On the $K$-theory of $\mathbf{Z}/p^n$

TL;DR

This paper provides an explicit algebraic description of the K-theory of rings like Z/p^n using prismatic cohomology, enabling practical computation and revealing structural properties of these K-groups.

Contribution

It introduces a new algebraic framework based on prismatic cohomology for computing and understanding the K-theory of rings Z/p^n, including algorithms and theoretical results.

Findings

Vanishing of even K-groups in high degrees

Determination of orders of odd K-groups in high degrees

Degree of nilpotence of v_1 on mod p syntomic cohomology

Abstract

We give an explicit algebraic description, based on prismatic cohomology, of the algebraic K-groups of rings of the form $O_K/I$ where $K$ is a p-adic field and $I$ is a non-trivial ideal in the ring of integers $O_K$; this class includes the rings $\mathbf{Z}/p^n$ where $p$ is a prime. The algebraic description allows us to describe a practical algorithm to compute individual K-groups as well as to obtain several theoretical results: the vanishing of the even K-groups in high degrees, the determination of the orders of the odd K-groups in high degrees, and the degree of nilpotence of $v_1$ acting on the mod $p$ syntomic cohomology of $\mathbf{Z}/p^n$.