A Dual Representation in Spectral Algebraic Geometry

TL;DR

This paper explores dual perceptions of spectral Deligne-Mumford stacks, linking geometric and algebraic representations via QCoh, dg Lie algebras, and Tannaka duality, extending spectral Artin representability.

Contribution

It introduces a dual framework for perceiving spectral stacks through affine and local morphisms, connecting geometric and algebraic data with new generalizations.

Findings

Extraction of affine perception from QCoh(X).

Recovery of local perception via functors representing X.

Extension of spectral Artin representability to new functors.

Abstract

Given a spectral Deligne-Mumford stack $X$, we define a perception of $X$ to be a collection of a certain class of morphisms $Y \rightarrow X$. For the class of affine morphisms in SpDM, we show that from QCoh($X$) on can extract the affine perception $\text{Aff}_X$ of $X$ on the one hand, and a subcategory of an $\infty$-category of representations $\text{Rep}_{\mathfrak{g}_*}$ of a dg Lie algebra $\mathfrak{g}_*$ associated with $X$ on the other. For the class of local morphisms $\text{Sp\'et } R \rightarrow X$, the local perception of $X$ is given by the functor $\mathbf{X} = \text{Hom}(\text{Sp\'et}(-), X)$ it represents. If $\mathbf{X}$ is a geometric stack, Tannaka duality allows us to recover $\mathbf{X}$ from $\text{QCoh}(\mathbf{X})$, from which we can also get, after base change, a subcategory of $\text{Rep}_{\mathfrak{g}_*}$. We generalize those results by considering functors $\mathbf{X}: \text{CAlg}^{\text{cn}} \rightarrow \mathcal{S}$ that are representable in accordance with the spectral Artin representability theorem of Lurie.