On the algebraic K-theory of Witt vectors of finite length

TL;DR

This paper investigates the algebraic K-theory of Witt vectors of finite length over perfect fields, demonstrating Galois descent and identifying the first p-torsion element in stable homotopy groups within specific K-groups.

Contribution

It proves Galois descent for the K-theory of Witt vectors and computes K-groups, revealing the detection of the first p-torsion element in stable homotopy groups.

Findings

K(W_n(k)) satisfies Galois descent.

Computed K-groups up to certain degrees.

First p-torsion element detected in K_{2p-3}(W_n(k)).

Abstract

Let k be a perfect field of characteristic p and let $W_n(k)$ denote the p-typical Witt vectors of length n. For example, $W_n(\mathbb{F}_p)=\mathbb{Z}/p^n$. We study the algebraic K-theory of $W_n(k)$, and prove that $K(W_n(k))$ satisfies "Galois descent". We also compute the K-groups through a range of degrees, and show that the first p-torsion element in the stable homotopy groups of spheres is detected in $K_{2p-3}(W_n(k))$ for all $n \geq 2$.