Prismatization via spherical loop spaces
Rok Gregoric
TL;DR
This work constructs even periodic enhancements of prismatization stacks that bridge Hodge theory in mixed characteristic with topological cyclic homology. By defining Frobenius-untwisted variants $\mathrm{TP}^{(-1)}$ and $\mathrm{TC}^{-(-1)}$ for qrsp rings and extending them via quasi-syntomic descent, the authors produce canonical even periodic spectral stacks $\pounds X$, $\textlira X$, and $\$ X$ whose underlying classical stacks recover the prismatization data $X^{\mathbbl\Delta}$, $X^{\mathcal N}$, and $X^{\mathrm{Syn}}$. The even periodic enhancements encode Breuil–Kisin twists and the Drinfeld formal group on the prismatization stacks, unifying prismatic cohomology with cyclic homology and revealing loop-space analogies through their $\mathrm L$-space interpretations. The results extend Bhatt–Morrow–Scholze by removing Nygaard completions and relaxing affineness, while providing descent-based constructions and connections to animated, global, and loop-space perspectives. Overall, the paper advances a cohesive spectral-geometric framework for prismatic cohomology and its Hodge-theoretic structures via Frobenius-twisted topological cyclic homology and even periodic enhancements.
Abstract
We introduce Frobenius-untwists of the variants of topological cyclic homology, following Manam. Using these, we construct modifications of the free loop space over the sphere spectrum, and show that they provide even periodic spectral enhancements of the prismatization stacks of Bhatt-Lurie and Drinfeld. We identify the extra structure on prismatization encoded in the even periodic enhancements with previously-known structures, such as the Breuil-Kisin twists and the Drinfeld formal group.