[Paper Review] Non-commutative derived moduli prestacks

J. P. Pridham|arXiv (Cornell University)|Aug 26, 2020

Algebraic structures and combinatorial models28 references4 citations

TL;DR

This paper introduces a derived non-commutative (NC) moduli formalism for differential graded associative algebras (DGAAs), establishing a framework for derived NC prestacks without relying on descent. It constructs atlases via idempotent parametrization and introduces stacky DGAAs as derived non-commutative analogues of Lie algebroids, enabling the recovery of invariants via rigid points and facilitating the extension of shifted bi-symplectic and double Poisson structures to derived NC prestacks.

ABSTRACT

We introduce a formalism for derived moduli functors on differential graded associative algebras, which leads to non-commutative enhancements of derived moduli stacks and naturally gives rise to structures such as Hall algebras. Descent arguments are not available in the non-commutative context, so we establish new methods for constructing various kinds of atlases. The formalism permits the development of the theory of shifted bi-symplectic and shifted double Poisson structures in the companion paper.

Motivation & Objective

• To develop a derived moduli formalism for non-commutative algebraic geometry based on differential graded associative algebras (DGAAs), extending derived algebraic geometry beyond the commutative setting.
• To overcome the absence of descent in the non-commutative setting by constructing atlases through idempotent parametrization and homotopy étale morphisms.
• To define and study derived NC Artin prestacks, particularly for moduli of perfect complexes and projective modules, using a new framework of stacky DGAAs.
• To enable the extension of shifted bi-symplectic and double Poisson structures to derived NC prestacks via the rigid core of stacky DGAAs.
• To show that invariants of derived NC prestacks can be recovered from their rigid points in the stacky DGAA framework, reducing structure classification to étale functoriality.

Proposed method

• Proposes a formalism for derived NC prestacks as functors on non-negatively graded DGAAs, restricting to commutative derived moduli functors on CDGAs and non-commutative deformation functors on Artinian DGAAs.
• Introduces stacky DGAAs—bigraded associative algebras with one grading for derived structure and one for stacky structure—where equivalences are defined relative to the derived grading.
• Defines the derived prestack $ D_*F $ associated to a derived NC prestack $ F $, and isolates its rigid core $ (D_*F)_{\mathrm{rig}} $, consisting of points corresponding to suitably étale maps from stacky DGAAs.
• Uses homotopy colimits and derived left adjoints to show that $ D_*F \simeq \mathbf{L}\theta^*(D_*F)_{\mathrm{rig}} $, establishing that $ D_*F $ is determined by its rigid part.
• Applies the theory to construct atlases for moduli of perfect complexes and projective modules by parametrizing idempotents in the DGAAs.
• Demonstrates that maps from $ F $ to a structure-classifying functor $ G $ correspond to structures on $ D_*F $, with the key result that only homotopy étale functoriality of $ D_*G $ is required.

Experimental results

Research questions

• RQ1 How can derived moduli functors be generalized to non-commutative settings without relying on descent, which fails in the non-commutative context?
• RQ2 What is the appropriate non-commutative analogue of a smooth atlas or Lie algebroid in derived algebraic geometry?
• RQ3 How can shifted bi-symplectic and double Poisson structures be defined on derived NC prestacks when descent is unavailable?
• RQ4 Can invariants of derived NC prestacks be recovered from their rigid points in the stacky DGAA framework?
• RQ5 What is the relationship between the derived NC prestack $ F $, its associated stacky DGAA prestack $ D_*F $, and the rigid core $ (D_*F)_{\mathrm{rig}} $?

Key findings

• The moduli functors of projective modules and perfect complexes over a DGAA are shown to be derived NC Artin ∞-prestacks, establishing foundational examples in the new formalism.
• A new method for constructing atlases is developed by parametrizing idempotents in DGAAs, circumventing the need for descent in the non-commutative setting.
• The prestack $ D_*F $ associated to a derived NC prestack $ F $ is shown to be equivalent to the derived left adjoint of its rigid core: $ D_*F \simeq \mathbf{L}\theta^*(D_*F)_{\mathrm{rig}} $.
• The rigid core $ (D_*F)_{\mathrm{rig}} $ captures all essential invariants of $ F $, allowing the classification of structures on $ F $ to be reduced to those on $ (D_*F)_{\mathrm{rig}} $.
• A natural weak equivalence is established between maps from $ F $ to a structure-classifying functor $ G $ and maps from $ D_*F $ to $ D_*G $, showing that only homotopy étale functoriality of $ D_*G $ is required.
• The framework enables the extension of shifted double Poisson and bi-symplectic structures to derived NC prestacks, as shown in the companion paper [Pri9], by reducing to the rigid stacky DGAA setting.