# Artin approximation for left-right equivalence of map-germs and for quivers
of map-germs

Dmitry Kerner  Department of Mathematics, Ben Gurion University of the Negev,
P.O.B. 653, Be’er Sheva 84105, Israel<EMAIL_ADDRESS>
(Date: filename: main.tex)

###### Abstract.

The standard Artin approximation (AP) is not readily applicable to the left-
right equivalence of map-germs
$Maps\big{(}(\Bbbk^{n},o),(\Bbbk^{m},o)\big{)}.$ Moreover, the naive extension
does not hold in the $\Bbbk$-analytic case, because of Osgood-Gabrielov-Shiota
examples.

The left-right version of Artin approximation ($\mathscr{L}\mathscr{R}$-AP)
was established by M. Shiota for map-germs that are either Nash or [real-
analytic and of finite singularity type]. We establish
$\mathscr{L}\mathscr{R}$-AP for $Maps(X,Y),$ where $X,Y$ are
$\Bbbk$-analytic/$\Bbbk$-Nash germs of schemes of any characteristic. More
precisely:

  * •

($\Bbbk$-Nash power series,
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.}$ and
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
y\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.}$.)
$\mathscr{L}\mathscr{R}$-AP holds for any map.

  * •

($\Bbbk$-analytic power series,
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{x\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.}$
and
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{y\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.}$.)
$\mathscr{L}\mathscr{R}$-AP holds for maps of weakly-finite singularity type.

This “weakly-finite singularity type” (which we introduce) is a natural
extension of the classical finite singularity type, and is of separate
interest.

As a trivial corollary we get: the inverse Artin approximation holds for
analytic maps of weakly-finite singularity type.

Then we extend the properties $\mathscr{R}$-AP, $\mathscr{L}$-AP,
$\mathscr{L}\mathscr{R}$-AP to the approximation results for quivers of maps,
$\Gamma$-AP. Finally, we establish the nested version, “$\Gamma$-AP with
parameters”. It is needed for families/unfoldings of maps.

###### Key words and phrases:

Artin approximation, nested Artin approximation, inverse Artin question,
finite singularity type, lifting automorphisms to the ambient space

###### 2020 Mathematics Subject Classification:

Primary 13B40.Secondary 13J05,13J07,13J15,14B05, 14B12,32A05

I was supported by the Israel Science Foundation (grant No. 1910/18)

###### Contents

  1. 1 Introduction
  2. 2 Definitions, notations, preliminaries
  3. 3 The left-right Artin approximation
  4. 4 Artin approximation for quivers of maps

##  1\. Introduction

###  1.1. The ordinary AP

Consider a system of implicit function equations, $F(x,y)=0.$ Here $x,y$ are
multi-variables, while $F$ is a (finite) vector of power series. Assume $F$ is
either $\Bbbk$-analytic or $\Bbbk$-Nash, i.e. its entries belong to one of the
rings:

  * •

$\Bbbk\\{x,y\\},$ the locally convergent power series (here $\Bbbk$ is a
normed field, complete with respect to its norm)

  * •

$\Bbbk\langle x,y\rangle,$ the algebraic power series (here $\Bbbk$ is any
field). For $\Bbbk=\mathbb{R}$ these are the Nash power series.

The classical approximation theorems [Artin.68],[Artin.69] read: any formal
solution, $F(x,\hat{y}(x))\\!=\\!0,$ is approximated by ordinary solutions,
$y(x)$ (analytic or algebraic power series).

The subject “Artin approximations” has been studied extensively, see e.g.
[Popescu.00], [Rond.18] for numerous results and further references.

###  1.2. The inverse AP

Fix some ($\Bbbk$-analytic, resp. $\Bbbk$-algebraic) power series,
$f(x)=f_{1}(x),\dots,$ $f_{m}(x).$ Consider the equation $\Phi(f(x))=0,$ with
the unknown $\Phi.$ The “inverse Artin question”, [Grothendieck.61], asks: is
any formal solution, $\hat{\Phi}(f(x))=0,$ approximated by ordinary solutions,
$\Phi(f(x))=0?$ (Here $\Phi$ is $\Bbbk$-analytic, resp. $\Bbbk$-algebraic.)

This inverse Artin approximation holds in the algebraic case, the proof is
based on the nested AP, [Popescu.85], [Popescu.86]. In the analytic case the
inverse AP holds for $\mathbb{C}\\{x\\}$ of dimension$\leq 2$, [Abhyankar-van
der Put.70], but fails in higher dimensions, [Gabrielov.71], [Gabrielov.73].
Yet it holds for the ring $\mathbb{R}\\{x\\},$ when the map $f$ is of finite
singularity type111i.e. $f$ is finite with respect to the contact equivalence,
i.e. the complexified subgerm $V(o)\subset(\mathbb{C}^{n},o)$ has an isolated
singularity, see §2.5, [Shiota.98], [Shiota.10].

This inverse Artin approximation was studied in several works, see the
references in [Izumi.07], and is still under investigation, see
[Belot.Curmi.Rond.21], [Alon.Cast-Jimé.Haus.Kout.18].

Observe that the classical Artin approximation is related to the
$\mathscr{R}$-AP question: when is the formal right equivalence of maps,
$f\circ\hat{\Phi}^{-1}={\tilde{f}},$ approximated by ordinary equivalence?
Similarly, the inverse Artin approximation can be stated as the
$\mathscr{L}$-AP question, when is the formal left equivalence of maps,
$\hat{\Phi}\circ f={\tilde{f}},$ approximated by ordinary equivalence? (See
remark 3.2.)

###  1.3. The left-right AP

The germs of maps, $Maps\big{(}(\Bbbk^{n},o),(\Bbbk^{m},o)\big{)},$ are
traditionally studied up to the left-right equivalence, $f\sim\Phi_{Y}\circ
f\circ\Phi_{X}^{-1}.$ Here $f$ is a vector of (analytic, resp. algebraic)
power series, while $\Phi_{X},\Phi_{Y}$ are (analytic, resp. algebraic)
coordinate changes in the source and the target. Hence the fundamental
question is the left-right Artin approximation ($\mathscr{L}\mathscr{R}$-AP):

Is every formal solution of the equation $\Phi_{Y}\circ
f\circ\Phi_{X}^{-1}={\tilde{f}}$ approximated by ordinary solutions?

Recall that the condition $\Phi_{Y}\circ f={\tilde{f}}\circ\Phi_{X}$ is not an
implicit function equation. Thus the property $\mathscr{L}\mathscr{R}$-AP is
not a direct consequence of the classical version of AP.

A remark, if a map $f:(\Bbbk^{n},o)\to(\Bbbk^{p},o)$ is finitely
$\mathscr{L}\mathscr{R}$-determined, then the property
$\mathscr{L}\mathscr{R}$-AP holds trivially for $f.$ (See [Kerner.21, §7].)

The property $\mathscr{L}\mathscr{R}$-AP was studied by M. Shiota for
$Map\big{(}(\mathbb{R}^{n},o),(\mathbb{R}^{m},o)\big{)}.$

  1. i.

For Nash-germs, $\mathbb{R}\langle x\rangle,$ the property
$\mathscr{L}\mathscr{R}$-AP holds for any map, [Shiota.10, Theorem 4].

Though stated for $\mathbb{R}\langle x\rangle$, the proof is actually
characteristic-free and works over any field.

  2. ii.

Over the ring $\mathbb{R}\\{x\\}$ the property $\mathscr{L}\mathscr{R}$-AP
does not hold in general, [Shiota.98, Fact.1.4], [Shiota.10, pg. 1061]. The
counterexample is based on Gabrielov’s counterexample to the “Inverse Artin
question”, [Gabrielov.71].

However, the property $\mathscr{L}\mathscr{R}$-AP holds for maps of finite
singularity type (as above), [Shiota.98, Fact.1.7]. That proof, [Shiota.98,
pg.119-126], is heavily analytic.

The global version of $\mathscr{L}\mathscr{R}$-AP, for $Maps(X,Y)$ with $X,Y$
compact Nash/real-analytic manifolds, is more complicated and is also treated
in [Shiota.98], [Shiota.10].

These approximation results were extended in [Fich.Quar.Shiot] to equations of
type $F(x,\\!\Phi(x),\\!\Psi\\!\circ\\!g(x))\\!=\\!0$. Here $F,g$ are
prescribed $\mathbb{R}$-Nash power series, while $\Phi,\Psi$ are unknowns.

###  1.4. Results

To study group orbits in singularity theory (in arbitrary characteristic),
[Kerner.21], and deformations/unfoldings, [Kerner.22], one needs such
approximation results in higher generality. This paper extends the results of
[Shiota.98], [Shiota.10] in several directions.

  1. i.

Instead of $Maps\big{(}(\Bbbk^{n},o),(\Bbbk^{m},o)\big{)}$ we study
$Maps(X,Y),$ where $X\\!=\\!Spec(R_{X})$ and $Y=Spec(R_{Y})$ are scheme-germs
with arbitrary singularities. Here
$\bullet R_{X}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.}\\!,R_{Y}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
y\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.}\\!,$ with $\Bbbk$ a
field or an excellent henselian local ring over a field;
or
$\bullet
R_{X}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{x\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.}\\!,R_{Y}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{y\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.}\\!,$
with $\Bbbk$ a complete normed field or a henselian DVR over a field.

For the pair
$R_{X}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.}$,
$R_{Y}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
y\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.},$ the property
$\mathscr{L}\mathscr{R}$-AP holds for any map, Theorem 3.3. In the analytic
case $\mathscr{L}\mathscr{R}$-AP holds for maps of “weakly finite singularity
type”, Theorem 3.4.

This class of maps is a large extension of those of the finite singularity
type. (For scalar-valued maps, i.e. $m=1,$ the two notions coincide.) Though
appearing naturally, this extension seems to be completely new. Thus, even in
the classical case, $Maps\big{(}(\Bbbk^{n},o),(\Bbbk^{m},o)\big{)},$ for
$\Bbbk=\mathbb{R},\mathbb{C}$, Theorem 3.4 is stronger than the known results.

As a trivial corollary we get (Corollary 3.6): the inverse Artin approximation
holds in the $\Bbbk$-analytic case for maps of weakly-finite singularity type.

The proofs are purely-algebraic and a bit more conceptual than the proofs in
[Shiota.98], [Shiota.10]. (Those used several clever ad-hoc tricks.)

The proof of $\Bbbk$-analytic case uses an auxiliary statement, useful on its
own. Given a subgerm $Z\subset X,$ denote by $Aut_{X,Z}$ the group of those
automorphisms of $X$ that act on $Z.$ The restriction $Aut_{X,Z}\to Aut_{Z}$
is non-surjective, i.e. not every automorphism of $Z$ extends to the ambient
space. However (Lemma 2.9) for $d$’th infinitesimal neighborhood, $Z\subset
Z_{d}\subset X,$ with $d\gg 1,$ the restriction $Aut_{X,Z_{d}}\to Aut_{Z_{d}}$
is surjective.

  2. ii.

For algebraic power series we establish also approximation results for
“quivers of maps”. The versions $\mathscr{R}$-AP, $\mathscr{L}$-AP,
$\mathscr{L}\mathscr{R}$-AP correspond to the simplest quivers:

(1) $\Phi_{X}\circlearrowright X\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f}}}$}\
Y,\quad\quad\quad\quad\quad X\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f}}}$}\
Y\circlearrowright\Phi_{Y},\quad\quad\quad\quad\quad\Phi_{X}\circlearrowright
X\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f}}}$}\
Y\circlearrowright\Phi_{Y}.$

More generally, take a directed graph $\Gamma,$ whose vertices, $\\{X_{v}\\},$
are scheme-germs, while the edges are germs of maps, $X_{v}\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f_{wv}}}}$}\
X_{w}.$ A morphism of such quivers,

(2) $(\Gamma,\\{{\tilde{X}}_{\bullet}\\},\\{{\tilde{X}}_{v}\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{{\tilde{f}}_{wv}}}}$}\
{\tilde{X}}_{w}\\}_{wv})\quad\quad\to\quad\quad(\Gamma,\\{X_{\bullet}\\},\\{X_{v}\
\lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f_{wv}}}}$}\
X_{w}\\}_{wv}),$

is a collection of morphisms of schemes $\\{\Phi_{v}:{\tilde{X}}_{v}\to
X_{v}\\}_{v}$ that ensure commutativity of diagrams “$\tilde{\Gamma}$ over
$\Gamma$”, see §4.1. This gives the system of equations
$\\{\Phi_{w}\circ{\tilde{f}}_{wv}=f_{wv}\circ\Phi_{v}\\}_{wv}.$ The maps
$\\{f_{v}\\},\\{{\tilde{f}}_{v}\\}$ are prescribed, while $\\{\Phi_{v}\\}$ are
unknowns, in different sets of variables. One can impose additional conditions
on $\\{\Phi_{v}\\}$, e.g. $\Phi_{i}$ is invertible, $\Phi_{v}=Id_{X_{v}},$ or
$\Phi_{v}$ is identity modulo higher order terms, or $\Phi_{v}$ sends a
subgerm ${\tilde{Z}}\subset{\tilde{X}}$ to a subgerm $Z\subset X.$

Theorem 4.5 establishes the “$\Gamma$-Artin approximation property” in the
$\Bbbk$-Nash case when $\Gamma$ is a rooted tree.

