2022/05/09 by Fedele, Francesca
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2205.04219
Let k be an algebraically closed field and Φ a finite dimensional k-algebra. The universal localization Φ→ ΦS of Φ with respect to a set of morphisms between finitely generated projective Φ-modules S always exists. Moreover, when Φ is hereditary, Krause and Šťovíček proved that the universal localizations of Φ are in bijective correspondence with various natural structures. Taking inspiration from an alternative definition of universal localizations involving a triangulated subcategory of Dperf(Φ), we introduce a higher analogue of universal localizations. That is, fixing a positive integer d, we define universal localizations of d-homological pairs (Φ,F) with respect to suitable wide subcategories U of Db(modΦ). When gldimΦ≤ d, we show that the result by Krause and Šťovíček has a (partial) higher analogue and that such universal localizations exist with respect to any choice of 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.