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Grauert's Approximation Theorem in any Characteristic and Applications

Gert-Martin Greuel, Gerhard Pfister

TL;DR

This work generalizes Grauert's division and nested Artin approximation from complex analytic germs to convergent power series over arbitrary real valued fields $K$ of any characteristic, formalizing a real-valued analog of standard bases and Weierstrass division. It proves a Grauert-type nested approximation theorem in this broader setting, enabling constructive, convergent solutions to systems of analytic equations and yielding a convergent semiuniversal deformation for isolated singularities. The results further establish a convergent splitting lemma for non-isolated singularities and provide a framework for convergent unfoldings, with explicit constructions for ICIS and hypersurface cases via the Tjurina/T$^1_R$ data. Collectively, the paper extends deformation theory to positive characteristic and non-Archimedean contexts, offering both theoretical insight and tools for singularity classification in broader algebraic-analytic settings.

Abstract

In his seminal Inventiones paper from 1972 Grauert proved the existence of a semiuniversal deformation of an arbitrary complex analytic isolated singularity. For the proof he invented an approximation theorem for solving a system of "nested" analytic equations, which is now called Grauert's approximation theorem. To prove this, Grauert introduced standard bases for ideals in power series rings and proved a generalized Weiertrass division theorem. All this was done for convergent power series over the complex numbers. The purpose of this article is to extend Grauert's division and approximation theorem to convergent power series over arbitrary real valued fields of any characteristic. As an application, which was actually the motivation for this article, we derive the existence of a convergent semiuniversal deformation for an isolated singularity and a splitting lemma for not necessarily isolated hypersurface singularities over any real valued field.

Grauert's Approximation Theorem in any Characteristic and Applications

TL;DR

This work generalizes Grauert's division and nested Artin approximation from complex analytic germs to convergent power series over arbitrary real valued fields of any characteristic, formalizing a real-valued analog of standard bases and Weierstrass division. It proves a Grauert-type nested approximation theorem in this broader setting, enabling constructive, convergent solutions to systems of analytic equations and yielding a convergent semiuniversal deformation for isolated singularities. The results further establish a convergent splitting lemma for non-isolated singularities and provide a framework for convergent unfoldings, with explicit constructions for ICIS and hypersurface cases via the Tjurina/T data. Collectively, the paper extends deformation theory to positive characteristic and non-Archimedean contexts, offering both theoretical insight and tools for singularity classification in broader algebraic-analytic settings.

Abstract

In his seminal Inventiones paper from 1972 Grauert proved the existence of a semiuniversal deformation of an arbitrary complex analytic isolated singularity. For the proof he invented an approximation theorem for solving a system of "nested" analytic equations, which is now called Grauert's approximation theorem. To prove this, Grauert introduced standard bases for ideals in power series rings and proved a generalized Weiertrass division theorem. All this was done for convergent power series over the complex numbers. The purpose of this article is to extend Grauert's division and approximation theorem to convergent power series over arbitrary real valued fields of any characteristic. As an application, which was actually the motivation for this article, we derive the existence of a convergent semiuniversal deformation for an isolated singularity and a splitting lemma for not necessarily isolated hypersurface singularities over any real valued field.
Paper Structure (6 sections, 16 theorems, 133 equations)

This paper contains 6 sections, 16 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Grauert's Division Theorem
  3. Cartan's Lemma
  4. Grauert's Approximation Theorem
  5. Unfoldings and Convergent Splitting Lemma
  6. Convergent Semiuniversal Deformation

Key Result

Theorem 1.1

Let $K$ be a real valued field. If the characteristic of $K$ is positive, we assume additionally that $K$ is quasi-complete $K$ is called quasi-complete if the completion $\bar{K}$ of $K$ is a separable field extension of $K$. Note that in characteristic 0 every real valued field is already quasi-co Then there exists for any integer $c>0$ a convergent solution $y_c(x)\in K\{ x\}^m$, such that

Theorems & Definitions (41)

  • Theorem 1.1: Analytic Artin Approximation
  • Theorem 1.2: Nested Algebraic Artin Approximation
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3: Division Theorem for real valued fields
  • proof
  • Remark 2.4
  • Definition 2.5
  • ...and 31 more