Deformation theory of abelian categories

TL;DR

This paper develops an infinitesimal deformation theory for abelian categories, generalizing algebra deformation theory, introducing flatness, and establishing preservation of properties and equivalences in deformations.

Contribution

It introduces a deformation theory for abelian categories, extending classical algebra deformation concepts and defining flatness within this broader context.

Findings

Flatness for abelian categories is defined and analyzed.

Basic properties are preserved under flat deformations.

Multiple equivalences between deformation problems are constructed.

Abstract

In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under flat deformations and we construct several equivalences between deformation problems.