Witt vectors and $\delta$-Cartier rings

TL;DR

This paper establishes a universal property for the construction of p-typical Witt vectors, extends it to derived rings, and characterizes derived Witt vectors within an infinity-category framework.

Contribution

It introduces a universal property for Witt vectors, generalizes to derived rings, and characterizes derived Witt vectors as an equivalence in an infinity-category.

Findings

Defined an infinity-category of p-typical derived δ-Cartier rings.

Showed the derived Witt vector construction is an object in this category.

Established an equivalence between derived rings and derived p-typical δ-Cartier rings.

Abstract

We give a universal property of the construction of the ring of $p$-typical Witt vectors of a commutative ring, endowed with Witt vectors Frobenius and Verschiebung, and generalize this construction to the derived setting. We define an $\infty$-category of $p$-typical derived $\delta$-Cartier rings and show that the derived ring of $p$-typical Witt vectors of a derived ring is naturally an object in this $\infty$-category. Moreover, we show that for any prime $p$, the formation of the derived ring of $p$-typical Witt vectors gives an equivalence between the $\infty$-category of all derived rings and the full subcategory of all derived $p$-typical $\delta$-Cartier rings consisting of $V$-complete objects.