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

TL;DR

This paper extends the equivalence between quasi-abelian categories and tilting torsion pairs into an $n$-hierarchy, explores derived categories with $n$-tilting pairs, and applies these concepts to perverse coherent sheaves in algebraic geometry.

Contribution

It introduces a hierarchy of $n$-quasi-abelian categories and $n$-tilting torsion classes, and generalizes Bridgeland's results to relative dimension 2.

Findings

Any $n$-quasi-abelian category admits a derived category with an $n$-tilting pair of $t$-structures.

Hearts of these $t$-structures are described as quotient categories of coherent functors.

Generalization of Bridgeland's theorem to relative dimension 2.

Abstract

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 admits a derived category 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 $2$.