The deformation theory of sheaves of commutative rings

TL;DR

This paper develops a deformation theory framework for sheaves of commutative rings using a sheaf of abelian groups, linking cohomology and obstructions in deformation problems.

Contribution

It introduces a sheaf of abelian groups representing cotangent complex cohomology and connects obstructions to deformation problems with torsors and gerbes.

Findings

Cohomology of the sheaf is represented by the cotangent complex.

Obstructions are characterized as classes of torsors and gerbes.

Provides a new perspective on deformation problems in algebraic geometry.

Abstract

We define a sheaf of abelian groups whose cohomology is represented by the cotangent complex. We show how obstructions to some standard deformation problems arise as the classes of torsors under and gerbes banded by this sheaf.