Deformation Theory of Monoid Schemes I

Abstract

The aim of this paper is to develop a deformation theory of monoid schemes, generalising the approach developed by Grillet. The core idea of this approach is to introduce the notion of a system of abelian groups, as the naive approach to exactness does not work for monoids. We first study the case of monoid sheaves (functors over a poset into the category of monoids) and prove a classification theorem in this setting, showing that the coextensions of a monoid functor with a system of abelian groups is a symmetric categorical group and equivalent to the one obtained by the abelian group homomorphism $[\mathcal{C}^0 \to \mathsf{ker}\partial^1]$, thereby linking with cohomology of certain types of complexes, as expected. We then move towards monoid schemes, which are a type of a monoid sheaf, but where localisations now allow us to develop our most noteworthy result: We show that coextensions can be seen in a natural way as a stack of symmetric categorical groups. We will mention a few mild implications of this, but leave the deeper uses of stack theory in this setting for later papers.