No analytic version of this result is given, as the proof inevitably uses the
nested Artin approximation. (Which is absent in the analytic case.)

  3. iii.

To study deformations/families of maps (and other objects) the base field
$\Bbbk$ is replaced by a base ring, e.g. $\Bbbk\\{t\\},\Bbbk\langle t\rangle,$
here $t=(t_{1},\dots,t_{r}).$ Then we have families of scheme-germs, $X_{t}$,
and families of maps-germs $f_{t}:X_{t}\to Y_{t}.$ Accordingly we have the
property “$\mathscr{L}\mathscr{R}$-AP with parameters”.

More generally, we have quivers of families of maps,
$(\Gamma,\\{X_{t,v}\\}_{v},\\{X_{t,v}\\!\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f_{t,wv}}}}$}\
\\!X_{t,w}\\}_{wv}).$ Proposition 4.8 gives the (nested) “$\Gamma$-AP with
parameters”.

###  1.5. A bit about the proofs

  * •

(The proof of Theorem 3.3) The key idea is to reformulate the
$\mathscr{L}\mathscr{R}$-condition, $\Phi_{Y}\circ
f={\tilde{f}}\circ\Phi_{X},$ as a certain homomorphism of algebras with base
change. This translates the initial functional equation problem into an
implicit function equation with nested solutions. Then, for $\Bbbk$-Nash case
(${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.},{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
y\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.}$), one uses the nested
Artin approximation.

  * •

(The proof of Theorem 3.4) In the analytic case,
(${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.},{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
y\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.}$), the nested Artin
approximation does not hold. Therefore we have to work with maps of weakly-
finite singularity type. First one restricts $f$ to its $r$’th critical locus,
$f_{|}:Crit_{r}(f)\to Y.$ By our assumption $f_{|}$ is a finite map. Hence,
instead of a homomorphism of algebras, we can look for a homomorphism of
(finitely-generated) modules, $mod$-$R_{Y}\ni R_{Crit_{r}(f)}\to
R_{Crit_{r}({\tilde{f}})}\in mod$-$R_{\tilde{Y}},$ with a base change
$R_{Y}\to R_{\tilde{Y}}.$ This homomorphism (once established) is promoted to
a homomorphism of algebras, ensuring $f_{|}\sim{\tilde{f}}_{|}.$

We have found an element of the group $Aut_{Crit_{r}}\times Aut_{Y}$. It is
lifted to an element of the group $Aut_{X}\times Aut_{Y}$ that ensures:
${\tilde{f}}\equiv f\ mod\ Jac_{r}(f)^{d},$ for $d\gg 1.$ Finally one applies
an implicit function argument of the type of Bourbaki-Artin-Tougeron (and
certain finiteness of possible factorizations) to achieve: ${\tilde{f}}\equiv
f\ mod\ Jac_{r+1}(f)^{d}.$ For $r=m$ this finishes the proof. For $r<m$ one
iterates this procedure to get ${\tilde{f}}\equiv f\ mod\ Jac_{r+2}(f)^{d},$
then ${\tilde{f}}\equiv f\ mod\ Jac_{r+3}(f)^{d},$ …, up to ${\tilde{f}}\equiv
f\ mod\ Jac_{m+1}(f)^{d}.$ The later means: ${\tilde{f}}=f.$

###  1.6. Acknowledgements

Thanks are to A.-F. Boix for valuable knowledge and remarks, to G. Rond for
the nice survey [Rond.18] and to D. Popescu for important advices.

##  2\. Definitions, notations, preliminaries

###  2.1. Germs of schemes and their mappings

  1. i.

We use the multivariable notation, $x=(x_{1},\dots,x_{n}),$
$y=(y_{1},\dots,y_{m}),$ $F=(f_{1},\dots,f_{r}).$ The notations
$F(x,y)\in(x,y)^{N}$ or $y(x)\in(x)^{N}$ mean: every component of the vector
$F$ (resp. $y(x)$) satisfies this condition.

  2. ii.

Consider ($\Bbbk$-analytic/$\Bbbk$-Nash) maps of scheme-germs, $Maps(X,Y)$.
Here:

     * •

$X\\!=\\!Spec(R_{X})$, where
$R_{X}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{x\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.},$
resp. $R_{X}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.}.$ (We assume
$J_{X}\\!\subseteq\\!(x)^{2}$.)

     * •

$Y=Spec(R_{Y})$, where
$R_{Y}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{y\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.},$
resp. $R_{Y}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
y\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.}.$ (We assume
$J_{Y}\\!\subseteq\\!(y)^{2}$.)

     * •

For analytic power series,
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{x\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.},$
$\Bbbk$ is either a normed field (complete with respect to its norm,) or a
henselian DVR over a field. The main relevant examples of DVR are the rings
$\Bbbk_{o}[[\tau]],$ $\Bbbk_{o}\langle\tau\rangle,$ $\Bbbk_{o}\\{\tau\\}.$
(Here $\Bbbk_{o}$ is another field.)

     * •

For algebraic power series,
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.},$ $\Bbbk$ is a(ny)
field or an excellent henselian local ring over a field. See [Moret-Bailly.21]
for more detail.

When $\Bbbk$ is not a field we denote its maximal ideal by
${\mathfrak{m}}_{\Bbbk}.$

For $J_{X}=0,J_{Y}=0$ one gets the maps of $\Bbbk$-smooth germs,
$Maps\big{(}(\Bbbk^{n},o),(\Bbbk^{m},o)\big{)}.$ Otherwise the singularities
of $X,Y$ are arbitrary.

  3. iii.

Fix an embedding $Y\subseteq(\Bbbk^{m},o)$ and some coordinates on
$(\Bbbk^{m},o).$ We get the generators of the ideal
$(y)=(y_{1},\dots,y_{m})\subset R_{Y}.$ If $\Bbbk$ is a field, then $(y)$ is
the maximal ideal.

Then a map $f:X\to Y$ is presented by the tuple
$(f_{1},\dots,f_{m})\in{\mathfrak{m}}\cdot R^{\oplus m}_{X}$. This embeds the
space of maps as a subset $Maps(X,Y)\subseteq{\mathfrak{m}}\cdot R^{\oplus
m}_{X}.$ If the target is $\Bbbk$-smooth, $Y=(\Bbbk^{m},o),$ then we get the
$R_{X}$-submodule $Maps\big{(}X,(\Bbbk^{m},o)\big{)}\cong{\mathfrak{m}}\cdot
R^{\oplus m}_{X}.$

  4. iv.

Suppose $\Bbbk$ is not a field, then $f\in Maps(X,Y)$ can be considered as a
family of maps, over the base $Spec(\Bbbk).$ Taking the maximal ideal,
${\mathfrak{m}}_{\Bbbk}\subset\Bbbk,$ one gets the central fibres:

     * •

of the schemes,
$X_{o}=Spec(R_{X}\otimes{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.}),$
$Y_{o}=Spec(R_{Y}\otimes{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.}).$

     * •

of the map,
$f_{o}:=f\otimes{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.}:X_{o}\to
Y_{o}.$ This is defined algebraically via
$f^{\sharp}_{o}:=f^{\sharp}\otimes{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.}\in
Hom_{{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.}}(R_{Y}\otimes{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.},R_{X}\otimes{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.}).$

If $\Bbbk$ is a field then (trivially) $f_{o}=f.$

  5. v.

Consider (formal/$\Bbbk$-analytic/$\Bbbk$-Nash) automorphisms of $X$ over
$Spec(\Bbbk),$ i.e. $Aut_{X}:=Aut_{\Bbbk}(R_{X}).$ Similarly one takes
$Aut_{Y}:=Aut_{\Bbbk}(R_{Y}).$ In the $\Bbbk$-smooth case,
$Maps\big{(}(\Bbbk^{n},o),(\Bbbk^{m},o)\big{)},$ these automorphisms are the
coordinate changes in the source and the target.

These automorphisms act on the space of maps, defining the right and left
actions:

     * •

$\mathscr{R}:=Aut_{X}=Aut_{\Bbbk}(R_{X})\circlearrowright Maps(X,Y)$, by
$f\rightsquigarrow f\circ\Phi^{-1}_{X}$.

     * •

$\mathscr{L}:=Aut_{Y}=Aut_{\Bbbk}(R_{Y})\circlearrowright Maps(X,Y)$, by
$f\rightsquigarrow\Phi_{Y}\circ f$;

Combining them we get the left-right equivalence of maps,
$\mathscr{L}\times\mathscr{R}:f\sim\Phi_{Y}\circ f\circ\Phi^{-1}_{X}.$

See [Mond. Nuñ.-Bal.] for the general introduction to
$Maps\big{(}(\Bbbk^{n},o),(\Bbbk^{m},o)\big{)}$ in the real/complex-analytic
case. See [Kerner.21] for some results over arbitrary fields.

###  2.2. The classical versions of Artin approximation

  1. i.

(The ordinary version for analytic power series, [Denef-Lipshitz, Theorem
1.1]) Let $\Bbbk$ be a complete normed field or a henselian DVR. Consider
equations $F(x,y)=0$ for $F(x,y)\in\Bbbk\\{x,y\\}.$ Then any formal solution,
$\hat{y}(x)\in{\hat{\Bbbk}}[[x]],$ is approximated by analytic solutions.

  2. ii.

(The nested version for algebraic power series, [K.P.P.R.M.78], [Popescu.86],
[Popescu.00, Prop.3.5], [Rond.18, Theorem 5.8]) Let $\Bbbk$ be a field or an
excellent henselian local ring. Consider equations $F(x,y)=0$ for
$F(x,y)\in\Bbbk\langle x,y\rangle.$ Suppose these equations have a formal
nested solution,

(3)
$\hat{y}_{1}(x)\in{\hat{\Bbbk}}[[x_{1},\dots,x_{i_{1}}]],\dots,\hat{y}_{r}(x)\in{\hat{\Bbbk}}[[x_{1},\dots,x_{i_{r}}]],\text{
for some }1\leq i_{1}\leq\cdots\leq i_{r}\leq n.$

This formal solution is approximated by the nested algebraic solution,
$y_{1}(x)\in\Bbbk\langle x_{1},\dots,x_{i_{1}}\rangle,$ …,
$y_{r}(x)\in\Bbbk\langle x_{1},\dots,x_{i_{r}}\rangle.$

  3. iii.

(The strong Artin approximation, [Pfister-Popescu.75], [Denef-Lipshitz,
Theorem 7.1])

Let $\Bbbk$ be a field or a complete DVR, and take the power series
$F(x,y)\in\Bbbk[[x,y]].$ There exists a function
$\beta_{F}:\mathbb{N}\to\mathbb{N}$ satisfying: for every $c\in\mathbb{N}$ any
approximate solution, $F(x,\bar{y}(x))\in{\mathfrak{m}}^{\beta_{F}(c)},$ is
approximated by an ordinary solution, i.e. $F(x,y(x))=0$ and
$y(x)-\bar{y}(x)\in{\mathfrak{m}}^{c}.$

######  Example 2.1.

Let $X,{\tilde{X}}$ be analytic (resp. Nash) scheme-germs, §2.1.ii. Any formal
morphism $\hat{\Phi}:\hat{\tilde{X}}\to{\hat{X}}$ (i.e. a homomorphism
$\hat{\Phi}^{\sharp}:{\widehat{R}}_{X}\to{\widehat{R}}_{\tilde{X}}$) is
approximated by ordinary morphisms, ${\tilde{X}}\to X$ (i.e. homomorphisms
$\Phi^{\sharp}:R_{X}\to R_{\tilde{X}},$ $x_{i}\to x_{i}({\tilde{x}})$).
Namely, for each $d$ there exists a (analytic, resp. Nash) morphism
$\Phi_{d}:{\tilde{X}}\to X$ satisfying
$\hat{\Phi}^{\sharp}(x)-\Phi^{\sharp}_{d}(x)\in(x)^{d}.$

In particular (for $X={\tilde{X}}$), any formal automorphism, $\hat{\Phi}\in
Aut_{\hat{\Bbbk}}({\widehat{R}}_{X})$ is approximated by ordinary
automorphisms.

