Foundations of superstack theory
Ugo Bruzzo, Daniel Hernández Ruipérez
TL;DR
This work develops a foundational framework for superstacks in algebraic supergeometry, defining stacks over the étale site of superschemes and constructing quotient superstacks, bosonic reductions, and a robust theory of algebraic and Deligne–Mumford superstacks. It extends classical stack theory to the super setting, establishing representability criteria, morphism properties, and topological/dimensional notions that mirror the bosonic case while incorporating odd (fermionic) directions. A major contribution is the characterization of Deligne–Mumford superstacks via unramified diagonals and the analysis of quasi-coherent sheaves, relative differentials, and descent in both Deligne–Mumford and algebraic contexts, enabling moduli space constructions in algebraic supergeometry. The paper also situates superstacks within a broader toolkit that includes quotient constructions, universal families, and the theory of group superschemes and principal superbundles, underscoring potential applications to supermoduli spaces and related moduli problems. Overall, the work provides a systematic, technically detailed foundation for moduli problems in supersymmetric settings, aligning supergeometric structures with their bosonic counterparts while handling the additional superstructure coherently.
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 étale 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.