Universal localizations of $d$-homological pairs

TL;DR

This paper extends the concept of universal localization from hereditary algebras to $d$-homological pairs, establishing existence and properties of these localizations in higher homological contexts.

Contribution

It introduces a higher analogue of universal localizations for $d$-homological pairs, generalizing classical results and connecting to triangulated subcategories.

Findings

Universal localizations exist for $d$-homological pairs under certain conditions.

The higher analogue coincides with classical localization when $d=1$.

Partial generalization of Krause and Šťovíček's results to higher homological dimensions.

Abstract

Let $k$ be an algebraically closed field and $\Phi$ a finite dimensional $k$-algebra. The universal localization $\Phi\rightarrow \Phi_\mathcal{S}$ of $\Phi$ with respect to a set of morphisms between finitely generated projective $\Phi$-modules $\mathcal{S}$ always exists. Moreover, when $\Phi$ is hereditary, Krause and \v{S}\v{t}ov\'i\v{c}ek proved that the universal localizations of $\Phi$ are in bijective correspondence with various natural structures. Taking inspiration from an alternative definition of universal localizations involving a triangulated subcategory of $\mathcal{D}^{\text{perf}}(\Phi)$, we introduce a higher analogue of universal localizations. That is, fixing a positive integer $d$, we define universal localizations of $d$-homological pairs $(\Phi,\mathcal{F})$ with respect to suitable wide subcategories $\mathcal{U}$ of $\mathcal{D}^b(\text{mod}\Phi)$. When gldim$\Phi\leq d$, we show that the result by Krause and \v{S}\v{t}ov\'i\v{c}ek has a (partial) higher analogue and that such universal localizations exist with respect to any choice of $\mathcal{U}$ with the required properties. Moreover, we show that in this setup, the base case of our definition and the definition of classic universal localization coincide.