$n$-quasi-abelian categories vs $n$-tilting torsion pairs

Luisa Fiorot

Documenta mathematica, Tome 26 (2021), pp. 149-197

Résumé

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 t-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 E admits a "derived" category D(E) endowed with a n-tilting pair of t-structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these t-structures as quotient categories of coherent functors, generalizing Auslander's 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 f∗​-acyclic coherent sheaves form a 1-tilting torsion class, whose associated heart is derived equivalent to D(Y). We generalize this theorem to relative dimension 2.

Détail

DOI : 10.4171/dm/812

Classification : 14F08, 16S90, 18E05, 18E20, 18E40
Mots-clés : perverse coherent sheaves, torsion pair, quasi-abelian category, t-structures, tilting objects, Bondal-Orlov conjecture