Animated $\lambda$-rings and Frobenius lifts

TL;DR

This paper introduces animated λ-rings as an integral analogue of animated δ-rings, defining them via compatible Frobenius lifts and connecting to prismatic cohomology, expanding the framework of λ-ring theory.

Contribution

It formalizes animated λ-rings using animated rings with Frobenius lifts, extending classical λ-ring concepts into the animated setting based on prismatic cohomology.

Findings

Defined animated λ-rings with compatible Frobenius lifts

Connected animated λ-rings to classical λ-ring theory via ∞-categories

Built on the theory of animated δ-rings by Bhatt and Lurie

Abstract

In this note, we study an integral analogue of animated $\delta$-rings: Animated $\lambda$-rings. We define animated $\lambda$-rings in terms of animated rings equipped with a structure of coherently compatible Frobenius lifts and show that the resulting $\infty$-category is obtained from animating the classical notion of a $\lambda$-ring. Our results build on the theory of animated $\delta$-rings developed by Bhatt and Lurie in the context of prismatic cohomology.