Prismatization via spherical loop spaces

TL;DR

This paper develops new spectral enhancements of prismatization stacks using Frobenius-untwists and spherical loop spaces, connecting them with known structures like Breuil-Kisin twists and Drinfeld formal groups.

Contribution

It introduces Frobenius-untwists of topological cyclic homology variants and constructs even periodic spectral enhancements of prismatization stacks.

Findings

Spectral enhancements relate to known structures like Breuil-Kisin twists.

Construction of modifications of the free loop space over the sphere spectrum.

Identification of extra structures with established mathematical objects.

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.