Witt Vectors, Polynomial Maps, and Real Topological Hochschild Homology

TL;DR

This paper explores the functorial properties of Witt vectors with respect to polynomial laws and applies these results to extend Witt vectors to Tambara functors, ultimately analyzing real topological Hochschild homology fixed points.

Contribution

It establishes new functoriality results for Witt vectors and extends their application to Tambara functors and real topological Hochschild homology.

Findings

Witt vectors are functorial under polynomial laws of finite degree.

Extension of Witt vectors to $ ext{Z}/2$-Tambara functors for odd primes.

Description of dihedral fixed-points in real topological Hochschild homology.

Abstract

We show that various flavors of Witt vectors are functorial with respect to multiplicative polynomial laws of finite degree. We then deduce that the $p$-typical Witt vectors are functorial in multiplicative polynomial maps of degree at most $p-1$. This extra functoriality allows us to extend the $p$-typical Witt vectors functor from commutative rings to $\mathbb{Z}/2$-Tambara functors, for odd primes $p$. We use these Witt vectors for Tambara functors to describe the components of the dihedral fixed-points of the real topological Hochschild homology spectrum at odd primes.