First order thickenings and cotorsors

TL;DR

This paper establishes an equivalence between the category of first order thickenings of a scheme by a quasi-coherent sheaf and the category of cotorsors, providing a new perspective on scheme deformations.

Contribution

It introduces a novel equivalence between first order thickenings and cotorsors, linking scheme theory with cogroup objects in a new way.

Findings

Category of first order thickenings is equivalent to the category of cotorsors.

Quasi-coherent sheaves can be viewed as cogroup objects in scheme categories.

Provides a new framework for understanding scheme deformations.

Abstract

Let $X$ be a scheme and let $\mathcal{M}$ be a quasi-coherent sheaf on $X$. Then $\mathcal{M}$ can be viewed as a cogroup object in the category of schemes under $X$. We show that the category of first order thickenings of $X$ by $\mathcal{M}$ is equivalent to the category of $\mathcal{M}$-cotorsors.