First order thickenings and cotorsors

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.