A Dual Representation in Spectral Algebraic Geometry

2020/06/28 by Gauthier, Renaud
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2006.15687

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 → X. For the class of affine morphisms in SpDM, we show that from QCoh(X) on can extract the affine perception Aff_X of X on the one hand, and a subcategory of an ∞-category of representations Rep_\mathfrakg_* of a dg Lie algebra \mathfrakg_* associated with X on the other. For the class of local morphisms Spét R → X, the local perception of X is given by the functor X = Hom(Spét(-), X) it represents. If X is a geometric stack, Tannaka duality allows us to recover X from QCoh(X), from which we can also get, after base change, a subcategory of Rep_\mathfrakg_*. We generalize those results by considering functors X: CAlg^cn → S that are representable in accordance with the spectral Artin representability theorem of Lurie.

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