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

Benjamin Antieau; Achim Krause; Thomas Nikolaus

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arXiv · 2405.04329

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

Benjamin Antieau·Achim Krause·Thomas Nikolaus

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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 $\mathbf{Z}/p^n$.

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BibTeX RIS

Benjamin Antieau, Achim Krause, Thomas Nikolaus. 2024-05-07. On the $K$-theory of $\mathbf{Z}/p^n$. https://arxiv.org/abs/2405.04329

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