Deformations of Modules Through Butterflies and Gerbes

TL;DR

This paper introduces a new cohomological approach to classify module extensions using a Grothendieck topology, avoiding the cotangent complex, and compares it with Illusie's classical obstruction theory.

Contribution

It develops an alternative cohomological framework for module extension obstructions via butterflies and gerbes, providing a new construction and answering a longstanding question.

Findings

Established a new cohomology class for module extensions

Compared and related to Illusie's obstruction theory

Circumvented the use of the cotangent complex

Abstract

Classifying obstructions to the problem of finding extensions between two fixed modules goes back at least to L. Illusie's thesis. Our approach, following in the footsteps of J. Wise, is to introduce an analogous Grothendieck Topology on the category $A$-mod of modules over a fixed ring $A$ in a topos $E$. The problem of finding extensions becomes a banded gerbe and furnishes a cohomology class on the site $A$-mod. We compare our obstruction and that coming from Illusie's work, giving another construction of the exact sequence Illusie used to obtain his obstruction. Our work circumvents the cotangent complex entirely and answers a question posed by llusie.