###  2.3. Presenting the condition $\Phi(I)\subseteq J$ as an implicit
function equation

Denote by $R$ the rings
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk[[x]]$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}$}\right.},$
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}$}\right.},$
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{x\\}$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}$}\right.}.$
Fix two ideals $I,J\subset R$ and a homomorphism $\Phi:R\to R.$ For the fixed
generators, $x=(x_{1},\dots,x_{n}),$ we present $\Phi$ by power series,
$\\{\Phi(x_{i})={\tilde{x}}_{i}(x)\\}_{i}.$ Fix some (finite set of)
generators, $\\{q^{I}_{i}\\}$ of $I,$ and $\\{q^{J}_{j}\\}$ of $J.$ Then the
condition $\Phi(I)\subseteq J$ transforms into the set of equations:

(4) $[\quad\Phi(q^{I}_{i}(x))=\joinrel=]\hskip
28.45274ptq^{I}_{i}({\tilde{x}}(x))=\sum{\tilde{z}}_{ji}\cdot
q^{J}_{j}(x),\quad\forall\ i.$

This is a system of implicit function equations for the unknowns
$\\{{\tilde{x}}_{i}\\},\\{{\tilde{z}}_{ji}\\}$ in $R.$

###  2.4. Presenting the condition $f\circ\Phi_{X}=\Phi_{Y}\circ{\tilde{f}}$
as a morphism of algebras with base-change

The map $f\in Maps(X,Y)$ induces on $R_{X}$ the structure of $R_{Y}$-algebra.
The structure map is $R_{Y}\ni g(y)\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f^{\sharp}}}}$}\
g(f)\in R_{X}$. We observe: $f\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\sim}\limits^{\scriptscriptstyle{\mathscr{L}\mathscr{R}}}}$}\
{\tilde{f}}$ iff the corresponding algebras admit an isomorphism with
(invertible) base-change, as on the diagram

(5) $\begin{matrix}f\in&R_{X}&\lower
1.72218pt\hbox{$\xrightarrow[\,\smash{\raisebox{4.95134pt}{$\scriptstyle\sim$}}\,]{x\to
x({\tilde{x}})}$}&R_{\tilde{X}}&\ni{\tilde{f}}\\\ \uparrow\quad&\uparrow
f^{\sharp}&&{\tilde{f}}^{\sharp}\uparrow&\quad\uparrow\\\ y\in&R_{Y}&\lower
1.72218pt\hbox{$\xrightarrow[\,\smash{\raisebox{4.95134pt}{$\scriptstyle\sim$}}\,]{y\to
y({\tilde{y}})}$}&R_{\tilde{Y}}&\ni{\tilde{y}}\end{matrix}\quad.$

Indeed, the commutativity of the diagram means
$f(x({\tilde{x}}))=y({\tilde{f}}({\tilde{x}}))$, i.e.:
$f(\Phi_{X}({\tilde{x}}))=\Phi_{Y}({\tilde{f}}(x)),$ where
$\Phi_{X}({\tilde{x}})=x({\tilde{x}})$ and
$\Phi_{Y}({\tilde{y}})=y({\tilde{y}}).$

In some cases one imposes the condition
$f\circ\Phi_{X}=\Phi_{Y}\circ{\tilde{f}}$ without invertibility of
$\Phi_{X},\Phi_{Y}.$ Then this diagram expresses just a homomorphism of
algebras with base-change.

###  2.5. The critical locus of a map

Fix a minimal presentation
$R_{Y}={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk[[y]]$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.},{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{y\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.},{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
y\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.}$ with
$J_{Y}\subseteq(y)^{2}.$ Geometrically we have a minimal embedding
$Y\subseteq(\Bbbk^{m},o).$ Below we consider the map $f\in Maps(X,Y)$ as a map
$X\to(\Bbbk^{m},o).$ Take the module of $\Bbbk$-linear derivations,
$Der_{X}:=Der_{\Bbbk}R_{X}.$ Applying this module to the map one gets the
submodule $Der_{X}(f)\subseteq R^{\oplus m}_{X}.$

The “Jacobian” ideal of the map is defined by the zeroth Fitting ideals of the
quotient,
$Jac(f):=Fitt_{0}[{\footnotesize\left.\raisebox{1.37775pt}{$R^{\oplus
m}_{X}$}\\!/\\!\raisebox{-1.37775pt}{$Der_{X}(f)$}\right.}]\subseteq R_{X}.$
This ideal defines the critical locus of the map,
$Crit(f):=Spec({\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$Jac(f)$}\right.})\subseteq
X.$

######  Example 2.2.

  1. i.

For maps of smooth germs, $f\in
Maps\big{(}(\Bbbk^{n},o),(\Bbbk^{m},o)\big{)},$ the module
$Der_{(\Bbbk^{n},o)}(f)$ is generated by all the partials of $f.$ The ideal
$Jac(f)$ is then the determinantal ideal $I_{m}[f^{\prime}].$ Here
$f^{\prime}\in Mat_{m\times n}(R_{X})$ is the matrix of partial derivatives.

  2. ii.

Recall: in many cases $rankDer_{X}\leq dim(R_{X}).$ Then for $dim(X)<m$ we
have necessarily $Jac(f)=0$, i.e. $Crit(f)=X.$

Recall the needed properties.

  1. i.

For $\Bbbk$ a local ring we take the central fibre, see §2.1.iv. Suppose the
(natural) map
$Der_{X}\otimes{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.}\to
Der_{X_{o}}$ is an isomorphism.

Fitting ideals of modules are compatible with base change,
$Fitt_{j}(M\otimes_{R}S)=Fitt_{j}(M)\cdot S\subset S$ for $0\leq j\leq m$ and
$S$ an $R$-algebra. Therefore the critical locus restricts (with its scheme
structure) to the central fibre:

(6) $X_{0}\supset
Crit(f_{o})=V(Fitt_{0}[{\footnotesize\left.\raisebox{1.37775pt}{$R^{\oplus
p}_{X_{o}}$}\\!/\\!\raisebox{-1.37775pt}{$Der_{X_{o}}(f_{o})$}\right.}])=V(Fitt_{0}[{\footnotesize\left.\raisebox{1.37775pt}{$R^{\oplus
m}_{X}$}\\!/\\!\raisebox{-1.37775pt}{$Der_{X}(f)$}\right.}\otimes{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.}])=$
$\hskip
85.35826ptV(Fitt_{0}[{\footnotesize\left.\raisebox{1.37775pt}{$R^{\oplus
m}_{X}$}\\!/\\!\raisebox{-1.37775pt}{$Der_{X}(f)$}\right.}]\otimes{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.})=Crit(f)\cap
X_{o}\subset X.$

  2. ii.

The restriction $Crit(f)\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f|}}}$}\
(\Bbbk^{m},o)$ is a finite map iff $f:X\to(\Bbbk^{m},o)$ is contact
(${\mathscr{K}}\\!$)-finite. The $\mathbb{C}$-analytic case is well known,
[Mond. Nuñ.-Bal., pg.224], [Mond-Montaldi.91]. For the case of arbitrary
characteristic see [Kerner.21, Lemma 3.14].

Observe: this finiteness condition does not depend on the choice of scheme
structure on $Crit(f)$, e.g. it is preserved if one passes to
$Spec({\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$\sqrt{Jac(f)}$}\right.})$
or to
$Spec({\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$Jac(f)^{d}$}\right.})$.

  3. iii.

Suppose the central fibre, $f_{o},$ is ${\mathscr{K}}\\!$-finite. Then the map
$Crit(f_{o})\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f_{o}|}}}$}\
(Y_{o},o)$ is finite. Assuming
$Der_{X}\otimes{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.}\lower
1.72218pt\hbox{$\xrightarrow[\,\smash{\raisebox{4.95134pt}{$\scriptstyle\sim$}}\,]{}$}Der_{X_{o}}$
we get: the map $Crit(f)\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f|}}}$}\
(Y,o)$ is finite. (This goes by the Weierstrass finiteness theorem, which
holds for W-systems, [Denef-Lipshitz, §1], in particular for our rings.)

###  2.6. Map-germs of weakly finite singularity type

Given a map $f:X\to Y\subseteq(\Bbbk^{m},o)$, we take its j’th Jacobian ideal,
$Jac_{j}(f):=Fitt_{m-j}[{\footnotesize\left.\raisebox{1.37775pt}{$R^{\oplus
m}_{X}$}\\!/\\!\raisebox{-1.37775pt}{$Der_{X}(f)$}\right.}]\subset R_{X}.$ It
defines the j’th critical locus,
$Crit_{j}(f):=Spec({\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$Jac_{j}(f)$}\right.})\subseteq
X.$ We get the chains of ideals and loci:

(7)
$R_{X}\\!=\\!Jac_{0}(f)\\!\supseteq\\!Jac_{1}(f)\\!\supseteq\\!\cdots\\!\supseteq\\!Jac_{m+1}(f)\\!=\\!0.\quad\quad
X\\!=\\!Crit_{m+1}(f)\\!\supseteq\\!\cdots\\!\supseteq\\!Crit_{0}(f)\\!=\\!\varnothing.$

######  Example 2.3.

For $f\\!\in\\!Maps\big{(}(\Bbbk^{n},o),(\Bbbk^{m},o)\big{)}$ we get:
$Jac_{j}(f)\\!=\\!I_{j}[f^{\prime}]\\!\subseteq\\!R_{X},$ the j’th
determinantal ideal of matrix of partial derivatives. In particular:

  * •

$Jac_{m}(f)$ is the ordinary Jacobian ideal of the map, and
$Crit_{m}(f)\\!=\\!Crit(f)$ is the usual critical locus.

  * •

In zero characteristic one gets: $Crit_{1}(f)=\cap_{i}V(Jac(f_{i}))\ \lower
1.72218pt\hbox{$\mathrel{\mathop{=}\limits^{\scriptscriptstyle{Br.Sk}}}$}\
\cap_{i}SingV(f_{i}).$ (The last equality holds by the Briançon-Skoda theorem,
[Huneke-Swanson, §13].) Therefore we get: $f(Crit_{1}(f))=\\{o\\}.$

If the restriction $f_{|}:Crit_{j}(f)\to Y$ is finite, then its image is a
well defined (formal/analytic/Nash) subgerm $f(Crit_{j}(f))\subset Y.$ This
image coincides with the subgerm $V(ker(f^{\sharp}))\subseteq Y$ for the
kernel ideal $ker(f^{\sharp})\subset R_{Y}\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f^{\sharp}}}}$}\
R_{X}.$

If the restriction $f_{|}:Crit_{j}(f)\to Y$ is non-finite (and $dim(Y)\geq
3$), then the image can be pathological (not a well defined germ, [Joiţţa-
Tibar], or not an analytic germ, [Gabrielov.71]). Instead of working with
$f(Crit_{j}(f))$ we take the kernel-ideal, $ker(f^{\sharp}_{|})\subseteq
R_{Y}\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f^{\sharp}_{|}}}}$}\
{\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$Jac_{j}(f)$}\right.}.$
Define the j’th discriminant as $\Delta_{j}:=V(ker(f^{\sharp}_{|})).$ Thus
$\Delta_{i}\subseteq Y$ is a (formal/analytic/Nash) subgerm.

Fix some coordinates in $(\Bbbk^{m},o),$ thus we have the components of the
map $f=(f_{1},\dots,f_{m}).$ For each $j$ we take the truncated map
$(f_{1},\dots,f_{j}):X\to(\Bbbk^{j},o).$ Then we take its (higher) critical
loci, $Crit_{\bullet}(f_{1}\dots f_{j})\subseteq X.$

######  Definition 2.4.

The (formal/$\Bbbk$-analytic/$\Bbbk$-Nash) map $f:X\to
Y\subseteq(\Bbbk^{m},o)$ is called of weakly-finite singularity type if the
following conditions hold.

  1. 1.

The restriction $f_{|}:Crit_{r}(f)\to Y$ is a finite morphism for some $1\leq
r\leq m.$

  2. 2.

$dim(\Delta_{i+1})\leq i$ for each $r\leq i<m.$

  3. 3.

For each $r\leq i<m$ there exists a coordinate choice in $(\Bbbk^{m},o)$ that
ensures (set-theoretically): $Crit_{i}(f)\supseteq Crit_{i+1}(f)\cap
Crit_{dim\Delta_{i+1}}(f_{1},\dots,f_{dim\Delta_{i+1}}).$

######  Example 2.5.

  1. i.

The case $r=m\geq 1.$ The conditions 2, 3. are empty. The condition 1. means
just the finite singularity type.

  2. ii.

For $r=1<m$ and $X=(\Bbbk^{n},o)$ of $char=0$ the condition 3. is implied by
1,2. Indeed:

(8) $Crit_{1}(f)=\cap^{m}_{i=1}Crit(f_{i})\ \lower
1.72218pt\hbox{$\mathrel{\mathop{=}\limits^{\scriptscriptstyle{Br.Sk.}}}$}\
\cap^{m}_{i=1}V(Sing(f_{i}))\ \lower
1.72218pt\hbox{$\mathrel{\mathop{=}\limits^{\scriptscriptstyle{Bert.}}}$}\
SingV(\sum c_{j}f_{j})\ \lower
1.72218pt\hbox{$\mathrel{\mathop{=}\limits^{\scriptscriptstyle{Br.Sk.}}}$}\
Crit(\sum c_{j}f_{j}).$

Here $c_{1},\dots,c_{m}$ are generic constants, while the equality $Bert.$
holds by Bertini theorem.

######  Remark 2.6.

  1. i.

The subgerms $Crit_{\bullet}(f)\subseteq X$ depend neither on the choice of
embedding $(Y,o)\subseteq(\Bbbk^{m},o)$ nor on the choice of coordinates in
$(\Bbbk^{m},o).$ Proof: any other embedding leads to the transition
$f\rightsquigarrow\Phi_{Y}\circ f.$ Then for the module of derivations one
gets $Der_{X}(\Phi_{Y}\circ f)=\Phi_{Y}^{\prime}|_{f}\cdot Der_{X}(f).$
Finally,
$Fitt_{\bullet}[{\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$Der_{X}(\Phi_{Y}\circ
f)$}\right.}]=Fitt_{\bullet}[{\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$Der_{X}(f)$}\right.}]\subset
R_{X}.$

Therefore the validity of conditions 1,2,3 does not depend on the choice of
embedding. Thus the notion of weakly-finite singularity type is well-defined.

  2. ii.

The set-theoretic condition 3 means:
$Jac_{i}(f)\subseteq\sqrt{Jac_{i+1}(f)+Jac_{dim\Delta_{i+1}}(f_{1},\dots,f_{dim\Delta_{i+1}})}.$

  3. iii.

