Cartier modules and cyclotomic spectra

TL;DR

This paper introduces a new t-structure on p-typical cyclotomic spectra, enabling the recovery of crystalline cohomology for smooth schemes over perfect fields through the novel concept of p-typical topological Cartier modules.

Contribution

It develops a new approach to p-typical cyclotomic spectra using p-typical topological Cartier modules and characterizes the heart of the cyclotomic t-structure.

Findings

The heart of the cyclotomic t-structure corresponds to derived V-complete objects.

A new framework for understanding crystalline cohomology via cyclotomic spectra.

Establishment of a full subcategory of p-typical Cartier modules as the heart.

Abstract

We construct and study a t-structure on p-typical cyclotomic spectra and explain how to recover crystalline cohomology of smooth schemes over perfect fields using this t-structure. Our main tool is a new approach to p-typical cyclotomic spectra via objects we call p-typical topological Cartier modules. Using these, we prove that the heart of the cyclotomic t-structure is the full subcategory of derived V-complete objects in the abelian category of p-typical Cartier modules.