Global Picard Spectra and Borel Parametrized Algebra
Phil Pützstück
TL;DR
The paper constructs global Picard spectra for ultra-commutative global ring spectra by assembling the Picard spectra of all underlying $G$-spectra into a global object, and proves a fixed-point formula $\text{pic}_{eq}(R)^G \simeq \text{pic}(\text{Mod}_{\text{res}_G R}(\text{Sp}_G))$ for finite $G$. It develops a parametrized higher-algebra framework with normed (global) categories and Borelification to compatibly transport equivariant/global ring spectra into normed algebras, connecting $G$-equivariant, $G$-global, and global stable homotopy theories through DK localization and monoidal Borelifications. The paper then constructs normed global categories of equivariant and global spectra, provides comparison functors from ultra-commutative ring spectra to normed algebras, and develops a parametrized module theory that yields parametrized Picard functors. A central outcome is that the global Picard spectrum of an ultra-commutative global ring spectrum can be understood levelwise via genuine fixed points, and the equivariant/global Picard theories are naturally equivalent through the restriction/induction framework, enabling a unified description across $G$-equivariant and $G$-global contexts.
Abstract
We answer a question of Schwede on the existence of global Picard spectra associated to his ultra-commutative global ring spectra; given an ultra-commutative global ring spectrum $R$, we show there exists a global spectrum $\mathrm{pic}_\mathrm{eq}(R)$ assembling the Picard spectra of all underlying $G$-equivariant ring spectra $\mathrm{res}_G R$ of $R$ into one object, in that for all finite groups $G$, the genuine fixed points are given by $\mathrm{pic}_\mathrm{eq}(R)^G \simeq \mathrm{pic}(\mathrm{Mod}_{\mathrm{res}_G R}(\mathrm{Sp}_G))$. Along the way, we develop a generalization of Borel-equivariant objects in the setting of parametrized higher algebra. We use this to assemble the symmetric monoidal categories of $G$-spectra for all finite groups $G$ together with all restrictions and norms into a single `normed global category', and build a comparison functor which allows us to import ultra-commutative $G$-equivariant or global ring spectra into the setting of parametrized higher algebra.