If the restriction $f_{|}:Crit_{1}(f)\to(\Bbbk^{m},o)$ is non-finite then
$Crit(f_{1})\cap\cdots\cap Crit(f_{m})$ is of dimension$>0.$ In $char=0$ we
get: $Sing(V(f_{1}))\cap\cdots\cap Sing(V(f_{m}))$ is of positive dimension.

  4. iv.

For $f:(\Bbbk^{n},o)\to(\Bbbk^{m},o)$ the expected codimension of
$Crit_{i}(f)$ is $(n-i+1)(m-i+1)$, while the codimension of $Crit_{i+1}(f)\cap
Crit(f_{1}\dots f_{i})$ is at most $(n-i)(m-i)+(n-i+1)=(n-i+1)(m-i+1)+i-m.$
Hence condition 3. is non-trivial.

###  2.7. Extending maps to automorphisms, the $jet_{0}$-assumption, and a
version of Newton’s lemma

Suppose the scheme germ is $\Bbbk$-smooth, $X\cong(\Bbbk^{n},o),$ i.e.
$J_{X}=0.$ Then any map of the form $x\to x+h(x),$ with $h(x)\in(x)^{2},$
defines a coordinate change, $\Phi\in Aut_{(\Bbbk^{n},o)}.$ If $X$ is not
$\Bbbk$-smooth, i.e. $J_{X}\neq 0,$ then such a map does not necessarily lift
to an automorphism of $R_{X}.$ Therefore in §3.2 we impose the following
$\boldsymbol{jet_{0}}$-assumption, cf. [Kerner.21, §2.3]:

(9) Any map $x\\!\to\\!x\\!+\\!\xi(x)$, with $\xi\in Der_{X}$ and
$\xi({\mathfrak{m}})\\!\subseteq\\!\mathfrak{a}^{2}\\!\cdot\\!{\mathfrak{m}}^{j}$
(for some ideal $\mathfrak{a}\\!\subseteq\\!{\mathfrak{m}}$ and
$j\\!\geq\\!0$),
---
extends to an automorphism $\Phi\in Aut_{X}$ that satisfies:
$\Phi(x)-x-\xi(x)\in\mathfrak{a}^{3}\cdot{\mathfrak{m}}^{j}$.
Moreover, presenting $\Phi=Id+\xi+\varphi_{\xi}$, one has (for
$c_{1},c_{2}\in\Bbbk$): $\varphi_{c_{1}\xi_{1}+c_{2}\xi_{2}}$ is a power
series in $c_{1},c_{2}$.

This assumption holds trivially for $J_{X}=0$ (one just takes
$\varphi_{\xi}=0$). It holds also if $char(\Bbbk)=0,$ [B.G.K.22, §3.2.2]. For
$char(\Bbbk)>0$ and $J_{X}\neq 0$ this is a non-trivial condition, [Kerner.21,
§2.5].

With this assumption the $\mathscr{R}$-orbit of a map is controlled via the
derivations by a version of Newton’s lemma (of type [Bourbaki], Artin,
Tougeron, Elkik and many others).

######  Theorem 2.7.

[Kerner.21, §4.1] (The “$\mathscr{R}$-implicit function theorem”)

Suppose the $jet_{o}$-assumption holds for $R_{X}.$ Then
$\mathscr{R}f\supseteq\\{f\\}+Jac(f)^{3}\cdot R^{\oplus m}_{X}.$

More precisely, if two maps $f,{\tilde{f}}:X\to(\Bbbk^{m},o)$ satisfy
${\tilde{f}}-f\in\mathfrak{a}^{3}\cdot R^{\oplus m}_{X},$ for some ideal
$\mathfrak{a}\subseteq Jac(f)^{3},$ then $f,{\tilde{f}}$ are
$\mathscr{R}$-equivalent by an automorphism $\Phi_{X}\in Aut_{X}$ that
satisfies: $\Phi(x)-x\in\mathfrak{a}^{2}.$

###  2.8. Extending automorphisms to the ambient space

Fix a (formal/$\Bbbk$-analytic/$\Bbbk$-Nash) scheme-germ, $X=Spec(R_{X}).$
Here
$R_{X}={\footnotesize\left.\raisebox{1.37775pt}{$S$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.},$
where $S$ is one of the rings $\Bbbk[[x]],$ $\Bbbk\\{x\\},$ $\Bbbk\langle
x\rangle.$ Fix a subgerm
$Z:=Spec({\footnotesize\left.\raisebox{1.37775pt}{$S$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}+I_{Z}$}\right.})\subset
X.$ (Below we denote the image of $I_{Z}$ by ${\bar{I}}_{Z}\subset R_{X}$.)

The pair $(X,Z)$ has three types of automorphisms:

  1. i.

Automorphisms of $X$ that restrict to identity on $Z$, i.e.

$Aut_{X/Z}:=Aut_{R_{X}/I_{Z}}:=\\{\Phi\in Aut_{\Bbbk}(R_{X})\ |\ \
x-\Phi(x)\in{\bar{I}}_{Z}\\}.$

  2. ii.

Automorphisms of $X$ that preserve the subgerm $Z\subset X$, i.e.

$Aut_{X,Z}:=Aut_{I_{Z}}(R_{X}):=\\{\Phi\in Aut_{\Bbbk}(R_{X})\ |\ \
\Phi({\bar{I}}_{Z})={\bar{I}}_{Z}\\}.$

  3. iii.

Automorphisms of $Z$, i.e.
$Aut_{Z}:=Aut_{\Bbbk}({\footnotesize\left.\raisebox{1.37775pt}{$S$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}+I_{Z}$}\right.}).$

The natural inclusion and projection of these groups define the sequence

(10) $1\to Aut_{X/Z}\to Aut_{X,Z}\to Aut_{Z}\to 1.$

For $J_{X}=0$, i.e. $R_{X}$ is regular over $\Bbbk$, this sequence is exact,
i.e. any automorphism of $Z$ comes from $Aut_{X,Z}.$ But in general this
sequence is not exact.

######  Example 2.8.

Let $\Bbbk={\bar{\Bbbk}}$ and take a generic homogeneous cubic polynomial
$f_{3}(x,y,z)\in\Bbbk[x,y,z].$ Take the (formal) germ
$V(f_{3})\subset(\Bbbk^{3},o)$, with the local ring
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk[[x,y,z]]$}\\!/\\!\raisebox{-1.37775pt}{$(f_{3})$}\right.},$
this surface has an isolated singularity. Take a generic line
$V(l)\subset\mathbb{P}^{2}$, then $Z:=V(l,f_{3})\subset(\Bbbk^{3},o)$ is the
union of three (germs of) lines in a plane. Present $Z$ as the cone over three
points in $\mathbb{P}^{1}.$ The group of automorphisms is $Aut_{Z}\supseteq
S_{3}$. But in general none of these permutations of $Z$ lifts to an
automorphism of
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk[[x,y,z]]$}\\!/\\!\raisebox{-1.37775pt}{$(f_{3})$}\right.}.$
Indeed, the group of $\mathbb{P}^{2}$-automorphisms that preserve the
(generic) cubic $V(f_{3})\subset\mathbb{P}^{2}$ is finite, as it must send the
(nine) flexes of the cubic to themselves. Thus this group cannot contain
permutations of the sections $V(l,f_{3})$ for all the lines $V(l).$

To correct the non-exactness of (10) we take the $d$’th infinitesimal
neighborhood of $Z\subset X,$ i.e.
$Z_{d}:=Spec({\footnotesize\left.\raisebox{1.37775pt}{$S$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}+I^{d}_{Z}$}\right.})\subset
X.$ One gets the corresponding subgroups:

  * •

Automorphisms of $X$ that restrict to identity on $Z_{d}$, i.e.

$Aut_{X/Z_{d}}:=Aut_{R_{X}/I_{Z_{d}}}:=\\{\Phi\in Aut_{\Bbbk}(R_{X})\ |\ \
\Phi(x)-x\in{\bar{I}}^{d}_{Z}\\}<Aut_{X/Z}.$

  * •

Automorphisms of $X$ that preserve the subgerms $Z_{d},Z\subset X$, i.e.

$Aut_{X,Z_{d}}:=Aut_{I_{Z}}(R_{X}):=\\{\Phi\in Aut_{\Bbbk}(R_{X})\ |\ \
\Phi({\bar{I}}_{Z})={\bar{I}}_{Z}\\}.$

  * •

Automorphisms of $Z_{d}$ that restrict to those of $Z$, i.e.

$Aut_{Z_{d}\supset Z}:=\\{\Phi\in
Aut_{\Bbbk}({\footnotesize\left.\raisebox{1.37775pt}{$S$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}+I^{d}_{Z}$}\right.})\
|\ \ \Phi({\bar{I}}_{Z})={\bar{I}}_{Z}\\}\leq Aut_{Z}.$

A remark: the assumption $\Phi({\bar{I}}_{Z})={\bar{I}}_{Z}$ is not needed in
many cases, e.g. when the ideal ${\bar{I}}_{Z}\subset R_{X}$ is radical.

######  Lemma 2.9.

The sequence $1\to Aut_{X/Z_{d}}\to Aut_{X,Z_{d}}\to Aut_{Z_{d}\supset Z}\to
1$ is exact, provided $d\gg 1$ (for a given $Z\subset X$).

Proof. The homomorphism $Aut_{X/Z_{d}}\to Aut_{X,Z_{d}}$ is an embedding by
its construction.

  * •

(Exactness at $Aut_{X,Z_{d}}$) Take $\Phi\in Aut_{X,Z_{d}},$ thus
$\Phi({\bar{I}}_{Z})={\bar{I}}_{Z}$ and
$\Phi({\bar{I}}^{d}_{Z})={\bar{I}}^{d}_{Z}.$ Hence $\Phi$ restricts to the
element $[\Phi]\in Aut_{Z_{d}\supset Z}.$ The later is identity iff
$\Phi(x)-x\in{\bar{I}}^{d}_{Z}\subset R_{X}.$ And this means exactly $\Phi\in
Aut_{X/Z_{d}}.$

  * •

(Exactness at $Aut_{Z_{d}\supset Z}$) Take $\Phi\in Aut_{Z_{d}\supset Z}.$
Lift it to ${\tilde{\Phi}}\in Aut_{\Bbbk}(S),$ which satisfies
${\tilde{\Phi}}(J_{X}+I_{Z})=J_{X}+I_{Z}$ and
${\tilde{\Phi}}(J_{X}+I^{d}_{Z})=J_{X}+I^{d}_{Z}.$ It is enough to ensure/to
verify ${\tilde{\Phi}}(J_{X})=J_{X},$ as then ${\tilde{\Phi}}$ descends to
$Aut_{X,Z_{d}}.$ The condition ${\tilde{\Phi}}(J_{X})=J_{X}$ is an implicit
function equation (see §2.3). And we have an approximate solution,
${\tilde{\Phi}}(J_{X})\equiv J_{X}\ modI^{d}_{Z},$ for (a fixed) $d\gg 1.$

The Strong Artin Approximation ensures an exact formal solution. In detail,
take the ${\mathfrak{m}}$-completion,
${\widehat{R}}_{X}={\footnotesize\left.\raisebox{1.37775pt}{${\hat{\Bbbk}}[[x]]$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}\cdot{\hat{\Bbbk}}[[x]]$}\right.}.$
By Cohen structure theorem
${\hat{\Bbbk}}={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk_{o}[[t]]$}\\!/\\!\raisebox{-1.37775pt}{$I_{t}$}\right.},$
where $\Bbbk_{o}$ is either a field or a complete DVR. Then apply the SAP,
§2.2.iii.

Finally, apply the ordinary AP, §2.2.i. (resp. §2.2.ii.) to get a
$\Bbbk$-analytic (resp. $\Bbbk$-Nash) solution. $\blacksquare$

###  2.9. Finite number of liftings over a finite morphism

Below $X,Y,Z$ are Noetherian scheme-germs, i.e. spectra of Noetherian local
rings.

######  Lemma 2.10.

Let $\pi:Z\to Y$ be a finite morphism of reduced scheme-germs. Any morphism
$f:X\to Y$ admits at most a finite number of factorizations $f=\pi\circ
f_{Z}.$ Proof. 1. $X$ can be non-reduced, with embedded components.
Accordingly we $\begin{matrix}X\dashrightarrow Z\\\ \hskip
8.5359ptf\searrow\hskip 8.5359pt\downarrow\pi\\\ \hskip
34.14322ptY\end{matrix}$
take the (finite) primary decomposition, $R_{X}=0=\cap{\mathfrak{q}\hskip
1.42271pt}_{i},$ and $X=\cup X_{i},$ where
$X_{i}=Spec({\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{q}\hskip
1.42271pt}_{i}$}\right.}).$ Thus each $X_{i}$ has no embedded components and
$(X_{i})_{red}$ is irreducible.

