# Disentangling mappings defined on ICIS

Alberto Fernández-Hernández, Juan J. Nuño-Ballesteros  Departament de
Matemàtiques, Universitat de València, Campus de Burjassot, 46100 Burjassot
SPAIN<EMAIL_ADDRESS>Departament de Matemàtiques, Universitat de
València, Campus de Burjassot, 46100 Burjassot SPAIN. Departamento de
Matemática, Universidade Federal da Paraíba CEP 58051-900, João Pessoa - PB,
BRAZIL<EMAIL_ADDRESS>
###### Abstract.

We study germs of hypersurfaces $(Y,0)\subset(\mathbb{C}^{n+1},0)$ that can be
described as the image of $\mathscr{A}$-finite mappings
$f:(X,S)\rightarrow(\mathbb{C}^{n+1},0)$ defined on an icis $(X,S)$ of
dimension $n$. We extend the definition of the Jacobian module given by
Fernández de Bobadilla, Nuño-Ballesteros and Peñafort-Sanchis when
$X=\mathbb{C}^{n}$, which controls the image Milnor number $\mu_{I}(X,f)$. We
apply these results to prove the case $n=2$ of the generalised Mond
conjecture, which states that
$\mu_{I}(X,f)\geq\textup{codim}_{\mathscr{A}_{e}}(X,f)$, with equality if
$(Y,0)$ is weighted homogeneous.

###### Key words and phrases:

Image Milnor number, the Mond conjecture, icis

###### 2000 Mathematics Subject Classification:

Primary 58K15; Secondary 32S30, 58K40

Work of J. J. Nuño-Ballesteros partially supported by Grant
PID2021-124577NB-I00 funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A
way of making Europe”

##  1\. Introduction

The main objects that are studied along this paper are hypersurface
singularities $(Y,0)\subset(\mathbb{C}^{n+1},0)$. It is well known that if a
hypersurface germ has isolated singularity, then one can define two different
invariants to encode information of the hypersurface, namely the Milnor and
Tjurina numbers, $\mu$ and $\tau$, respectively. The first of them, $\mu$, has
a more topological flavour, since it counts the number of spheres in the
homotopy type of a Milnor fibre of $Y$. On the other hand, the Tjurina number
has a much more rigid nature, and counts the number of parameters of a versal
deformation of the hypersurface. An inspection of the algebraic formulas that
define them makes clear that $\mu\geq\tau$, with equality in the case that
$(Y,0)$ is weighted homogeneous.

In [9], D. Mond extended the definition of $\mu$ to a particular kind of
hypersurfaces which could be described as the image of a mapping
$f:(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ that has isolated
instability, or, equivalently, that has finite codimension. In this case, $f$
is the normalisation mapping of the hypersurface $(Y,0)$. He then introduced
the concept of image Milnor number $\mu_{I}(f)$ of the mapping $f$, which is
well defined if the dimensions $(n,n+1)$ are Mather’s nice dimensions, or if
$f$ has corank one, as will be commented on in the following lines. This new
invariant played the role of the Milnor number for this kind of hypersurfaces
that do not have isolated singularity anymore. Moreover, the codimension of
$f$, denoted by $\textup{codim}_{\mathscr{A}_{e}}(f)$, is an analytical
invariant that measures the minimum number of parameters required to fully
deform $f$ through versal unfoldings, in the same way the Tjurina number does.
In terms of this invariant, one says that a mapping is $\mathscr{A}$-finite
provided it has finite codimension, or that is stable whenever it has
codimension 0. Although the invariants are written and described in terms of
the mapping $f$, they can be seen to depend only on the analytic class of the
image $(Y,0)$, and hence $\mu_{I}(f)$ and
$\textup{codim}_{\mathscr{A}_{e}}(f)$ are the natural invariants that
correspond to $\mu$ and $\tau$ in this different context. The question is
therefore natural:

######  Conjecture 1.1 (Mond conjecture).

Let $f:(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ be an
$\mathscr{A}$-finite mapping, where $(n,n+1)$ are Mather’s nice dimensions.
Then, $\mu_{I}(f)\geq\textup{codim}_{\mathscr{A}_{e}}(f)$, with equality if
$f$ is weighted homogeneous.

