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

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arxiv 2405.04329 v1 pith:R6RLSEVT submitted 2024-05-07 math.KT math.AGmath.AT

Benjamin Antieau , Achim Krause , Thomas Nikolaus This is my paper · ORCID
classification math.KTmath.AGmath.AT
keywords k-groupsalgebraicmathbfcohomologydegreesdescriptionhighrings
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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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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the integral algebraic K-theory of Morava K-theory

    math.AT 2026-07 conditional novelty 8.0 of 10

    For connective Morava K-theory, the paper determines algebraic K-theory group cardinalities in all degrees outside two congruence classes over finite fields, and proves even-degree groups vanish over algebraically clo...

  2. On the motivic cohomology of some singular rings

    math.AG 2026-08 conditional novelty 6.0 of 10

    The non-A1-invariant motivic cohomology groups are now explicitly computed for finite chain rings, truncated polynomial rings, perfect and semiperfect rings, valuation rings, and commutative C*-algebras.

  3. A note on prismatic sites for p-quasisyntomic rings

    math.AG 2026-07 accept novelty 5.0 of 10

    Transversal objects and relatively quasiregular semiperfectoid covers of a p-quasisyntomic ring R produce objects in R_Δ that cover the final object and admit finite self-coproducts.

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