An Alternative to Spherical Witt Vectors

TL;DR

The paper constructs spherical Witt vectors in the spectral setting and proves they arise as the $I$-adic completion $\mathbb{S}[R]^{\wedge}_I$ of the spherical monoid algebra $\mathbb{S}[R]$, with $I = \mathrm{fib}(\mathbb{S}[R] \to R)$. It proves this by identifying the completion with the Amitsur complex of $\mathbb{S}[R] \to R$ and showing the map $\mathbb{S}[R]^{\wedge}_p \to \mathbb{S}_{{\mathrm{W}\xspace}(R)}$ is a $p$-complete homological epimorphism; the Adams tower (a variant of Bousfield–Kan completion) ensures convergence, and the construction extends to general $p$-divisible monoids. The work then describes the $\infty$-category of $p$-complete modules over $\mathbb{S}_{{\mathrm{W}\xspace}(R)}$ as the restriction-full image inside $\mathbb{S}[R]$-modules, with a concrete $I$-complete criterion for membership, and gives a universal property characterizing spherical Witt vectors as the universal $p$-complete $\mathbb{E}_1$-ring receiving a multiplicative map from $R$ that becomes additive mod $p$. These results parallel the classical Cuntz–Deninger description for ordinary Witt vectors and extend to higher $\mathbb{E}_n$ contexts, providing a robust framework for applications in chromatic homotopy theory.

Abstract

We give a direct construction of the ring spectrum of spherical Witt vectors of a perfect $\mathbb{F}_p$-algebra R as the completion of the spherical monoid algebra $\mathbb{S}[R]$ of the multiplicative monoid $(R,\cdot)$ at the ideal $I = \mathrm{fib}(\mathbb{S}[R] \to R)$. This generalizes a construction of Cuntz and Deninger. We also use this to give a description of the category of p-complete modules over the spherical Witt vectors and a universal property for spherical Witt vectors as an $\mathbb{E}_1$-ring.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Corollary 1
• Remark 2
• Proposition 3
• Lemma 1.1
• proof
• Lemma 1.2
• proof
• proof : Proof of Theorem \ref{['thm:spherical Witt']}