The only known cases are $n=1,2$, and it is still an open problem if $n\geq
3$. This conjecture is therefore the extension of the $\mu\geq\tau$ statement
for hypersurfaces that admit an $\mathscr{A}$-finite normalisation mapping
with normal space $(\mathbb{C}^{n},S)$. Nevertheless, the amount of
hypersurfaces satisfying this property is limited. In particular, one would
have interest in studying this statement if the normal space of $(Y,0)$ is
singular. More precisely, in this article we study the case of hypersurfaces
whose normalisation is $\mathscr{A}$-finite and whose normal space is an
isolated complete intersection singularity (icis).

In [10], D. Mond and J. Montaldi settled the deformation theory of mappings
$f:(X,S)\rightarrow(\mathbb{C}^{p},0)$ with $(X,S)$ being an icis of dimension
$n$, and they took the first steps to study an extension of the Mond
conjecture in this general setting.

It turns out that the notions of $\textup{codim}_{\mathscr{A}_{e}}(X,f)$ and
$\mu_{I}(X,f)$ can be defined in this wider context, as Mond and Montaldi
showed. The former invariant, namely the codimension of $(X,f)$, equals the
minimum number of parameters of a versal unfolding of $(X,f)$, where now
unfoldings are allowed to deform both the mapping $f$ and the space $X$. On
the other hand, the definition of $\mu_{I}(X,f)$ is given in the same way as
it is done for mappings with smooth source. In order to properly define it,
one requires the existence of stabilisations, which are deformations $f_{t}$
of $f$ with the property that $f_{t}$ is stable for every $t\neq 0$ small
enough. These particular deformations do only exist provided $(n,n+1)$ are
nice dimensions or provided $f$ has corank one. In these terms, it can be
checked that the images $Y_{t}=\textup{im}\,f_{t}$, which are called
disentanglements of $Y$, all have the homotopy type of a wedge of $n$-spheres.
The number of such spheres is therefore defined as the image Milnor number
$\mu_{I}(X,f)$.

This settles the framework to study in this general case whether
$\mu_{I}(X,f)\geq\textup{codim}_{\mathscr{A}_{e}}(X,f)$, with equality in the
weighted homogeneous case. This question is what we refer to as the
generalised Mond conjecture. The only case that was known before this article
is studied in [1] by D. Henrique and J.J. Nuño-Ballesteros, and provides a
postive answer for the generalised Mond conjecture if $n=1$ and
$(X,S)\subset(\mathbb{C}^{2},0)$ is a plane curve.

Our main contribution in this article is that we prove the generalised Mond
conjecture in the case that $n=2$ in its whole generality. Therefore, we show
that the $\mu\geq\tau$ statement can be extended to surfaces $(Y,0)$ in
$(\mathbb{C}^{3},0)$ with $\mathscr{A}$-finite normalisation mapping on an
icis. Formally, the main theorem of this paper is the following result:

######  Theorem 1.2.

Let $f:(X,S)\rightarrow(\mathbb{C}^{3},0)$ be an $\mathscr{A}$-finite mapping
where $(X,S)$ is an icis of dimension $2$ and with image $(Y,0)$. Then,

$\mu_{I}(X,f)\geq\textup{codim}_{\mathscr{A}_{e}}(X,f),$

with equality if $(Y,0)$ is weighted homogeneous.

In order to prove it, we adapt the construction of the module $M(g)$ and its
relative version for unfoldings $M_{\textup{rel}}(G)$ that were defined in [4]
by J. Fernández de Bobadilla, J. J. Nuño-Ballesteros and G. Peñafort-Sanchis.
We extend to this general framework their main result, which states that the
length of $M(g)$ equals $\mu_{I}(X,f)$ if and only if $M_{\textup{rel}}(G)$ is
a Cohen-Macaulay module, and that, if each of these cases hold, the
generalised Mond conjecture holds for the mapping $(X,f)$.

##  2\. Mappings on icis

Singularities of smooth mappings between manifolds is a classical subject in
Singularity Theory. The infinitesimal methods were developed by Thom and
Mather in the late sixties and by this reason it is known as Thom-Mather
theory. We refer to the book [11] for a modern presentation of the theory,
which also includes the extension to holomorphic germs between complex
manifolds.

In [10] Mond and Montaldi extended the Thom-Mather theory of singularities of
mappings $f\colon(X,S)\to(\mathbb{C}^{p},0)$ defined on an icis $(X,S)$. The
crucial point here is that they consider deformations not only of the mapping
$f$, but also of the icis $(X,S)$. Here we summarise some of the basic
definitions and properties, in order to make this paper more self-contained.
For a more detailed account we refer to the original paper by Mond and
Montaldi [10].

