On the K-theory of truncated polynomial algebras over the integers

TL;DR

This paper computes the algebraic K-theory groups of truncated polynomial algebras over integers, revealing their finiteness and free abelian structures, and connects these to topological Hochschild homology spectra.

Contribution

It provides explicit formulas for K-theory groups of truncated polynomial rings over integers, linking algebraic K-theory to topological Hochschild homology.

Findings

K_{2i}(Z[x]/(x^m),(x)) is finite with order (mi)!(i!)^{m-2}

K_{2i+1}(Z[x]/(x^m),(x)) is free abelian of rank m-1

Equivariant homotopy groups of THH(Z) are finite in odd degrees and free abelian in even degrees

Abstract

We show that the K_{2i}(Z[x]/(x^m),(x)) is finite of order (mi)!(i!)^{m-2} and that K_{2i+1}(Z[x]/(x^m),(x)) is free abelian of rank m-1. This is accomplished by showing that the equivariant homotopy groups of the topological Hochschild spectrum THH(Z) are finite, in odd degrees, and free abelian, in even degrees, and by evaluating their orders and ranks, respectively.