N-Quasi-Abelian Categories vs N-Tilting Torsion Pairs

It is a well established fact that the notions of quasi-abelian categories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of (formula-presented)-structures. Firstly, we extend this picture into a hierarchy of n-quasi-abelian categories and n-tilting torsion classes. We prove that any n-quasi-abelian category formula-presented admits a “derived” category D(formula-presented) endowed with a n-tilting pair of formula-presented-structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these formula-presented-structures as quotient categories of coherent functors, generalizing Auslanderformula-presenteds Formula. Thirdly, we apply our results to Bridgeland’s theory of perverse coherent sheaves for flop contractions. In Bridgeland’s work, the relative dimension 1 assumption guaranteed that fformula-presented-acyclic coherent sheaves form a 1-tilting torsion class, whose associated heart i...