Along this section we work with holomorphic map germs
$f\colon(X,S)\to(\mathbb{C}^{p},0)$, where $(X,S)$ is an icis of dimension
$n$. It is usual to denote such a map germs by a pair $(X,f)$, although
sometimes we may omit the base set of the germ if it does not provide relevant
information or it is clear from the context. The two first definitions declare
the type of equivalences and deformations we are dealing with in this theory.

######  Definition 2.1.

Two holomorphic map germs $f,g:(X,S)\rightarrow(\mathbb{C}^{p},0)$ are called
$\mathscr{A}$-equivalent if we have a commutative diagram

${(X,S)}$${(\mathbb{C}^{p},0)}$${(X,S)}$${(\mathbb{C}^{p},0)}$$\scriptstyle{f}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$$\scriptstyle{g}$

where the columns are biholomorphisms.

######  Definition 2.2.

An unfolding of the pair $\left(X,f\right)$ is a map germ
$F\colon(\mathcal{X},S^{\prime})\rightarrow\left(\mathbb{C}^{p}\times\mathbb{C}^{r},0\right)$
together with a flat projection
$\pi\colon(\mathcal{X},S^{\prime})\rightarrow(\mathbb{C}^{r},0)$ and an
isomorphism $j\colon(X,S)\to\big{(}\pi^{-1}(0),S^{\prime}\big{)}$ such that
the following diagram commutes

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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.83388pt}{-48.02083pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\hbox{$\scriptstyle{\pi_{2}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where $\pi_{2}:\mathbb{C}^{p}\times\mathbb{C}^{r}\rightarrow\mathbb{C}^{r}$ is
the Cartesian projection.

In Definition 2.2, $\mathbb{C}^{r}$ is called the parameter space of the
unfolding. It is common to denote the unfolding by $(\mathcal{X},\pi,F,j)$.
For each parameter $u\in\mathbb{C}^{r}$ in a neighbourhood of the origin, we
have a mapping $f_{u}\colon X_{u}\rightarrow\mathbb{C}^{p}$, where
$X_{u}:=\pi^{-1}(u)$, denoted also by $(X_{u},f_{u})$.

######  Definition 2.3.

Two unfoldings $(\mathcal{X},\pi,F,j)$ and
$(\mathcal{X}^{\prime},\pi^{\prime},F^{\prime},j^{\prime})$ over
$\mathbb{C}^{r}$ are isomorphic if the following diagram commutes:

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where $\Phi$ and $\Psi$ are biholomorphisms and also $\Psi$ is an unfolding of
the identity over $\mathbb{C}^{d}$.

If $(\mathcal{X},\pi,F,j)$ is an unfolding of $(X,f)$ over
$(\mathbb{C}^{r},0)$, a germ
$\rho:(\mathbb{C}^{s},0)\rightarrow(\mathbb{C}^{r},0)$ induces and unfolding
$(\mathcal{X}_{\rho},\pi_{\rho},F_{\rho},j_{\rho})$ of $(X,f)$ by a base
change or, in other words, by the fibre product of $F$ and
$\text{id}_{\mathbb{C}^{p}}\times\rho$:

$\leavevmode\hbox to161.9pt{\vbox
to52.37pt{\pgfpicture\makeatletter\hbox{\hskip
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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{59.91838pt}{-14.03864pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{$\scriptstyle{\text{id}_{\mathbb{C}^{p}}\times\rho}$}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{F}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where we omit the points of the germs for simplicity.

The unfolding $(\mathcal{X},\pi,F,j)$ is versal if every other unfolding, for
example $(\mathcal{X}^{\prime},\pi^{\prime},F^{\prime},j^{\prime})$, is
isomorphic to an unfolding induced from the former by a base change,
$(\mathcal{X}_{\rho},\pi_{\rho},F_{\rho},j_{\rho})$. A versal unfolding is
called miniversal if it has a parameter space with minimal dimension.

######  Definition 2.4.

A germ $(X,f)$ is _stable_ if any unfolding is _trivial_ , that is, isomorphic
to the constant unfolding
$(X\times\mathbb{C}^{r},\pi_{2},f\times\textnormal{id}_{\mathbb{C}^{r}},i)$.

We say that $(X,f)$ has _isolated instability_ if there exists a
representative $f\colon X\to\mathbb{C}^{p}$ such that the restriction $f\colon
X\setminus f^{-1}(0)\to\mathbb{C}^{p}\setminus\\{0\\}$ has only stable
singularities.

