Note

Torsion in Nil K-groups of schemes

Author: Wei Yang · Added: September 29, 2026 · Updated: September 29, 2026

Latest version: v2 (September 29, 2026)

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Abstract

Let $X$ be a Noetherian quasi-excellent scheme and let $p$ be a prime such that $X[1/p]$ is regular. For every integer $N$, one power of $p$ annihilates its polynomial Nil $K$-groups in every degree at most $N$ and every finite number of variables. For $X=\operatorname{Spec} A$, the exponent can also be chosen uniformly over all ind-smooth $A$-algebras. The dependence on the degree bound is necessary even for reduced excellent curves. The proof combines cdh descent and derived Hochschild homology. Products of presheaves pass from finite-dimensional local rings to open neighbourhoods. The corresponding assertion for all Noetherian schemes fails already for one-dimensional domains in degree zero. We give explicit counterexamples in characteristic $p$ and in mixed characteristic; the latter are local and flat over $\mathbb{Z}_{(p)}$ with nonempty regular generic fibre.

Version History

• v2 · September 29, 2026 — Adds a result showing that dependence on the degree bound is necessary even for reduced excellent curves. PDF (25 pages)
• v1 · September 29, 2026 — Initial version. PDF (24 pages)