Restrict the map $f:X\to Y$ to these components, $f_{i}:X_{i}\to Y$ is defined
via $f^{\sharp}_{i}:R_{Y}\to
R_{X}\to{\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{q}\hskip
1.42271pt}_{i}$}\right.}.$ These restrictions $\\{f_{\bullet}\\}$ determine
$f.$ Indeed, suppose two maps $f,{\tilde{f}}:X\to Y$ satisfy
$\\{f_{i}={\tilde{f}}_{i}\\}_{i}.$ Then
$(f^{\sharp}-{\tilde{f}}^{\sharp})(R_{Y})\subseteq\cap{\mathfrak{q}\hskip
1.42271pt}_{i}=0\subset R_{X}.$

Therefore, it is enough to prove the statement for
$X=Spec({\footnotesize\left.\raisebox{1.37775pt}{$R$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{q}\hskip
1.42271pt}$}\right.}),$ where $R$ is a local domain, while
${\mathfrak{q}\hskip 1.42271pt}$ is a primary ideal. Accordingly we assume:
$Y$ is irreducible. And then $Z$ is irreducible.

2. Take the generic point $\eta_{Y}=Spec(Frac(R_{Y}))\in Y.$ Moreover, we pass to the algebraic closure, $\overline{\eta_{Y}}:=Spec(\overline{Frac(R_{Y})}).$ Any factorization of the initial morphism, $X\ \lower 1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f_{Z}}}}$}\ Z\ \lower 1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{\pi}}}$}\ Y,$ leads to the factorization of the generic fibre,

(11)
$X_{\overline{\eta_{Y}}}=Spec(\overline{Frac(R_{Y})}\otimes_{R_{Y}}R_{X})\to
Z_{\overline{\eta_{Y}}}=Spec(\overline{Frac(R_{Y})}\otimes_{R_{Y}}R_{Z})\
\lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{\pi_{\overline{\eta_{Y}}}}}}$}\
\overline{\eta_{Y}}.$

The morphism $\pi_{\overline{\eta_{Y}}}$ is still finite. Moreover, as $X$ has
no embedded components (and thus
$R_{X}\hookrightarrow\overline{Frac(R_{Y})}\otimes_{R_{Y}}R_{X}$), the
factorization $f_{Z}$ is determined by its generic fibre,
$f_{Z}|_{\overline{\eta_{Y}}}.$

Finally, $Z_{\overline{\eta_{Y}}}$ is a reduced scheme over the algebraically
closed field $\overline{Frac(R_{Y})},$ and
$dimZ_{\overline{\eta_{Y}}}=dim\overline{\eta_{Y}}=0.$ Therefore
$Z_{\overline{\eta_{Y}}}$ is the disjoint union of copies of
$\overline{\eta_{Y}},$ and the restriction of $\pi_{\overline{\eta_{Y}}}$ to
each of these copies is an isomorphism.

Hence, for a fixed map $X_{\overline{\eta_{Y}}}\to\overline{\eta_{Y}},$ there
exists at most a finite number of factorizations $X_{\overline{\eta_{Y}}}\to
Z_{\overline{\eta_{Y}}}\to\overline{\eta_{Y}}.$ And thus there is only a
finite number of factorizations of $f:X\to Y$ through $Z.$ $\blacksquare$

##  3\. The left-right Artin approximation

######  Definition 3.1.

The left-right Artin approximation, $\mathscr{L}\mathscr{R}$-AP, is said to
hold for a map $f\in Maps(X,Y)$ if for any map ${\tilde{f}}\in Maps(X,Y),$ and
the equation $\Phi_{Y}\circ{\tilde{f}}=f\circ\Phi_{X},$ any formal solution
$\hat{\Phi}_{Y}\circ{\tilde{f}}=f\circ\hat{\Phi}_{X}$ is approximated by
ordinary solutions.

Namely, for any $d\in\mathbb{N}$ there exist ring endomorphisms
$\Phi_{X}\circlearrowright R_{X}$ and $\Phi_{Y}\circlearrowright R_{Y}$,
satisfying:

(12)
$\Phi_{X}(x)-\hat{\Phi}_{X}(x)\in(x)^{d},\quad\quad\quad\quad\quad\Phi_{Y}(y)-\hat{\Phi}_{Y}(y)\in(y)^{d},\quad\quad\quad\quad\quad\Phi_{Y}\circ{\tilde{f}}=f\circ\Phi_{X}.$

######  Remark 3.2.

  1. i.

If $\mathscr{L}\mathscr{R}$-AP holds for a map $f,$ then the properties
$\mathscr{R}$-$AP$ and $\mathscr{L}$-$AP$ hold as well. Indeed, to the
equation $\Phi_{Y}\circ{\tilde{f}}=f\circ\Phi_{X}$ one just appends the
condition $\Phi_{Y}(y)=y$ (resp. $\Phi_{X}(x)=x$). And the later condition is
an implicit function equation.

  2. ii.

We did not impose invertibility of $\Phi_{X},\Phi_{Y}$ here. But if
$\hat{\Phi}_{X},\hat{\Phi}_{Y}$ are invertible, then their approximations
$\Phi_{X},\Phi_{Y}$ are ring automorphisms, i.e. $\Phi_{X}\in\mathscr{R}$ and
$\Phi_{Y}\in\mathscr{L}$.

  3. iii.

(For $Y=(\Bbbk^{m},o).$) The property $\mathscr{L}$-AP implies the inverse
Artin approximation for $f.$
Proof: Take a formal relation $\hat{Q}(f)=0.$ If $\hat{Q}(y)\subseteq(y)^{2}$
then define $\hat{\Phi}:=Id+\hat{Q}\in\hat{\mathscr{L}}.$ Thus
$\hat{\Phi}(f)=f.$ Approximate $\hat{\Phi}$ by an ordinary automorphism,
$\Phi(f)=f.$ Then $Q:=\Phi-Id$ is the needed approximation of $\hat{Q}.$

If $\hat{Q}(y)\not\subseteq(y)^{2}$ then apply the previous argument to
$\hat{Q}(y)^{2}$ or to $\\{y_{i}\cdot\hat{Q}(y)\\}.$ $\blacksquare$

###  3.1. $\mathscr{L}\mathscr{R}$-AP for algebraic power series,
$R_{X}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.}$ and
$R_{Y}={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
y\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.}$

######  Theorem 3.3.

The property $\mathscr{L}\mathscr{R}$-AP holds for any map $f\in Maps(X,Y)$.

Proof. In view of §2.4 we present $R_{X}$ as an $R_{Y}$-algebra,
$R_{X}={\footnotesize\left.\raisebox{1.37775pt}{$R_{Y}\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}+(y-f(x))$}\right.}.$ Similarly
$R_{\tilde{X}}={\footnotesize\left.\raisebox{1.37775pt}{$R_{\tilde{Y}}\langle{\tilde{x}}\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{\tilde{X}}+({\tilde{y}}-{\tilde{f}}({\tilde{x}}))$}\right.}.$
We have formal homomorphisms in the diagram (5), and want to approximate these
by ordinary homomorphisms of the $R_{Y}$-algebras $R_{X},R_{\tilde{X}}$ with
the base change. The needed homomorphism is a map $\psi:R_{X}\to
R_{\tilde{X}}$ presentable by (algebraic) power series,
$\psi(y)=y({\tilde{y}})$ and $\psi(x)=x({\tilde{x}}).$ And $\psi$ must
satisfy:

(13) $\psi(J_{Y})\subseteq J_{\tilde{Y}},\hskip
28.45274pt\psi(J_{X}+(y-f(x)))\subseteq
J_{\tilde{X}}+({\tilde{y}}-{\tilde{f}}({\tilde{x}})).$

These inclusions of ideals are implicit function equations, see §2.3. For
example, take $J_{X}=0$ and $J_{Y}=0$, then the condition is:
$y({\tilde{y}})-f(x({\tilde{x}}))=z({\tilde{x}},{\tilde{y}})\cdot({\tilde{y}}-{\tilde{f}}({\tilde{x}})),$
with the unknowns $y({\tilde{y}}),x({\tilde{x}}),z({\tilde{x}},{\tilde{y}}).$

By our assumption these conditions admit a formal solution,

(14)
$\hat{\psi}(y)=\hat{y}({\tilde{y}})\in\Bbbk[[{\tilde{y}}]],\quad\quad\hat{\psi}(x)=\hat{x}({\tilde{x}})\in\Bbbk[[{\tilde{x}}]],\quad\quad\hat{z}({\tilde{x}},{\tilde{y}})\in\Bbbk[[{\tilde{x}},{\tilde{y}}]].$

We can think of it as a formal nested solution,
$\hat{\psi}(y)=\hat{y}({\tilde{y}})$ and
$\hat{\psi}(x)=\hat{x}({\tilde{x}},{\tilde{y}})\in\Bbbk[[{\tilde{x}},{\tilde{y}}]].$

Apply the nested Artin approximation, §2.2.ii., to approximate this formal
nested solution by an algebraic nested solution,
$y({\tilde{y}})\in\Bbbk\langle{\tilde{y}}\rangle$ and
$x({\tilde{x}},{\tilde{y}})\in\Bbbk\langle{\tilde{x}},{\tilde{y}}\rangle.$
Explicitly:
$y({\tilde{y}})-f(x({\tilde{x}},{\tilde{y}}))\in({\tilde{y}}-{\tilde{f}}({\tilde{x}}))\subset\Bbbk\langle{\tilde{x}},{\tilde{y}}\rangle.$

Then $y\to y({\tilde{y}})$ defines the needed endomorphism of the target,
$\Phi_{Y}\circlearrowright Y.$ Substituting
${\tilde{y}}={\tilde{f}}({\tilde{x}})$ we get $x\to
x({\tilde{x}},{\tilde{f}}({\tilde{x}})),$ which is the needed endomorphism of
the source, $\Phi_{X}\circlearrowright X.$ Altogether we get:
$y({\tilde{y}})-f(x({\tilde{x}},{\tilde{f}}({\tilde{x}})))\in({\tilde{y}}-{\tilde{f}}({\tilde{x}}))\subset\Bbbk\langle{\tilde{x}},{\tilde{y}}\rangle.$
$\blacksquare$

###  3.2. $\mathscr{L}\mathscr{R}$-AP for analytic power series,
$R_{X}\\!=\\!{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{x\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.},$
$R_{Y}={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{y\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{Y}$}\right.}$

If [$J_{X}\neq 0$ and $char(\Bbbk)>0$] then we impose the $jet_{0}$-assumption
on $X$, see (9).

######  Theorem 3.4.

The property $\mathscr{L}\mathscr{R}$-AP (with $\Phi_{X},\\!\Phi_{Y}$
automorphisms) holds for $\Bbbk$-analytic map germs $f:X\to Y$ of weakly-
finite singularity type.

######  Example 3.5.

Take $f\in Maps(X,Y)$ whose central fibre, $f_{o}\in Maps(X_{o},Y_{o}),$ is of
finite singularity type, §2.5.ii. Suppose
$Der_{X}\otimes{\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}_{\Bbbk}$}\right.}\lower
1.72218pt\hbox{$\xrightarrow[\,\smash{\raisebox{4.95134pt}{$\scriptstyle\sim$}}\,]{}$}Der_{X_{o}}.$
(E.g. this holds if $X\cong X_{o}\times Spec(\Bbbk).$) Then the property
$\mathscr{L}\mathscr{R}$-AP (with $\Phi_{X},\\!\Phi_{Y}$ automorphisms) holds
for $f.$

Indeed, $f_{o}$ being of finite singularity type, the restriction
$f_{o}|:Crit(f_{o})\to(\Bbbk^{m},o)$ is a finite morphism. Therefore (cf.
§2.5) $f|:Crit\to(\Bbbk^{m},o)$ is a finite morphism. Hence $f$ is of finite
singularity type. Now apply theorem 3.4.

Remark 3.2 then extends the results of [Shiota.98], [Shiota.10]:

######  Corollary 3.6.

The $\mathscr{L}$-AP and the inverse Artin approximations hold for
$\Bbbk$-analytic maps of weakly-finite singularity type.

####  3.2.1. The proof of theorem 3.4 in the particular case, $f:X\to Y$ is a
finite morphism

Suppose the map-germs $f,{\tilde{f}}:X\to Y$ are formally
$\mathscr{L}\mathscr{R}$-equivalent. Thus ${\tilde{f}}$ is a finite morphism
as well. Therefore $R_{X}$ is a finitely-generated module over $R_{Y},$ via
$f,$ while $R_{\tilde{X}}$ is a finitely-generated module over
$R_{\tilde{Y}},$ via ${\tilde{f}}.$

  1. Step 1.

The (formal) $\mathscr{L}\mathscr{R}$-equivalence $f\sim{\tilde{f}}$ implies
the (formal) isomorphism of these modules with the base change, as on the
diagram (5). Explicitly, fix the minimal presentation matrices,
$R_{X}=coker[A(y)]$, $R_{\tilde{X}}=coker[\tilde{A}({\tilde{y}})]$. (Here
$A(y)\in Mat(y)$, $\tilde{A}({\tilde{y}})\in Mat({\tilde{y}})$.) Then $f\
\lower
1.72218pt\hbox{$\mathrel{\mathop{\sim}\limits^{\scriptscriptstyle{\mathscr{L}\mathscr{R}}}}$}\
{\tilde{f}}$ implies $R_{Y}\lower
1.72218pt\hbox{$\xrightarrow[\,\smash{\raisebox{4.95134pt}{$\scriptstyle\sim$}}\,]{}$}R_{\tilde{Y}}$
and $coker[A(y)]\cong coker[\tilde{A}({\tilde{y}})],$ isomorphism with base
change. By the uniqueness of minimal resolution (over local rings) we get
$A(y({\tilde{y}}))=U({\tilde{y}})\cdot\tilde{A}({\tilde{y}})\cdot
V({\tilde{y}})^{-1}$, here $U({\tilde{y}}),V({\tilde{y}})$ are invertible
matrices over $R_{\tilde{Y}}$. (In particular, the matrices
$A(y),\tilde{A}({\tilde{y}})$ are of the same size.) This is a system of
implicit function equations in the unknowns $U,V,y$. The equations are
quadratic in $U,V$ and analytic power series in $y$.