A _stabilisation_ of $(X,f)$ is a 1-parameter unfolding
$(\mathcal{X},\pi,F,j)$ with the property that for any small enough
$s\in\mathbb{C}\setminus\\{0\\}$, $(X_{s},f_{s})$ has only stable
singularities. Such a mapping $(X_{s},f_{s})$, with $s\neq 0$, is called a
_stable perturbation_ of $(X,f)$.

A crucial fact is that any germ $(X,f)$ with isolated instability admits a
stabilisation, provided that $(n,p)$ are nice dimensions in the sense of
Mather or $f$ has only kernel rank one singularities (that is, $f$ admits an
extension whose differential has kernel rank $\leq 1$ everywhere). A proof in
the case $X=\mathbb{C}^{n}$ can be found in [11] and the extension to the case
of mappings on icis appears in [6].

Next, we recall the notion of $\mathscr{A}_{e}$-codimension of a germ $(X,f)$.
In order to do this, we introduce the following notation:

  * •

$\mathscr{O}_{p}$ is the local ring of holomorphic functions
$(\mathbb{C}^{p},0)\to\mathbb{C}$,

  * •

$\mathscr{O}_{X,S}$ is the (semi-)local ring of holomorphic functions
$(X,S)\to\mathbb{C}$,

  * •

$f^{*}:\mathscr{O}_{p}\to\mathscr{O}_{X,S}$ is the induced ring morphism
$f^{*}(h)=h\circ f$,

  * •

$\theta_{\mathbb{C}^{p},0}$ is the $\mathscr{O}_{p}$-module of germs of vector
fields on $(\mathbb{C}^{p},0)$,

  * •

$\theta_{X,S}$ is the $\mathscr{O}_{X,S}$-module of germs of vector fields on
$(X,S)$,

  * •

$\theta(f)$ is the module of vector fields along $f$,

  * •

$\omega f:\theta_{\mathbb{C}^{p},0}\rightarrow\theta(f)$ is the mapping
$\omega(\eta)=\eta\circ f$,

  * •

$tf:\theta_{X,S}\rightarrow\theta(f)$ is mapping $tf(\xi)=d\tilde{f}\circ\xi$,
for some analytic extension $\tilde{f}$ of $f$.

######  Definition 2.5.

The _$\mathscr{A}_{e}$ -codimension_ of $(X,f)$ is defined as

$\textup{codim}_{\mathscr{A}_{e}}(X,f)=\dim_{\mathbb{C}}\frac{\theta(f)}{{tf(\theta_{X,S})+\omega
f(\theta_{p})}}+\sum_{x\in S}\tau(X,x),$

where $\tau(X,x)$ is the Tjurina number of $(X,x)$. When
$\textup{codim}_{\mathscr{A}_{e}}(X,f)<\infty$, the germ $(X,f)$ is called
$\mathscr{A}$-finite.

The versality theorem holds also for mappings on icis, as the reader can find
in [10]: $(X,f)$ is $\mathscr{A}$-finite if and only if it admits a versal
unfolding and in case it is $\mathscr{A}$-finite, then
$\textup{codim}_{\mathscr{A}_{e}}(X,f)$ is equal to the number of parameters
in a miniversal unfolding. As a consequence, $(X,f)$ is stable if and only if
$(X,S)$ is smooth and $f$ is stable in the usual sense (see [11]).

Another important issue with $\mathscr{A}$-finiteness is the extension of the
Mather-Gaffney geometric criterion for mappings on icis: $(X,f)$ is
$\mathscr{A}$-finite if and only if it has isolated instability (see [6]).

Finally, we will recall the definition of image Milnor number in the case
$p=n+1$. The original definition when $X=\mathbb{C}^{n}$ is due to Mond (see
[9, 11]), but it can be adapted quite easily to mappings on icis (see [6]). In
the case $p\leq n$, the analogous invariant is called the discriminant Milnor
number, considered for the first time by Damon and Mond in [2] in the case
$X=\mathbb{C}^{n}$ and extended to mappings on icis by Mond and Montaldi in
[10]. The definition is clearly inspired in the classical Milnor number and is
motivated by the following theorem.

We denote by $B_{\epsilon}$ the closed ball in $\mathbb{C}^{n+1}$ of radius
$\epsilon>0$ centered at the origin. We assume
$f\colon(X,S)\to(\mathbb{C}^{n+1},0)$ is $\mathscr{A}$-finite and that either
$(n,n+1)$ are nice dimensions of Mather or $f$ has only corank one
singularities. We take a stabilisation of $(X,f)$ with stable perturbation
$(X_{s},f_{s})$.

