$
\newcommand\nil{\mathrm{nil}}
\newcommand\gfrak{\mathfrak{g}}
\newcommand\A{\mathrm{A}}
\newcommand\B{\mathrm{B}}
\newcommand\C{\mathrm{C}}
\newcommand\D{\mathrm{D}}
\newcommand\E{\mathrm{E}}
\newcommand\F{\mathrm{F}}
\newcommand\G{\mathrm{G}}
\newcommand\H{\mathrm{H}}
\newcommand\h{\mathrm{h}}
\newcommand\K{\mathrm{K}}
\newcommand\L{\mathrm{L}}
\newcommand\M{\mathrm{M}}
\newcommand\R{\mathrm{R}}
\newcommand\t{\mathrm{t}}
\newcommand\T{\mathrm{T}}
\newcommand{\bA}{\mathbf{A}}
\newcommand{\bF}{\mathbf{F}}
\newcommand{\bG}{\mathbf{G}}
\newcommand{\bH}{\mathbf{H}}
\newcommand{\bT}{\mathbf{T}}
\newcommand{\bW}{\mathbf{W}}
\newcommand{\Gm}{\bG_m}
\newcommand\Ascr{\mathcal{A}}
\newcommand\Cscr{\mathcal{C}}
\newcommand\Dscr{\mathcal{D}}
\newcommand\Escr{\mathcal{E}}
\newcommand\Fscr{\mathcal{F}}
\newcommand\Kscr{\mathcal{K}}
\newcommand\Lscr{\mathcal{L}}
\newcommand\Oscr{\mathcal{O}}
\newcommand\Perf{\mathrm{Perf}}
\newcommand\Perfscr{\mathcal{P}\mathrm{erf}}
\newcommand\Acscr{\mathcal{A}\mathrm{c}}
\newcommand\heart{\heartsuit}
\newcommand\cn{\mathrm{cn}}
\newcommand\op{\mathrm{op}}
\newcommand\gr{\mathrm{gr}}
\newcommand\Gr{\mathrm{Gr}}
\newcommand\fil{\mathrm{fil}}
\newcommand\Ho{\mathrm{Ho}}
\newcommand\dR{\mathrm{dR}}
\newcommand\dRhat{\widehat{\dR}}
\newcommand\we{\simeq}
\newcommand\Sym{\mathrm{Sym}}
\newcommand\HH{\mathrm{HH}}
\newcommand\HC{\mathrm{HC}}
\newcommand\HP{\mathrm{HP}}
\newcommand\TC{\mathrm{TC}}
\newcommand\TR{\mathrm{TR}}
\newcommand\THH{\mathrm{THH}}
\newcommand{\bMap}{\mathbf{Map}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\Mod}{\mathrm{Mod}}
\newcommand{\coMod}{\mathrm{coMod}}
\newcommand{\Fun}{\mathrm{Fun}}
\newcommand{\bMap}{\mathbf{Map}}
\newcommand\bE{\mathbf{E}}
\newcommand\bZ{\mathbf{Z}}
\newcommand\bS{\mathbf{S}}
\newcommand\bQ{\mathbf{Q}}
\newcommand\bC{\mathbf{C}}
\newcommand\bN{\mathbf{N}}
\newcommand\bAM{\mathbf{AM}}
\newcommand\bLM{\mathbf{LM}}
\newcommand\Spec{\mathrm{Spec}\,}
\newcommand\CAlg{\mathrm{CAlg}}
\newcommand\aCAlg{\mathfrak{a}\CAlg}
\newcommand\dCAlg{\mathfrak{d}\CAlg}
\newcommand{\Cat}{\mathrm{Cat}}
\newcommand{\Sscr}{\mathcal{S}}
\newcommand{\poly}{\mathrm{poly}}
\newcommand{\perf}{\mathrm{perf}}
\newcommand\Sp{\mathrm{Sp}}
\newcommand\CycSp{\mathrm{CycSp}}
\newcommand\TCart{\mathrm{TCart}}
\newcommand\Fr{\mathrm{Fr}}
\newcommand\Br{\mathrm{Br}}
$
After a long delay, Achim Krause, Thomas Nikolaus, and I finally uploaded our paper [3]
on the $\K$-groups of $\bZ/p^n$, and more generally of $\Oscr_K/\varpi^n$ for $K$ a $p$-adic local field and $\varpi$ a uniformizer, to the arXiv. And, I have finally written this blog post, since the
paper has been up for six months now. These results were originally announced in
[1] and our previous post. See the latter
for an outline of the main results. Most importantly, our implementation of the algorithm which
computes these groups is available publicly at kzpn @ GitHub.
Several related papers have appeared in the meantime, most notably the paper [5] of Jeremy Hahn, Ishan
Levy, and Andrew Senger which computes the $\K$-groups of $\bZ/p^n$ modulo $p$ and $v_1$.
Remarkably, their computation shows that the answer is independent of $n$ for $n\geq 2$.
In a short paper using both the techniques of [3] and [5], Krause and Senger have given the exact
vanishing bound for the even $\K$-groups of $\bZ/p^n$ in [6], strengthening one of the main theorems in
[3].
Finally, our work uses our paper [2] on prismatic cohomology relative to $\delta$-rings, which
introduces in particular a relative form of syntomic cohomology.
References
[1] Antieau, Krause, and Nikolaus, The K-theory of $\bZ/p^n$ – announcement,
arXiv:2204.03420.
[2] Antieau, Krause, and Nikolaus, Prismatic cohomology relative to $\delta$-rings,
arXiv:2310.12770.
[3] Antieau, Krause, and Nikolaus, The K-theory of $\bZ/p^n$,
arXiv:2405.04329.
[4] Bhatt, Scholze, Prisms and prismatic cohomology,
arXiv:1905.08229.
[5] Hahn, Levy, Senger, Crystallinity for reduced syntomic cohomology and the mod
$(p,v_1^{p^{n-2}})$ $\K$-theory of $\bZ/p^n$,
arXiv:2409.20543.
[6] Krause, Senger, Exact bounds for even vanishing of \(\K_*(\bZ/p^n)\),
arXiv:2409.20523.