Constructing derived moduli stacks

TL;DR

This paper develops three comprehensive frameworks for constructing global derived moduli stacks, enabling the study of various geometric and algebraic objects within a unified derived setting.

Contribution

It introduces three new approaches—using dg-Lie algebras, cosimplicial groups, and quasi-comonoids—for building derived moduli stacks, broadening the scope of problems that can be addressed.

Findings

Successfully constructed derived moduli stacks for finite schemes and polarised projective schemes.

Provided explicit examples including torsors, coherent sheaves, and finite group schemes.

Bridged the gap between concrete and abstract derived moduli concepts.

Abstract

We introduce frameworks for constructing global derived moduli stacks associated to a broad range of problems, bridging the gap between the concrete and abstract conceptions of derived moduli. Our three approaches are via differential graded Lie algebras, via cosimplicial groups, and via quasi-comonoids, each more general than the last. Explicit examples of derived moduli problems addressed here are finite schemes, polarised projective schemes, torsors, coherent sheaves, and finite group schemes.