Foundations of superstack theory

TL;DR

This paper develops a foundational framework for superstack theory in algebraic supergeometry, defining key concepts like quotient superstacks and bosonic reduction, and exploring their properties and structures.

Contribution

It introduces the systematic study of superstacks, including definitions, structures, and criteria for properties like being Deligne-Mumford, extending classical algebraic geometry concepts to the supergeometric setting.

Findings

Defined superstacks and quotient superstacks.

Established criteria for Deligne-Mumford superstacks.

Analyzed properties of morphisms between superstacks.

Abstract

In view of applications to the construction of moduli spaces of objects in algebraic supergeometry, we start a systematic study of stacks in that context. After defining a superstack as a stack over the \'etale site of superschemes, we define quotient superstacks, and, based on previous literature, we see that, in analogy with superschemes, every superstack has an underlying ordinary stack, which we call its bosonic reduction. Then we progressively introduce more structure, considering algebraic superspaces, Deligne-Mumford superstacks and algebraic superstacks. We study the topology of algebraic superstacks and several properties of morphisms between them. We introduce quasi-coherent sheaves, and the sheaves of relative differentials. An important issue is how to check that an algebraic superstack is Deligne-Mumford, and we generalize to this setting the usual criteria in terms of the unramifiedness of the diagonal of the stack. Two appendices are devoted to collecting the basic definitions of group superschemes and principal superbundles, and to stating and analyzing some properties of morphisms of superschemes, that are at the basis of the study of morphisms of superstacks in the main text.