0, T_1 \in \operatorname{Add}T$ and vanishing $\operatorname{Hom}_R(T_1, T_0)$.
• Leverage properties of Prüfer domains to show that every universal localization at finitely presented cyclic modules arises from a perfect Gabriel topology.
• Use the fact that over a Prüfer domain, tilting classes are in bijection with perfect Gabriel topologies of finite type, and that $\mathcal{L}$-divisible modules coincide with $\operatorname{Gen}T$.
• Prove that $\operatorname{pd}(Q_{\mathcal{L}}) \leq 1$ if and only if $T \simeq Q_{\mathcal{L}} \oplus Q_{\mathcal{L}}/R$, recovering Salce's result as a corollary.

Experimental results

Research questions

• RQ1 Can every tilting module of projective dimension one over a ring $R$ be realized as arising from a universal localization at a set of finitely presented modules of projective dimension one?
• RQ2 Under what conditions does a tilting module $S \oplus S/R$ arise from a perfect Gabriel topology $\mathcal{G}$?
• RQ3 How are universal localization and perfect Gabriel localization related over Prüfer domains?
• RQ4 When is the localization $Q_{\mathcal{L}}$ of a Prüfer domain $R$ at a Gabriel topology $\mathcal{L}$ of finite type of projective dimension at most one?
• RQ5 What characterizes tilting modules of the form $S \oplus S/R$ in terms of homological and categorical properties of the associated module sequence?

Key findings

• Every tilting module $T$ of projective dimension one over a ring $R$ is associated to a universal localization $R \to R_{\mathcal{U}}$ at a set $\mathcal{U}$ of finitely presented modules of projective dimension one.
• For Artin algebras, every finitely generated tilting module of the form $S \oplus S/R$ arises from universal localization at a set of finitely presented modules.
• Over a Prüfer domain, every tilting module $T$ of projective dimension one such that $\operatorname{pd}(Q_{\mathcal{L}}) \leq 1$ is equivalent to $Q_{\mathcal{L}} \oplus Q_{\mathcal{L}}/R$, recovering Salce’s result.
• Over a Prüfer domain, every universal localization at a set of finitely presented cyclic modules arises from a perfect Gabriel topology.
• A tilting module $T$ of the form $S \oplus S/R$ arises from a perfect Gabriel topology $\mathcal{G}$ if and only if there exists an exact sequence $0 \to R \to T_0 \to T_1 \to 0$ with $T_0, T_1 \in \operatorname{Add}T$, $\operatorname{Hom}_R(T_1, T_0) = 0$, and $\mathcal{X}_{T_1}$ a Giraud subcategory.
• The conditions $\operatorname{pd}(Q_{\mathcal{L}}) \leq 1$, $T$ arising from perfect localization, $T$ arising from universal localization, and the existence of such a sequence are all equivalent over a Prüfer domain.