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Further results on Artin approximation, for group-actions on mapping-germs Maps(X,Y) and for quivers of maps

Dmitry Kerner

TL;DR

This work strengthens Artin approximation for mapping-germs by establishing Strong Artin approximation ($SAP$) and Ploski-type ($APP$) results for $\,\mathscr{G}$-equivalence acting on Maps$(X,Y)$, including cases with singular targets. It extends these approximation principles to quivers of map-germs, represented by directed rooted trees, and to base-change scenarios via unfoldings and parameterized families. The authors define precise $SAP\!\!\mathscr{G}$ and $APP\!\!\mathscr{G}$ properties, prove them for several group actions ($\mathscr{R},\mathscr{L},\mathscr{L}\mathscr{R},\mathscr{K}$), and provide refined filtrations and unipotent-subgroup versions. A key methodological feature is converting equivalence conditions into systems of implicit-function equations and applying nested SAP and nested APP to obtain both ordinary and parameterized solutions. The results generalize existing approximation frameworks to arbitrary singular maps, quivers, and base-change contexts, enabling robust normal-form and unfolding analyses in Singularity Theory and related deformation theories.

Abstract

Consider (analytic, resp. algebraic) map-germs, Maps((k^n,o),(k^m,o)). These germs are traditionally studied up to the right, let-right and contact equivalences. Below G is one of these groups. An important tool in this study is the Artin approximation: any formal G-equivalence of maps is approximated by ordinary (i.e. analytic, resp. algebraic) G-equivalence. We consider maps of (analytic, resp. algebraic) scheme-germs, with arbitrary singularities, Maps(X,Y), and establish stronger versions of this property (for G): the Strong Artin approximation and the Płoski approximation. As a preliminary step we study the contact equivalence for maps with singular targets. In many cases one works with multi-germs of spaces, and with their ``muti-maps". More generally, ``quivers of map-germs" occur in various applications. The needed tools are the Strong Artin approximation for quivers and the Płoski version. We establish these for directed rooted trees.

Further results on Artin approximation, for group-actions on mapping-germs Maps(X,Y) and for quivers of maps

TL;DR

This work strengthens Artin approximation for mapping-germs by establishing Strong Artin approximation () and Ploski-type () results for -equivalence acting on Maps, including cases with singular targets. It extends these approximation principles to quivers of map-germs, represented by directed rooted trees, and to base-change scenarios via unfoldings and parameterized families. The authors define precise and properties, prove them for several group actions (), and provide refined filtrations and unipotent-subgroup versions. A key methodological feature is converting equivalence conditions into systems of implicit-function equations and applying nested SAP and nested APP to obtain both ordinary and parameterized solutions. The results generalize existing approximation frameworks to arbitrary singular maps, quivers, and base-change contexts, enabling robust normal-form and unfolding analyses in Singularity Theory and related deformation theories.

Abstract

Consider (analytic, resp. algebraic) map-germs, Maps((k^n,o),(k^m,o)). These germs are traditionally studied up to the right, let-right and contact equivalences. Below G is one of these groups. An important tool in this study is the Artin approximation: any formal G-equivalence of maps is approximated by ordinary (i.e. analytic, resp. algebraic) G-equivalence. We consider maps of (analytic, resp. algebraic) scheme-germs, with arbitrary singularities, Maps(X,Y), and establish stronger versions of this property (for G): the Strong Artin approximation and the Płoski approximation. As a preliminary step we study the contact equivalence for maps with singular targets. In many cases one works with multi-germs of spaces, and with their ``muti-maps". More generally, ``quivers of map-germs" occur in various applications. The needed tools are the Strong Artin approximation for quivers and the Płoski version. We establish these for directed rooted trees.

Paper Structure

This paper contains 32 sections, 5 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Approximation problems for mapping-germs
  3. The results
  4. Strong Artin approximation for $\mathscr G$-equivalence
  5. Pł oski version of Artin approximation
  6. Remarks on Theorem \ref{['Thm.SAP.G.and.APP.G']}
  7. Artin approximations for quivers of maps
  8. Artin approximations for (quivers of) maps with base change
  9. Further applications
  10. Acknowledgements
  11. Definitions and auxiliary results
  12. Notation and conventions (Germs of schemes and their morphisms)
  13. The right, left, and left-right equivalence on ${\rm Maps}\!\left(X,Y\right)$
  14. The contact equivalence of ${\rm Maps}\!\left(X,Y\right)$ and its linearized version
  15. The classical versions of Artin approximation
  16. ...and 17 more sections

Key Result

Lemma 2.6

Fix a map $f\!\in\! {\rm Maps}\!\left(X,Y\right).$ Then ${\tilde{f}}\!\ \hbox{$\mathrel{\mathop{\sim}\limits^{{\mathscr K}\!}}$}\ \!f$ iff ${\tilde{f}}\!\in\! {\mathscr K}\!_{lin,f} f .$ [Thus ${\mathscr K}\! f\!=\!{\mathscr K}\!_{lin,f} f.$]

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Lemma 2.6
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Example 3.4
  • ...and 11 more