Approximation results of Artin-Tougeron-type for general filtrations and for $C^r$-equations

TL;DR

This paper extends Artin approximation results to general filtrations and $C^r$-rings, including the surjectivity of the completion map for rings of smooth functions, broadening their applicability beyond traditional settings.

Contribution

It generalizes Artin approximation to non-Noetherian filtrations and $C^r$-rings, and proves surjectivity of the completion map for $C^ fty$ function rings.

Findings

Extended approximation results to general filtrations.

Proved surjectivity of the completion map for $C^ fty$ rings.

Broadened the scope of Artin approximation beyond Noetherian rings.

Abstract

Artin approximation and other related approximation results are used in various areas. The traditional formulation of such results is restricted to filtrations by powers of ideals, $\{I^j\}$, and to Noetherian rings. In this paper we extend several approximation results both to rather general filtrations and to $C^r$-rings, for $2\le r\le\infty$. As an auxiliary step we establish the surjectivity of the completion map for rings of $C^\infty$ functions, for a very broad class of filtrations.