Exact bounds for even vanishing of K *(ℤ/ p n )

Source abstract

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.