全局Picard谱与Borel参数化代数
Global Picard Spectra and Borel Parametrized Algebra
AI总结：
本文回答了Schwede关于超交换全局环谱的全局Picard谱存在性问题，构造了组装所有底层$G$-等变环谱Picard谱的全局谱，并发展了参数化高代数中的Borel-等变对象推广及比较函子。
摘要：
我们回答了Schwede关于与其超交换全局环谱相关的全局Picard谱存在性的问题；给定一个超交换全局环谱$R$，我们证明存在一个全局谱$\mathrm{pic}_\mathrm{eq}(R)$，将$R$的所有底层$G$-等变环谱$\mathrm{res}_G R$的Picard谱组装成一个对象，即对所有有限群$G$，其真不动点由$\mathrm{pic}_\mathrm{eq}(R)^G \simeq \mathrm{pic}(\mathrm{Mod}_{\mathrm{res}_G R}(\mathrm{Sp}_G))$给出。在此过程中，我们在参数化高代数的背景下发展了Borel-等变对象的推广。我们利用它将所有有限群$G$的$G$-谱对称幺半范畴连同所有限制与范数组装成一个单一的“范数化全局范畴”，并构建了一个比较函子，使我们能将超交换$G$-等变或全局环谱导入参数化高代数的背景中。
英文摘要：
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.
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