An Alternative to Spherical Witt Vectors

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.

7 pages; added Remark 3 that generalizes main results; final version to appear in the Proceedings of the American Mathematical Society