Floor, ceiling, slopes, and $K$-theory

Abstract

We calculate ${\mathrm{K}}_{\ast }(k[x]/x^e;{\mathbf{Z}}_p)$ by evaluating the syntomic cohomology ${\mathbf{Z}}_p(i)(k[x]/x^e)$ introduced by Bhatt-Morrow-Scholze and Bhatt-Scholze. This recovers calculations of Hesselholt-Madsen and Speirs, and generalizes an example of Mathew treating the case $e=2$ and $p>2$. Our main innovation is systematic use of the floor and ceiling functions, which clarifies matters substantially even for $e=2$. We furthermore observe a persistent phenomenon of slopes. As an application, we answer some questions of Hesselholt.

v2: expanded exposition; added link to interactive figures/source code; fixed wildly incorrect mod-p multiplicative structure. 24 pages, 8 figures

References in corpus (3)

• Some recent advances in topological Hochschild homology
• The tower of K-theory of truncated polynomial algebras
• Verschiebung maps among $K$-groups of truncated polynomial algebras