A formal $\mathscr{L}\mathscr{R}$-equivalence implies a formal solution to
this system, $\hat{y}({\tilde{y}}),\hat{U}(y),\hat{V}(y)$. Invoking the Artin
approximation over the analytic ring $R_{Y},$ §2.2.i., we approximate this
formal solution by an analytic solution, $y({\tilde{y}})$, $U({\tilde{y}})$,
$V({\tilde{y}})$. This gives an analytic isomorphism of modules,
$coker[A(y)]\cong coker[\tilde{A}({\tilde{y}})].$ But this isomorphism of
$R_{Y}$-modules does not necessarily lift to an isomorphism of
$R_{Y}$-algebras of the diagram (5).

  2. Step 2.

To ensure the lifting from modules to algebras we impose additional equations.
Choose the generators of $R_{\tilde{X}}\in mod\textrm{-}R_{\tilde{Y}}$ as
monomials in ${\tilde{x}}$, i.e.
$R_{\tilde{X}}={\footnotesize\left.\raisebox{1.37775pt}{$R_{\tilde{Y}}\langle\\{\underline{{\tilde{x}}}^{\underline{m}}\\}\rangle$}\\!/\\!\raisebox{-1.37775pt}{$(relations)$}\right.}$.
Then, besides the relations of a module, we have the additional relations of
the algebra:
$\\{\prod_{\sum{\underline{m}}={\underline{d}}}\underline{{\tilde{x}}}^{\underline{m}}=\prod_{\sum{\underline{m}}^{\prime}={\underline{d}}}\underline{{\tilde{x}}}^{{\underline{m}}^{\prime}}\\}$.
(Here ${\underline{d}}\in\mathbb{N}^{n}$ is any multi-index.) We ask that the
isomorphism of modules $coker[A(y)]\lower
1.72218pt\hbox{$\xrightarrow[\,\smash{\raisebox{4.95134pt}{$\scriptstyle\sim$}}\,]{}$}coker\tilde{A}[{\tilde{y}}]$
preserves these relations. This gives the equations of type:

(15)
$\prod_{\sum{\underline{m}}={\underline{d}}}(U({\tilde{y}})\cdot\underline{{\tilde{x}}}^{\underline{m}})=\prod_{\sum{\underline{m}}^{\prime}={\underline{d}}}(U({\tilde{y}})\cdot\underline{{\tilde{x}}}^{{\underline{m}}^{\prime}}).$

(And similar equations on $V$.)

These conditions give polynomial equations on the entries of $U({\tilde{y}}).$
The set of such equations is infinite, but the ideal of these equations is
generated by a finite subset. (Because $R_{\tilde{Y}}$ is Noetherian, while
$R_{\tilde{X}}$ is a finitely generated module.)

We add this (finite) set of equations to the equations
$y(J_{\tilde{Y}})=J_{\tilde{Y}},$
$A(y({\tilde{y}}))=U({\tilde{y}})\cdot\tilde{A}({\tilde{y}})\cdot
V({\tilde{y}})^{-1}$. We still get a (finite) system of implicit-function
equations. They are polynomial in $U,V$ and analytic power series in $y$.

As the algebras $(R_{X},R_{Y},f)$, $(R_{\tilde{X}},R_{\tilde{Y}},{\tilde{f}})$
are formally isomorphic, these equations have a formal solution,
$\hat{U}({\tilde{y}}),\hat{V}({\tilde{y}}),\hat{y}({\tilde{y}})$. By the Artin
approximation, §2.2.i., this solution is approximated by an analytic solution.
And this ordinary solution is the needed analytic isomorphism of algebras.
$\blacksquare$

####  3.2.2. The proof of theorem 3.4 in the general case, $f:X\to Y$ is of
weakly finite singularity type

Suppose the maps germs $f,{\tilde{f}}:X\to Y$ are formally
$\mathscr{L}\mathscr{R}$-equivalent.

  1. Step 1.

Then the subschemes $Crit_{r}(f),Crit_{r}({\tilde{f}})\subseteq X$ are related
by a formal automorphism, $\hat{\Phi}_{X}\in Aut_{\hat{X}}.$ Namely, the
implicit-function equation $Jac_{r}(f)=\Phi(Jac_{r}{\tilde{f}})$ has a formal
solution. Applying the ordinary Artin approximation in $R_{X}$ we can assume:
$Crit_{r}(f)=Crit_{r}({\tilde{f}}),$ i.e.
$Jac_{r}(f)=Jac_{r}({\tilde{f}})\subset R_{X}.$ We denote this locus by
$Crit_{r}.$

The restrictions $f_{|},{\tilde{f}}_{|}:Crit_{r}\to Y$ are formally
$\mathscr{L}\mathscr{R}$-equivalent, and $f_{|}$ is finite. Thus
$f_{|},{\tilde{f}}_{|}$ are equivalent by an element of $Aut_{Crit_{r}}\times
Aut_{Y}$, by §3.2.1. More precisely, the formal equivalence is approximated
(to any order) by analytic equivalences.

The elements of $Aut_{Crit_{r}}$ do not necessarily lift to elements of
$Aut_{X,Crit_{r}},$ see §2.8. However, we could take the “fat” structure on
$Crit_{r},$ with the defining ideal $Jac_{r}(f)^{d}$ for $d\gg 1.$ Then the
restriction $Aut_{X,Crit_{r}}\to Aut_{Crit_{r}}$ is surjective, by lemma 2.9.
Repeating the previous argument, we get an element of $Aut_{Crit_{r}}\times
Aut_{Y},$ which now lifts to an element of $Aut_{X,Crit_{r}}\times Aut_{Y}.$

Therefore below we assume: $f_{|}={\tilde{f}}_{|},$ i.e. ${\tilde{f}}-f\in
Jac_{r}(f)^{d}\cdot R^{\oplus m}_{X}$ for some $d\gg 1.$ Here $d$ is chosen
large, to ensure that all the subsequent exponents $\\{d_{j}\\}$ (below) are
large as well.

If $r=m,$ i.e. $f$ is of finite singularity type, then we apply theorem 2.7 to
get $f\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\sim}\limits^{\scriptscriptstyle{\mathscr{R}}}}$}\
{\tilde{f}}.$ (Here one uses the $jet_{o}$-assumption.) Altogether: we have
constructed an element $(\Phi_{Y},\Phi_{X})\in\mathscr{L}\times\mathscr{R}$
that approximates $(\hat{\Phi}_{Y},\hat{\Phi}_{X}).$ This finishes the proof.

Below we assume $r<m$ and describe the iterations on $j,$ from $j=r$ to $j=m.$

  2. Step 2.

Suppose ${\tilde{f}}-f\in Jac_{j}(f)^{d_{j}}\cdot R^{\oplus m}_{X}$ for some
$d_{j}\gg 1.$ (After Step 1. we have this for $j=r.$) Denote
$Crit_{j}:=Crit_{j}(f)=Crit_{j}({\tilde{f}})\subseteq X.$

The subgerms $Crit_{j+1}(f),Crit_{j+1}({\tilde{f}})\subseteq X$ are related by
a formal automorphism $\hat{\Phi}_{X}\in Aut_{\hat{X}}.$ As $d_{j}\gg 1,$ we
can assume: $\hat{\Phi}_{X}$ preserves the subgerm $Crit_{j}.$ Moreover, we
can assume $\hat{\Phi}_{X}(x)-x\in Jac_{j}(f).$ Then the ordinary AP provides
an analytic automorphism $\Phi_{X}\in Aut_{X}$ satisfying
$\Phi_{X}(Crit_{j+1}({\tilde{f}}))=Crit_{j+1}(f),$ preserving $Crit_{j},$ and
approximating $\hat{\Phi}_{X}.$ Therefore, as before, we assume
$Crit_{j+1}({\tilde{f}})=Crit_{j+1}(f)$ and denote this subgerm by
$Crit_{j+1}.$

Take the restrictions $f_{|},{\tilde{f}}_{|}$ to $Crit_{j+1}$ and consider the
analytic images
$V(ker(f^{\sharp}_{|})),V(ker({\tilde{f}}^{\sharp}_{|}))\subset Y,$ see §2.6.
They are formally-$\mathscr{L}$-equivalent. The ordinary AP (in $R_{Y}$)
approximates this formal equivalence by an analytic equivalence. Therefore we
assume $V(ker(f^{\sharp}_{|}))=V(ker({\tilde{f}}^{\sharp}_{|}))\subset Y,$ and
denote this subgerm by $\Delta_{j+1}.$

  3. Step 3.

By the condition 3 of definition 2.4 we can choose the coordinates in $Y$ to
ensure $Crit_{j}(f)\supseteq Crit_{j+1}(f)\cap
Crit_{dim\Delta_{j+1}}(f_{1},\dots,f_{dim\Delta_{j+1}}).$ Therefore:

(16) ${\tilde{f}}-f\in Jac_{dim\Delta_{j+1}}(f_{1}\dots
f_{dim\Delta_{j+1}})^{a_{j}}+Jac_{j+1}(f)^{b_{j}}.\hskip 42.67912pt\text{Here
$a_{j},b_{j}\gg 1$ as $d_{j}\gg 1.$}$

Apply theorem 2.7 (with the $jet_{o}$-assumption on $X$). Thus we can assume:
${\tilde{f}}-f=(h_{1}\dots h_{m}),$ where $h_{1},\dots,h_{j}\in
Jac_{j+1}(f)^{b_{j}}$ and $h_{j+1},\dots,h_{m}\in
Jac_{dim\Delta_{j+1}}(f_{1}\dots
f_{dim\Delta_{j+1}})^{a_{j}}+Jac_{j+1}(f)^{b_{j}}.$

By our assumption $dim\Delta_{j+1}\leq j.$ Thus we take the (finite)
projection $\pi:\Delta_{j+1}\to(\Bbbk^{j},o).$ The restriction to $Crit_{j+1}$
satisfy: $\pi\circ f_{|}=\pi\circ{\tilde{f}}_{|}.$ Therefore the maps
$f,{\tilde{f}}$ are both liftings of the map $(f_{1}\dots
f_{j}):Crit_{j+1}\to(\Bbbk^{j},o).$ By lemma 2.10 the number of such liftings
is finite (for the specified $f_{|}:Crit_{j+1}\to\Delta_{j+1}$). Therefore, as
$b_{j}\gg 1,$ we get: ${\tilde{f}}|_{Crit_{j+1}}=f|_{Crit_{j+1}},$ i.e.
${\tilde{f}}-f\in Jac_{j+1}(f)^{d_{j+1}}\cdot R^{\oplus m}_{X}$ for some
$d_{j+1}\gg 1.$

If $j+1=m$ then we have reached: ${\tilde{f}}-f\in Jac_{m}(f)^{d_{m}}\cdot
R^{\oplus m}_{X}$ for $d_{m}\gg 1.$ Apply theorem 2.7 on $X$ to get:
${\tilde{f}}=f.$ The composition of all the automorphisms in $X,Y,$ that we
have constructed in the proof, gives the needed analytic approximation to the
initial formal equivalence $\hat{\Phi}_{X},\hat{\Phi}_{Y}.$

If $j+1<m$ then iterate Steps 2,3. $\blacksquare$

##  4\. Artin approximation for quivers of maps

Below $\Gamma$ is always a finite directed tree with at least two vertices.

###  4.1. Encoding approximation problems by quivers of maps

As in §1.4, the vertices of $\Gamma$ are $\Bbbk$-analytic/$\Bbbk$-Nash scheme-
germs, $\\{X_{v}=Spec(R_{v})\\}_{v}.$ The arrows are morphisms,
$\\{f_{wv}:X_{v}\to X_{w}\\}_{wv}.$ Call this data “a quiver on the graph
$\Gamma$”.

The morphism of two such quivers on the same graph (and over the same base
$\Bbbk$),
$(\Gamma,\\{{\tilde{X}}_{\bullet}\\},\\{{\tilde{f}}_{wv}\\})\to(\Gamma,\\{X_{\bullet}\\},\\{f_{wv}\\}),$
is a collection of maps $\Phi_{v}:{\tilde{X}}_{v}\to X_{v},$ making all the
vertical rectangles to commute:

(17) $\begin{matrix}\underset{\nearrow}{\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{\searrow}}}$}\
}\quad{\tilde{X}}_{i}\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{{\tilde{f}}_{wv}}}}$}\
{\tilde{X}}_{j}\quad\underset{\searrow}{\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{\nearrow}}}$}\
}\\\ \Phi_{v}\downarrow\quad\quad\downarrow\Phi_{w}\\\ \underset{\nearrow}{\
\lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{\searrow}}}$}\
}\quad X_{v}\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f_{wv}}}}$}\
X_{w}\quad\underset{\searrow}{\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{\nearrow}}}$}\
}\end{matrix}$

The commutativity of vertical rectangles gives the system of equations:
$\\{\Phi_{w}\circ{\tilde{f}}_{wv}=f_{wv}\circ\Phi_{v}\\}_{v,w},$ with unknowns
$\\{\Phi_{\bullet}\\}.$ Each $\Phi_{v}$ corresponds to a (local) homomorphism
of local rings $\Phi^{\sharp}_{v}:R_{v}\to{\tilde{R}}_{v}.$ Fixing the
generators of maximal ideals, $\Phi^{\sharp}_{v}$ can be presented as a vector
of (analytic/algebraic) power series.

