Floor, ceiling, slopes, and K-theory. Yuri Sulyma.

We calculate K_*(k[x]/x^e;Z_p) by evaluating the syntomic cohomology 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.