$\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.