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

Yuri J. F. Sulyma

Vol. 8 (2023), No. 3, 331–354
Abstract

We calculate K (k[x]xe; p) by evaluating the syntomic cohomology

p(i)(k[x]xe)

introduced by Bhatt, Morrow and 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.

Keywords
K-theory, truncated polynomials, prismatic cohomology, syntomic cohomology, crystalline cohomology
Mathematical Subject Classification
Primary: 14F30, 19D55, 19E15
Supplementary material

Times tables for K

Milestones
Received: 14 December 2021
Revised: 22 January 2023
Accepted: 31 May 2023
Published: 27 August 2023
Authors
Yuri J. F. Sulyma
Brooklyn, NY
United States