######  Theorem 2.6.

[9, 6] For all $\epsilon,\eta$, with $0<\eta\ll\epsilon\ll 1$ and for all
$s\in\mathbb{C}$, with $0<|s|<\eta$, $f_{s}(X_{s})\cap B_{\epsilon}$ has the
homotopy type of a bouquet of spheres of dimension $n$. Moreover, the number
of such spheres, denoted by $\mu_{I}(X,f)$, is independent of the choice of
the stabilisation, the parameter $s$ and the numbers $\epsilon,\eta$.

######  Definition 2.7.

With the notation of Theorem 2.6, $f_{s}(X_{s})\cap B_{\epsilon}$ is called
the _disentanglement_ and $\mu_{I}(X,f)$ is called the _image Milnor number_
of $(X,f)$.

The proof of Theorem 2.6 is based on arguments by Lê [8] and by Siersma [15].
In fact, the original formulation in [15] gives a recipe of how to compute
$\mu_{I}(X,f)$ in a more algebraic way:

######  Theorem 2.8.

[15] With the notation of Theorem 2.6, let
$G\colon(\mathbb{C}^{n+1}\times\mathbb{C},0)\to(\mathbb{C},0)$ be a reduced
equation of the image of the stabilistation. Then,

$\mu_{I}(X,f)=\sum_{y\in B_{\epsilon}\setminus f_{s}(X_{s})}\mu(g_{s};y),$

where $g_{s}(y)=G(y,s)$ and $\mu(g_{s};y)$ is the Milnor number of $g_{s}$ at
$y$.

######  Remark 2.9.

In order to compute the image Milnor number $\mu_{I}(X,f)$, sometimes is more
convenient to consider a stable unfolding (i.e., an unfolding which is stable
as a germ) instead of a stabilisation. In such a case, the bifurcation set
$\mathcal{B}$ is the set of parameters $u\in\mathbb{C}^{r}$ in a neighbourhood
of the origin, such that $f_{u}\colon X_{u}\to\mathbb{C}^{n+1}$ has only
stable singularities. When $(n,n+1)$ are nice dimensions or when $f$ has only
corank one singularities, $\mathcal{B}$ is a proper closed analytic subset
germ in a neighbourhood of $0$ in $\mathbb{C}^{r}$ (see [11] or [6]). A
stabilisation can be constructed easily by taking a line
$L\subset\mathbb{C}^{r}$ with the property that $L\cap\mathcal{B}=\\{0\\}$. If
$u\notin\mathcal{B}$, then $f_{u}(X_{u})\cap B_{\epsilon}$ has the homotopy
type of a bouquet of $n$-spheres and the number of such spheres is
$\mu_{I}(X,f)$. Analogously, if
$G\colon(\mathbb{C}^{n+1}\times\mathbb{C}^{r},0)\to(\mathbb{C},0)$ is a
reduced equation of the image of the unfolding, then

$\mu_{I}(X,f)=\sum_{y\in B_{\epsilon}\setminus f_{u}(X_{u})}\mu(g_{u};y),$

where $g_{u}(y)=G(y,u)$ and $\mu(g_{u};y)$ is the Milnor number of $g_{u}$ at
$y$.

##  3\. The generalised version of the Jacobian module $M(g)$

Let $f:(X,S)\rightarrow(\mathbb{C}^{n+1},0)$ be an $\mathscr{A}$-finite
mapping defined on an icis $(X,S)\subset(\mathbb{C}^{n+k},S)$ of dimension
$n$. As $f$ is finite, its image $(Y,0)$ is a hypersurface of
$(\mathbb{C}^{n+1},0)$, and hence it can be described by a reduced equation
$g\in\mathscr{O}_{n+1}$. Let
$h:(\mathbb{C}^{n+k},S)\rightarrow(\mathbb{C}^{k},0)$ be a mapping so that
$(X,S)=h^{-1}(0)$. Consider an analytic extension
$\tilde{f}:(\mathbb{C}^{n+k},S)\rightarrow(\mathbb{C}^{n+1},0)$ of $f$, and
write
$\hat{f}=(\tilde{f},h):(\mathbb{C}^{n+k},S)\rightarrow(\mathbb{C}^{n+1+k},0)$.
It is therefore clear that the restriction of $\hat{f}$ to $(X,S)$ is
precisely the mapping $(f,0)$. Hence, $\hat{f}$ is a finite mapping, since
$\hat{f}^{-1}(0)=f^{-1}(0)=S$. Moreover, the diagram

