Obstruction theory for objects in abelian and derived categories

TL;DR

This paper develops an obstruction theory for lifting complexes and objects in derived categories, providing tools for understanding deformations of abelian categories and their objects.

Contribution

It introduces a new obstruction theory for lifting complexes in derived categories and proves the existence of miniversal derived deformations.

Findings

Obstruction theory for lifting complexes up to quasi-isomorphism.

Obstruction theory for lifting objects via Yoneda Ext-groups.

Existence of miniversal derived deformations of complexes.

Abstract

In this paper we develop the obstruction theory for lifting complexes, up to quasi-isomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular case we also obtain the corresponding obstruction theory for lifting of objects in terms of Yoneda Ext-groups. In appendix we prove the existence of miniversal derived deformations of complexes.