New paper: the K-theory of Z$/p^n$

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.