######  Remark 4.1.

One can impose various additional conditions on $\\{\Phi_{\bullet}\\},$ e.g.:

  1. i.

Some $\Phi_{v}$ vanish at the origin in a prescribed manner, i.e.
$\Phi^{\sharp}_{v}({\mathfrak{m}}_{X_{v}})\subseteq\tilde{\mathfrak{a}}_{v}\subset{\tilde{R}}_{v}.$
(The ideals $\tilde{\mathfrak{a}}_{\bullet}$ are prescribed.)

  2. ii.

Some $\Phi_{v}$ send subgerms ${\tilde{Z}}_{v}\subset{\tilde{X}}_{v}$ to
subgerms $Z_{v}\subset X_{v},$ i.e. $\Phi^{\sharp}_{v}(I_{Z_{v}})\subseteq
I_{{\tilde{Z}}_{v}}\subset{\tilde{R}}_{v}.$

  3. iii.

Some of $\Phi_{v}$ are isomorphisms.

  4. iv.

(Assuming $X_{v}={\tilde{X}}_{v}$.) Some $\Phi_{v}$’s are identities up to
“higher-order-terms”, i.e. $\Phi_{v}(x_{v})-x_{v}\in\mathfrak{a}_{v}$ for some
ideals $\mathfrak{a}_{v}\subset R_{v}.$

In the particular case $\mathfrak{a}_{v}=0$ this means $\Phi_{v}=Id_{X_{v}}.$
In this case $\Phi_{v}$ is just removed from the list of unknowns.

Observe that all these additional conditions on $\\{\Phi_{\bullet}\\}$ are
presentable as equations of implicit function type.

######  Example 4.2.

Take the simplest quiver $X\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f}}}$}\ Y,$
the corresponding equation is $\Phi_{Y}\circ{\tilde{f}}=f\circ\Phi_{X}.$

  * •

Impose the condition “$\Phi_{X},\Phi_{Y}$ are automorphisms”, then one gets
the left-right equivalence of maps $f,{\tilde{f}}.$

  * •

Imposing in addition $\Phi_{X}=Id_{X}$ (resp. $\Phi_{Y}=Id_{Y}$), one gets the
left (resp. the right) equivalence of maps $f,{\tilde{f}}.$

  * •

Imposing $\Phi_{X}=Id_{X}$, and no invertibility condition on $\Phi_{Y},$ one
gets the inverse Artin type problem.

  * •

Fix some subgerms $Z_{X}\subset X$ and $Z_{Y}\subset Y,$ and impose the
conditions $\Phi_{X}(Z_{X})=Z_{X},$ $\Phi_{Y}(Z_{Y})=Z_{Y}.$ We get the
$\mathscr{L}\mathscr{R}$-equivalence relative to these subgerms.

######  Definition 4.3.

($\boldsymbol{\Gamma}$-AP) The Artin approximation holds for the quiver
$(\Gamma,\\{X_{v}\\},\\{f_{wv}\\})$ if, given any quiver
$(\Gamma,\\{{\tilde{X}}_{v}\\},\\{{\tilde{f}}_{wv}\\})$ (over the same base
$\Bbbk$), any set of formal morphisms
$\\{\hat{\Phi}_{v}:\hat{{\tilde{X}}}_{i}\to{\hat{X}}_{v}\\}$ satisfying the
equations $\\{\Phi_{w}\circ{\tilde{f}}_{wv}=f_{wv}\circ\Phi_{v}\\}_{wv},$ is
approximated by sets of ordinary solutions $\\{\Phi_{v}:{\tilde{X}}_{v}\to
X_{v}\\}.$

Depending on the context one adds the conditions of remark 4.1.

######  Remark 4.4.

From the very beginning we exclude graphs with loops. Any loop would produce
the Schröder-type equation $f\circ\Phi_{X}=\Phi_{X}\circ{\tilde{f}}.$ Imposing
the condition “$\Phi_{X}$ is an automorphism”, we would get the conjugation
equivalence, ${\tilde{f}}=\Phi_{X}\circ f\circ\Phi^{-1}_{X}.$ Recall that in
this case the formal and $\mathbb{C}$-analytic classifications differ
essentially even in the one-dimensional case. See Siegel’s linearization
theorem, [Milnor, pg.114], [A.G.V., pg.356].

###  4.2. $\Gamma$-AP for rooted trees

Below $\Gamma$ is a directed rooted tree. Namely, every vertex $v\in\Gamma$ is
connected by a directed path to a particular vertex, $v\rightsquigarrow
o\in\Gamma,$ the later is called the root of $\Gamma.$

######  Proposition 4.5.

The property $\Gamma$-AP holds in the $\Bbbk$-Nash case, i.e.
$\\{R_{v}={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x^{(v)}\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{v}$}\right.}\\}_{v},$ for
directed rooted trees.

Proof. The proof is an extension of the proof of Theorem 3.3.

  1. 1.

Given two such trees we get the diagram (17), with the system of equations

(18) $\Phi_{v}\circ{\tilde{f}}_{wv}=f_{wv}\circ\Phi_{v},\hskip
85.35826pt\text{ for each edge }[v\to w]\subset\Gamma.$

and possibly additional equations of Remark 4.1.

As in the particular case of §3.1, each arrow $f_{wv}:X_{v}\to X_{w}$ presents
the (Nash) ring $R_{v}$ as an $R_{w}$-algebra,
$R_{v}={\footnotesize\left.\raisebox{1.37775pt}{$R_{w}\langle
x^{(v)}\rangle$}\\!/\\!\raisebox{-1.37775pt}{$(x^{(w)}-f_{wv}(x^{(v)}))$}\right.}.$
And each condition $\Phi_{w}\circ{\tilde{f}}_{wv}=f_{wv}\circ\Phi_{v}$ becomes
a homomorphism of algebras with base change. Namely, as in §3.1, we should
find maps $R_{v}\ni x^{(v)}\to x^{(v)}({\tilde{x}}^{(v)})\in{\tilde{R}}_{v}$
that ensure the conditions

(19)
$x^{(w)}({\tilde{x}}^{(w)})-f_{wv}(x^{(v)}({\tilde{x}}^{(v)}))\in({\tilde{x}}^{(w)}-{\tilde{f}}_{wv}({\tilde{x}}^{(v)})),\hskip
28.45274pt\text{ for each edge }[v\to w]\subset\Gamma.$

All these conditions (together with those of Remark 4.1) are implicit function
equations.

We have a formal solution to this system of equations,
$\\{\hat{x}^{(v)}({\tilde{x}}^{(v)})\\}_{v}.$ To approximate it by Nash
solutions we will use the nested Artin approximation.

The nested structure on the set of variables is defined as follows. Define the
grade of a vertex $v\in\Gamma$ as the length of the (unique) directed path to
the root, $v\rightsquigarrow o.$ Thus the grade of the root is 0. Denote by
$\Gamma_{\leq j}\subset\Gamma$ the subgraph of all the vertices (and their
edges) of grades$\leq j.$ Thus $\Gamma_{\leq 0}=\\{o\\}.$

Define the nested structure on the set of variables $\\{x^{\bullet}\\}$ by the
chain of subgraphs $\Gamma_{\leq 0}\subset\Gamma_{\leq
1}\subset\cdots\subset\Gamma.$ Observe: the subgraph $\Gamma_{\leq
j}\setminus\Gamma_{\leq j-1}$ is totally disconnected. (I.e. if
$v,w\in\Gamma_{\leq j}\setminus\Gamma_{\leq j-1}$ then $v,w$ are not connected
by an edge.)

  2. 2.

We have the formal nested solution,
$\\{\hat{x}^{(v)}({\tilde{x}}^{(v)})\\}_{v}.$ Take its nested approximation,
i.e. Nash power series
$\\{\chi^{(v)}(\\{{\tilde{x}}^{(\bullet)}\\}_{gr\bullet\leq grv})\\}_{v}.$
These power series are “non-pure”, thus they do not define automorphisms of
the schemes $\\{X_{v}\\}_{v}.$

By our construction, for each edge of the graph one has the inclusion:

(20) $[v\to w]\subset\Gamma\hskip
56.9055pt\chi^{(w)}(\\{{\tilde{x}}^{(\bullet)}\\}_{gr\bullet\leq
grw})-f_{wv}(\chi^{(w)}(\\{{\tilde{x}}^{(\bullet)}\\}_{gr\bullet\leq
grv}))\in({\tilde{x}}^{(w)}-{\tilde{f}}_{wv}({\tilde{x}}^{(v)})).$

This is a power series identity. The variables
$\\{{\tilde{x}}^{\bullet}\\}_{\bullet\neq v,w}$ are free parameters. Therefore
this particular condition $[v\to w]$ remains true under any specialization of
these parameters.

We will specialize the parameters iteratively, so that the functions
$\chi^{(v)}(\\{{\tilde{x}}^{(\bullet)}$ are transformed into “pure” solutions,
$\\{x^{(v)}({\tilde{x}}^{(v)})\\}_{v}.$ Observe that the solution for the root
of $\Gamma$ is already pure,
$\chi^{(o)}(\\{{\tilde{x}}^{(\bullet)}\\}_{gr\bullet\leq
gro})=\chi^{(o)}({\tilde{x}}^{(o)}).$

The iterative step. Suppose the solutions
$\chi^{(v)}({\tilde{x}}^{(\bullet)})$ are pure for vertices of grade $grv\leq
j,$ while for $grv>j$ all the functions $\chi^{(v)}({\tilde{x}}^{(\bullet)})$
depend on ${\tilde{x}}^{(\bullet)}$ with $gr\bullet\geq j$ only. (Initially
this holds for $j=0.$) We specialize certain parameters to pass from $j$ to
$j+1.$

Fix any vertex $w$ of $grade=j$ and any $v$ of $grade>j$ that is connected to
$w$ by a directed path, $v\rightsquigarrow w.$ Compose the maps along this
path, $f_{w\cdots v}:=f_{wv_{1}}\circ f_{v_{1}v_{2}}\circ\cdots\circ
f_{v_{n}v}.$ Specialize the power series $\chi^{(v)}({\tilde{x}}^{(\bullet)})$
to
$\chi^{(v)}({\tilde{x}}^{(\bullet)})|_{{\tilde{x}}^{(w)}={\tilde{f}}_{w\cdots
v}({\tilde{x}}^{(v)})}.$ We claim: this specialization preserves all the
conditions (20). For the condition of the edge $[v\to w]$ this is verified by
Taylor expansion of $f_{w\cdots v}(x^{(v)}).$ For all the other edges the
preservation is trivial.

Make such substitutions (repeatedly) for all vertices of the complement
$\Gamma_{\leq j+1}\setminus\Gamma_{\leq j}.$ At each step we replace all the
power series $\chi^{(v)}({\tilde{x}}^{(\bullet)})$ by their specializations.
This does not change the power series $\chi^{(v)}({\tilde{x}}^{(\bullet)})$
for $grv\leq j.$

By now, for each $v\in\Gamma_{j+1}$ the power series
$\chi^{(v)}({\tilde{x}}^{(\bullet)})$ depends only on
${\tilde{x}}^{(\bullet)}$ with $gr\bullet=j+1.$ It is not yet pure. Finally,
we specialize
$\chi^{(v)}({\tilde{x}}^{(\bullet)})\to\chi^{(v)}({\tilde{x}}^{(\bullet)})|_{{\tilde{x}}^{(\bullet\neq
v)}=0},$ to get a pure power series, $\chi^{(v)}({\tilde{x}}^{(v)}).$ Do this
for all the descendants of $v$, i.e.:
$\chi^{(v^{\prime})}({\tilde{x}}^{(\bullet)})\to\chi^{(v^{\prime})}({\tilde{x}}^{(\bullet)})|_{{\tilde{x}}^{(\bullet\neq
v)}=0},$ whenever $[v^{\prime}\rightsquigarrow v].$ Again, this preserves all
the conditions of (20).

The specialized power series satisfy now:
$\chi^{(v)}({\tilde{x}}^{(\bullet)})$ are pure for $grv\leq j+1,$ while for
$grv>j+1$ all the functions $\chi^{(v)}({\tilde{x}}^{(\bullet)})$ depend on
${\tilde{x}}^{(\bullet)}$ with $gr\bullet\geq j+1$ only.

