Graded loopspaces in mixed characteristics and de Rham-Witt algebra

TL;DR

This paper develops a theory of graded loop spaces for mixed characteristic derived schemes with Frobenius lifts, aiming to define the de Rham-Witt complex in derived algebraic geometry.

Contribution

It introduces a new graded loop space construction for derived schemes with Frobenius lifts, serving as a foundation for the de Rham-Witt complex in this setting.

Findings

Defined derived Frobenius lifts as homotopy-theoretic delta-structures.

Established the graded loop space as an action of the crystalline circle.

Connected derived Dieudonné complexes to actions of the crystalline circle.

Abstract

This PhD manuscript focuses on the study of a variation of the graded loop space construction for mixed graded derived schemes endowed with a Frobenius lift. We developed a theory of derived Frobenius lifts on a derived stack which are homotopy theoretic analogues of delta-structures for commutative rings. This graded loop space construction is the first step towards a definition of the de Rham-Witt complex for derived schemes. In this context, a loop is given by an action of the "crystalline circle", which is a formal analogue of the topological circle, endowed with its natural endomorphism given by multiplication by p. In this language, a derived Dieudonn\'e complex can be seen as a graded module endowed with an action of the crystalline circle.