On the K-theory of Z/pⁿ

2024/05/07 by Antieau, Benjamin, Krause, Achim, Nikolaus, Thomas
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2405.04329

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 Z/pⁿ 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 v₁ acting on the mod p syntomic cohomology of Z/pⁿ.

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