Exact bounds for even vanishing of $K_* (\mathbb{Z}/p^n)$

TL;DR

This paper precisely characterizes when the even K-theory groups of rac{rac{p^n}{} }{ ext{p}} are nonzero, refining previous vanishing results, and determines the nilpotence order of a specific element in algebraic K-theory.

Contribution

It provides exact bounds for the non-vanishing of even K-theory groups of rac{rac{p^n}{} }{ ext{p}} and links this to syntomic cohomology and recent crystallinity results.

Findings

Exact bounds for non-vanishing of K_{2i}(rac{rac{p^n}{} }{ ext{p}}).

Determination of the nilpotence order of v_1 in rac{ ext{pi}_* K(rac{rac{p^n}{} }{ ext{p}})}{ ext{p}}.

Refinement of the even vanishing theorem for algebraic K-theory.

Abstract

In this note, we prove that $K_{2i} (\mathbb{Z}/p^n) \neq 0$ if and only if $p-1$ divides $i$ and $0 \leq i \leq (p-1) p^{n-2}$, refining the even vanishing theorem of Antieau, Nikolaus and the first author in this case. As a corollary of our proof, we determine that the nilpotence order of $v_1$ in $\pi_* K(\mathbb{Z}/p^n)/p$ is equal to $\frac{p^n-1}{p-1}$. Our proof combines the recent crystallinity result for reduced syntomic cohomology of Hahn, Levy and the second author with the explicit complex computing the syntomic cohomology of $\mathcal{O}_K /\varpi^n$ constructed by Antieau, Nikolaus and the first author.