Repeat this iterative step. $\blacksquare$

###  4.3. A variation: unfoldings of map-germs (i.e. $\Gamma$-AP with the
change of base $\Bbbk$)

In definition 4.3 we dealt with quivers of mappings over the fixed base,
$Spec(\Bbbk).$ The maps $\Phi_{v}:R_{v}\to{\tilde{R}}_{v}$ were homomorphisms
of $\Bbbk$-algebras. When studying (induced) unfoldings/deformations, it is
important to allow also morphisms of the base ring $\Bbbk.$

######  Example 4.6.

Consider the map $X\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f}}}$}\ Y$ as
an unfolding of its central fibre, $X_{o}\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f_{o}}}}$}\
Y_{o},$ see §2.1.v. This is a family of maps over the base $Spec(\Bbbk).$
Another unfolding, ${\tilde{X}}\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{{\tilde{f}}}}}$}\
{\tilde{Y}},$ is called
induced from $f$ if there is a commutative diagram of maps. (See e.g. [Mond-
Montaldi.91].) In some cases one insists that the central fibres coincide,
i.e. $\Phi_{X_{o}}=Id_{X_{o}}$ and $\Phi_{Y_{o}}=Id_{Y_{o}}.$ The cases
relevant for us are (below $\Bbbk_{o}$ is a field): • ($\Bbbk$-analytic)
$\Bbbk={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk_{o}\\{t\\}$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}$}\right.},$
${\tilde{\Bbbk}}={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk_{o}\\{{\tilde{t}}\\}$}\\!/\\!\raisebox{-1.37775pt}{$\tilde{\mathfrak{a}}$}\right.},$
$R_{X}={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{x\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.},$
… • ($\Bbbk$-Nash)
$\Bbbk={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk_{o}\langle
t\rangle$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}$}\right.},$
${\tilde{\Bbbk}}={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk_{o}\langle{\tilde{t}}\rangle$}\\!/\\!\raisebox{-1.37775pt}{$\tilde{\mathfrak{a}}$}\right.},$
$R_{X}={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{X}$}\right.},$ …
$\begin{matrix}\hskip 14.22636pt{\tilde{X}}\quad\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\longrightarrow}\limits^{\scriptscriptstyle{\Phi_{X}}}}$}\
\quad X\hskip 14.22636pt\\\ \pi_{\tilde{X}}\searrow{\tilde{f}}\hskip
19.91684pt\pi_{X}\searrow f\\\ \hskip 28.45274pt{\tilde{Y}}\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\longrightarrow}\limits^{\scriptscriptstyle{\Phi_{Y}}}}$}\
\hskip 22.76228ptY\\\ \hskip 19.91684pt\swarrow\pi_{\tilde{Y}}\hskip
25.6073pt\swarrow\pi_{Y}\\\ Spec(\tilde{\Bbbk})\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\longrightarrow}\limits^{\scriptscriptstyle{\Phi_{\Bbbk}}}}$}\
Spec(\Bbbk)\end{matrix}\put(-95.0,5.0){\Big{\downarrow}}\put(-35.0,5.0){\Big{\downarrow}}$

Accordingly we write $f$ as $f_{t}(x),$ resp. ${\tilde{f}}$ as
${\tilde{f}}_{\tilde{t}}({\tilde{x}}),$ and $\Phi_{X}$ as
$\Phi_{X,{\tilde{t}}}({\tilde{x}}),$ and so on. Then the diagram is encoded by
the equation
$\Phi_{Y,{\tilde{t}}}({\tilde{f}}_{\tilde{t}}({\tilde{x}}))=f_{\Phi_{\Bbbk}({\tilde{t}})}(\Phi_{X,{\tilde{t}}}({\tilde{x}})).$
Here $\Phi_{\Bbbk}({\tilde{t}}),\Phi_{X,{\tilde{t}}},\Phi_{Y,{\tilde{t}}}$ are
unknowns.

More generally, for $\Bbbk$ as above, one considers quivers of families of
(analytic/Nash) map-germs over $Spec(\Bbbk),$ i.e. $\\{X_{t,v}\ \lower
1.72218pt\hbox{$\mathrel{\mathop{\to}\limits^{\scriptscriptstyle{f_{t,wv}}}}$}\
X_{t,w}\\}_{vw},$ with
$R_{t,v}={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{x^{(v)}\\}$}\\!/\\!\raisebox{-1.37775pt}{$J_{v}$}\right.},$
resp. ${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x^{(v)}\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J_{v}$}\right.}.$ Here
$J_{v}\subseteq(x^{(v)})+(t).$ We call this “a quiver $\Gamma$ over
$Spec(\Bbbk)$.”

######  Definition 4.7.

$\boldsymbol{(\Gamma,t)}$-AP. The Artin approximation with parameters holds
for the quiver-family $(\Gamma,\\{X_{t,v}\\},\\{f_{t,wv}\\})$ if for any
quiver-family
$(\Gamma,\\{{\tilde{X}}_{{\tilde{t}},v}\\}_{v},\\{{\tilde{f}}_{{\tilde{t}},wv}\\}_{wv},$
any set of formal morphisms
$\\{\hat{\Phi}_{v}:\hat{\tilde{X}}_{t,v}\to{\hat{X}}_{t,v}\\},$
$\hat{\Phi}_{\Bbbk}:Spec(\hat{\tilde{\Bbbk}})\to Spec({\hat{\Bbbk}}),$ solving
the equations
$\\{\Phi_{w}\circ{\tilde{f}}_{t,wv}=f_{\Phi_{\Bbbk}(\Bbbk),wv}\circ\Phi_{v}\\}_{v,w},$
is approximated by a set of ordinary morphisms
$\\{\Phi_{v}:X_{t,v}\to{\tilde{X}}_{{\tilde{t}},v}\\}_{v},$
$\Phi_{\Bbbk}:Spec(\Bbbk)\to Spec({\tilde{\Bbbk}}).$

######  Corollary 4.8.

$(\Gamma,t)$-AP holds in the Nash case for rooted trees.

Proof. Adjoin to the quiver $\Gamma$ the additional vertex
$\\{Spec(\Bbbk)\\},$ and one directed edge, $\Gamma\ni o\to\\{Spec(\Bbbk)\\}.$
The “extended” quiver $\Gamma^{\prime}=\Gamma\cup\\{Spec(\Bbbk)\\}$ is still a
rooted graph without cycles. Apply to it proposition 4.5. $\blacksquare$

######  Example 4.9.

The simplest quivers $\Phi_{X}\circlearrowright X\to Y$ and
$\Phi_{X}\circlearrowright X\to Y\circlearrowleft\Phi_{Y}$ are particularly
important for the study of families/unfoldings, [Kerner.22]. Suppose the
target is smooth, $Y=(\Bbbk^{m},o),$ then we identify $Maps(X,Y)\cong(x)\cdot
R^{\oplus m}_{X}.$ Each map $f:X\to Y$ is an unfolding of its central fibre,
$f_{o}:X_{o}\to Y_{o}.$ Fix a finite set of vectors, $v_{\bullet}\in R^{\oplus
m}_{X}.$

We want to bring an unfolding $f_{t}$ to the “normal form”, $f_{o}+\sum
c_{\bullet}(t)\cdot v_{\bullet}.$ Corollary 4.8 implies: if $f_{t}$ is
$\hat{\mathscr{R}}$ (resp. $\widehat{\mathscr{L}\mathscr{R}}$) equivalent to a
form $f_{o}+\sum\hat{c}_{\bullet}(t)\cdot v_{\bullet}$ (with formal
coefficients $c_{\bullet}\in\Bbbk[[t]]$), then $f_{t}$ is $\mathscr{R}$ (resp.
$\mathscr{L}\mathscr{R}$) equivalent to the form $f_{o}+\sum
c_{\bullet}(t)\cdot v_{\bullet}$ (with the coefficients
$c_{\bullet}\in\Bbbk\langle t\rangle$).

## References

  * [Abhyankar-van der Put.70] S. S. Abhyankar, S. S. van der Put, Homomorphisms of analytic local rings. J. Reine Angew. Math. 242 (1970), 26–-60.
  * [Alon.Cast-Jimé.Haus.Kout.18] M. E. Alonso, F. J. Castro-Jiméénez, H. Hauser, C. Koutschan, Echelons of power series and Gabrielov’s counterexample to nested linear Artin approximation. Bull. Lond. Math. Soc. 50, No. 4, 649–662 (2018).
  * [A.G.V.] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of differentiable maps. Volume 1. Classification of critical points, caustics and wave fronts. Reprint of the 1985 edition. Modern Birkhäuser Classics. Birkhäuser/Springer, New York, 2012
  * [Artin.68] M.Artin, On the solutions of analytic equations. Invent. Math. 5, 1968, 277–291.
  * [Artin.69] M. Artin, Algebraic approximation of structures over complete local rings, Publ. Math. IHES, 36, (1969), 23-58.
  * [Belot.Curmi.Rond.21] A. Belotto da Silva, O. Curmi, G. Rond, A proof of A. Gabrielov’s rank theorem. J. Éc. polytech. Math. 8 (2021), 1329–1396.
  * [B.G.K.22] A.-F. Boix, G.-M. Greuel, D. Kerner, Pairs of Lie-type and large orbits of group actions on filtered modules. (A characteristic-free approach to finite determinacy.), Math. Z. 301 (2022), no. 3, 2415–2463.
  * [Bourbaki] N. Bourbaki, Algèbre commutative, Fasc. XXVIII, Chap. III, §4, No. 5 (1961).
  * [Denef-Lipshitz] J. Denef, L. Lipshitz, Ultraproducts and approximation in local rings. II. Math. Ann. 253 (1980), no. 1, 1–28.
  * [Fich.Quar.Shiot] G. Fichou, R. Quarez, M Shiota, Artin approximation compatible with a change of variables. Canad. Math. Bull. 59 (2016), no. 4, 760–-76.
  * [Gabrielov.71] A. M. Gabrielov, Formal relations among analytic functions. Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 1056–1090. Translation from Funkts. Anal. Prilozh. 5, No. 4, 64-65 (1971).
  * [Gabrielov.73] A. M. Gabrielov, Formale Relationen zwischen analytischen Funktionen. Izv. Akad. Nauk SSSR, Ser. Mat. 37, 1056–1090, (1973).
  * [Grothendieck.61] A. Grothendieck, Techniques de construction en gééoméétrie analytique. VI. ÉÉtude locale des morphismes: germes d’espaces analytiques, platitude, morphismes simples, Sééminaire Henri Cartan, Volume 13 (1960-1961) p. 1-13.
  * [Huneke-Swanson] C. Huneke, I. Swanson, Integral closure of ideals, rings, and modules, London Math. Soc. Lecture Note Ser., 336 Cambridge University Press, Cambridge, 2006, xiv+431 pp.
  * [Izumi.07] S. Izumi, Fundamental properties of germs of analytic mappings of analytic sets and related topics. Real and complex singularities. Proceedings of the 1st Australian-Japanese workshop, University of Sydney, Australia, September 5–8, 2005. Hackensack, NJ: World Scientific (ISBN 978-981-270-551-8/hbk). 109-123 (2007).
  * [Joiţţa-Tibar] C. Joiţţa, M. Tibăăr, The local image problem for complex analytic maps, Ark. Mat. 59 (2021), no. 2, 345-–358.
  * [Kerner.21] D. Kerner, Orbits of the left-right equivalence of maps in arbitrary characteristic, arXiv:2111.02715.
  * [Kerner.22] D. Kerner, Unfoldings of maps, the first results on stable maps, and results of Mather-Yau/Gaffney-Hauser type in arbitrary characteristic, arXiv:2209.05071.
  * [K.P.P.R.M.78] H. Kurke, G. Pfister, D. Popescu, M. Roczen, T. Mostowski, Die Approximationseigenschaft lokaler Ringe, Lectures Notes in Math., 634, Springer-Verlag, Berlin-New York, (1978).
  * [Milnor] J. Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999, 257 pp.
  * [Mond-Montaldi.91] D. Mond, J. Montaldi, Deformations of maps on complete intersections, Damon’s ${\mathscr{K}}\\!_{V}$-equivalence and bifurcations. Brasselet, Jean-Paul (ed.), Singularities. Papers of the international congress ‘Singularities in geometry and topology’, held in Lille (France), 3-8 June, 1991. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 201, 263-284 (1994).
  * [Mond. Nuñ.-Bal.] D. Mond, J.-J. Nuño-Ballesteros, Singularities of Mappings. The Local Behaviour of Smooth and Complex Analytic Mappings, Grundlehren Math. Wiss., 357[Fundamental Principles of Mathematical Sciences] Springer, Cham, 2020, xv+567 pp.
  * [Moret-Bailly.21] L. Moret-Bailly, A Henselian preparation theorem, arXiv:2112.13552.
  * [Pfister-Popescu.75] G. Pfister, D. Popescu, Die strenge Approximationseigenschaft lokaler Ringe. Invent. Math. 30 (1975), no. 2, 145–174.
  * [Popescu.85] D. Popescu, General Nééron desingularization. Nagoya Math. J. 100, 97–126 (1985).
  * [Popescu.86] D. Popescu, General Nééron desingularization and approximation. Nagoya Math. J. 104, 85–115 (1986).
  * [Popescu.00] D. Popescu, Commutative Rings and Algebras. Artin approximation, in Handbook of algebra, Vol 2, pages 321-356, 2000.
  * [Rond.18] G. Rond, Artin approximation, J. Singul. 17 (2018), 108–192.
  * [Shiota.98] M. Shiota, Relation between equivalence relations of maps and functions. Real analytic and algebraic singularities (Nagoya/Sapporo/Hachioji, 1996), 114–144, Pitman Res. Notes Math. Ser., 381, Longman, Harlow, 1998.
  * [Shiota.10] M. Shiota, Analytic and Nash equivalence relations of Nash maps. Bull. Lond. Math. Soc. 42 (2010), no. 6, 1055–1064.