Achim Krause, Thomas Nikolaus, and I have uploaded our research announcement [1] on the $\K$-groups of $\bZ/p^n$ to the arXiv. We constructed an algorithm to compute the syntomic cohomology $\bZ_p(i)$ in the sense of Bhatt–Morrow–Scholze of $\bZ/p^n$ or more generally $\Oscr_K/\varpi^n$ where $K$ is a finite extension of $\bQ_p$ and $\varpi$ is a uniformizer of the ring of integers $\Oscr_K$.

The BMS spectral sequence in this case collapses entirely and hence the algorithm gives a way to compute the $p$-adic $\K$-groups:

$\K_{2i-1}(\Oscr_K/\varpi^n;\bZ_p)\cong\H^1(\bZ_p(i)(\Oscr_K/\varpi^n))$

for $i\geq 1$ and

$\K_{2i-2}(\Oscr_K/\varpi^n;\bZ_p)\cong\H^2(\bZ_p(i)(\Oscr_K/\varpi^n))$

for $i\geq 2$. The $\K$-groups are torsion and the prime-to-$p$ information is governed by Quillen’s computation of the $\K$-theory of finite fields.

See the announcement for tables of computations and references. After running our algorithm in a bunch of cases, we conjectured that in fact the even groups vanish for $i$ sufficiently large. This is indeed the case.

Theorem (Even vanishing theorem). If

$i\geq\frac{p^2}{(p-1)^2}\left(p^{\lceil\tfrac{n}{e}\rceil}-1\right),$

then $\K_{2i-2}(\Oscr_K/\varpi^n)=0$.

Recall that Bhatt and Scholze proved the odd vanishing conjecture, namely that odd $\K$-groups vanish and even $\K$-groups are $p$-torsion free quasisyntomic-locally . (This had been proved in characteristic $p$ first in BMS2.) Bhatt and Scholze also proved a more precise statement which showed in particular that $\K_*(\Oscr_{\bC_p}/p^n;\bZ_p)$ is concentrated in even degrees. Combined with our theorem, one obtains the following consequence.

Corollary. If $i\geq\frac{p^2(p^n-1)}{(p-1)^2}$, then $\K_{2i-2}(\Oscr_{\bC_p}/p^n;\bZ_p)$ is $p$-torsion free.

Added 03 May 2022: Bhargav Bhatt pointed out that the stronger form of the odd vanishing conjecture can be used to make the proof below easier.

Added 14 May 2022: Bhargav pointed out again that there is an easier argument! I was worried about something the even groups in $\K_{2i-2}(\Oscr_K/\varpi^m)$ somehow accumulating to contribute to $\H^2(\bZ_p(i)(\Oscr_{\bC_p}/p^n))$, which is not possible since for any quasiregular semiperfectoid ring $R$ the $p$-adic syntomic complexes $\bZ_p(i)(R)$ have cohomology concentrated in degrees $[0,1]$. So, this `corollary’ is really an easier corollary of the results of Bhatt–Morrow–Scholze and Bhatt–Scholze.

# References

[1] Antieau, Krause, and Nikolaus, The K-theory of $\bZ/p^n$, arXiv:2204.03420.

[2] Bhatt, Scholze, Prisms and prismatic cohomology, arXiv:1905.08229.