An Alternative to Spherical Witt Vectors

TL;DR

This paper introduces a new construction of spherical Witt vectors for perfect F_p-algebras, generalizing previous work, and explores their module categories and universal properties.

Contribution

It provides a direct construction of spherical Witt vectors as a completed monoid algebra, extending prior approaches and establishing their universal property as an E_1-ring.

Findings

Construction as completion of spherical monoid algebra

Description of p-complete modules over spherical Witt vectors

Universal property as an E_1-ring

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.