Kernels of Completed Locally Nilpotent Derivations

Definitions and mathematical statement

Let $B$ be a factorial affine domain over ${\mathbb{C}}$, meaning a finitely generated commutative ${\mathbb{C}}$-algebra which is a domain and a unique factorization domain. Let $\delta:B\to B$ be a locally nilpotent ${\mathbb{C}}$-derivation: it obeys the Leibniz rule and, for every $b\in B$, some power $\delta^r(b)$ is zero. Put $A=\ker\delta$. Let $M$ be a maximal ideal of $B$ and $m=M\cap A$. Use the local rings $B_M,A_m$ and their adic completions \[ \widehat B_M=\varprojlim_{r\geq1}B_M/(MB_M)^r, \qquad \widehat A_m=\varprojlim_{r\geq1}A_m/(mA_m)^r. \] This makes explicit the usual interpretation of completion of the local rings.

The localized derivation is defined by \[ \delta_M(b/s)=\frac{\delta(b)s-b\delta(s)}{s^2} \quad(s\notin M). \] Retain both source hypotheses: there is an extension $\widehat\delta_M$ of this derivation to $\widehat B_M$, in the intended completed-topology sense, and \[ \ker(\delta_M:B_M\to B_M)=A_m. \] The question is whether \[ \ker(\widehat\delta_M)=\widehat A_m. \] Local nilpotence is assumed on $B$; it is not silently imposed on the localized or completed derivations, where it need not persist.

The equality of completed rings needs its natural-map interpretation. The inclusion $A_m\hookrightarrow B_M$ sends $(mA_m)^r$ into $(MB_M)^r$, inducing a homomorphism $\widehat\iota:\widehat A_m\to\widehat B_M$. For the intended continuous extension, its image lies in $\ker\widehat\delta_M$. A strong precise reading of the requested equality is that \[ \widehat\iota:\widehat A_m\xrightarrow{\ \sim\ }\ker\widehat\delta_M \] is an isomorphism, rather than merely that its image equals the kernel without injectivity. The source does not explicitly establish that $\widehat\iota$ is injective or identify the two adic topologies. Those issues cannot be assumed in order to interpret a literal subring equality.

No finite-generation or Noetherian hypothesis on the invariant ring $A$ is supplied by the record, and none is added from the affineness of $B$. The chosen ideals, topology of the extension and intended canonical identification therefore require confirmation. The literature triage checked 2026-08-21 reports no authoritative complete resolution found. Its open label and notation-only verification remain dated metadata, not independently checked current results.

Short English statement

If the localized invariant ring is exactly $A_m$, does taking invariants still commute with completion? The equality must use the natural completion map, whose injectivity is not supplied. The original local-nilpotence condition is not extended to the localized or completed derivations, and the invariant ring is not assumed affine.

Sources

• Local UnsolvedMath registry, permanent ID 30000664, source code catalogue entry, "Kernels of Completed Locally Nilpotent Derivations," including all original and cleaned ring, localization, extension and kernel hypotheses.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, pp. 5--82, DOI 10.4171/owr/2007/01: https://doi.org/10.4171/owr/2007/01. This is also the sole supplied literature reference.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external source has been visited or independently checked here.