A proof of A. Gabrielov’s rank theorem

André Belotto Da Silva,Guillaume Rond,Octave Curmi

doi:10.5802/jep.173

Abstract

This article contains a complete proof of Gabrielov’s rank theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung theorem. We include, furthermore, new extensions of the rank theorem (concerning the Zariski main theorem and elimination theory) to commutative algebra.

Highlights

This article contains a complete and self-contained proof of Gabrielov’s rank theorem, a fundamental result in the study of analytic map germs
Osgood gave in [Osg16] an example for which the dimension of the smallest germ of analytic set containing the image is greater than r and, subsequently, Abhyankar generalized this example in a systematic way, see [Abh64]
Injective morphisms. — The problem raised by Grothendieck has been generalized to the following problem: given a morphism of C-analytic algebras φ : A → B, when does φ(A) ∩ B = φ(A) hold true? If the equality is verified, we say that φ is strongly injective. This terminology was introduced by Abhyankar and van der Put [AvdP70] who were the first ones to investigate this question. In particular they proved that φ is always strongly injective when A is a ring of convergent power series in two variables over any valued field

Summary

Introduction

This article contains a complete and self-contained proof of Gabrielov’s rank theorem, a fundamental result in the study of analytic map germs. In particular they proved that φ is always strongly injective when A is a ring of convergent power series in two variables over any valued field Without this assumption on the dimension, the equality φ(A) ∩ B = φ(A) does not hold in general (see Example 1.16 and (4)). Such a result would allow us to adapt arguments from elimination theory to the more general context of convergent power series From this perspective, Theorem 1.7(III) provides a formal characterization of the above condition. – In [Paw89], Pawłucki provides an example of a subanalytic set (given by a non regular morphism) which is neither formally nor analytically semi-coherent This contradicted a result previously announced by Hironaka [Hir86]. Spivakovsky for the fruitful discussions he had about this problem

Ranks and transformations

Gabrielov’s rank theorem

Proof of the low-dimensional Gabrielov theorem

Semi-global extension of convergent factors

The Abhyankar-Jung theorem