${{(\mathbb{C}^{k},0)}}$${{(\mathbb{C}^{n+k},S)}}$${{(\mathbb{C}^{n+1+k},0)}}$${{(X,S)}}$${{(\mathbb{C}^{n+1},0)}}$$\scriptstyle{f}$$\scriptstyle{i}$$\scriptstyle{\hat{f}}$$\scriptstyle{j}$$\scriptstyle{h}$

commutes, where $i$ is the inclusion and $j$ is the natural immersion. In
particular, $\hat{f}$ is an unfolding of $f$ deforming both the mapping and
the domain. After taking representatives, the induced deformations of $f$ are
the mappings
$\hat{f}_{t}:X_{t}\subset\mathbb{C}^{n+k}\rightarrow\mathbb{C}^{n+1}$ defined
as $\hat{f}_{t}(x)=\tilde{f}(x,t)$, where $X_{t}=h^{-1}(t)$ for
$t\in\mathbb{C}^{k}$ small enough.

The key idea is that $\hat{f}$ is the simplest unfolding of $f$ with smooth
source. Hence, the definition of the module for the mapping $f$ will be
performed through a specialisation of the one from $\hat{f}$.

Since $\hat{f}$ is a finite mapping, its image $(\hat{Y},0)$ is a hypersurface
of $(\mathbb{C}^{n+1+k},0)$. Furthermore, the restriction of $\hat{f}$ to its
image $(\mathbb{C}^{n+k},S)\rightarrow(\hat{Y},0)$ is the normalisation
mapping of $(\hat{Y},0)$, and hence it induces a monomorphism of rings
$\mathscr{O}_{\hat{Y},0}\hookrightarrow\mathscr{O}_{n+k}$ that lets us
consider $\mathscr{O}_{\hat{Y},0}$ to be a subring of $\mathscr{O}_{n+k}$. In
this case, the diagram

${\mathscr{O}_{n+1+k}}$${\mathscr{O}_{n+k}}$${\mathscr{O}_{\hat{Y},0}}$$\scriptstyle{\hat{f}^{*}}$$\scriptstyle{\pi}$

commutes, where $\pi$ is the epimorphism associated to the natural inclusion
of $(\hat{Y},0)$ in $(\mathbb{C}^{n+1+k},0)$. We consider both
$\mathscr{O}_{\hat{Y},0}$ and $\mathscr{O}_{n+k}$ to be
$\mathscr{O}_{n+1+k}$-modules via the corresponding morphisms $\pi$ and
$\hat{f}^{*}$, respectively.

Let us consider $\hat{g}\in\mathscr{O}_{n+1+k}$ to be a reduced equation of
$(\hat{Y},0)$ in such a way that $\hat{g}\circ j=g$. The following result from
R. Piene [14] relates the conductor ideal of $\hat{f}$, given by

$C(\hat{f})=\\{h\in\mathscr{O}_{\hat{Y},0}:h\cdot\mathscr{O}_{n+k}\subset\mathscr{O}_{\hat{Y},0}\\},$

with the partial derivatives of $\hat{g}$ and with the minors of the Jacobian
matrix $d\hat{f}$.

######  Theorem 3.1 ([14]).

There exists a unique $\lambda\in\mathscr{O}_{n+k}$ such that, for every
$l\in\\{1,\ldots,n+1+k\\}$,

$\partial_{l}\hat{g}\circ\hat{f}=(-1)^{l}\cdot\lambda\cdot\det(d\hat{f}_{1},\ldots,d\hat{f}_{l-1},d\hat{f}_{l+1},\ldots,d\hat{f}_{n+1+k}),$

where $\partial_{l}\hat{g}$ denotes the partial derivative of $\hat{g}$ with
respect to the $l$-th variable. Furthermore, the ideal $C(\hat{f})$ is
principal, and generated by the element $\lambda$.

Let us denote by $J(\hat{g})$ to the Jacobian ideal of $\hat{g}$, which is
generated by the partial derivatives $\partial_{l}\hat{g}$. This result
therefore shows that $J(\hat{g})\cdot\mathscr{O}_{n+k}\subset C(\hat{f})$,
where $J(\hat{g})\cdot\mathscr{O}_{n+k}=\hat{f}^{*}(J(\hat{g}))$.

On the other hand, since $\hat{f}$ is a finite mapping with degree 1 onto its
image, an application of Theorem 3.4 of [12] of D. Mond and R. Pellikaan shows
that $\mathscr{F}_{1}(\hat{f})\cdot\mathscr{O}_{n+k}=C(\hat{f})$, where
$\mathscr{F}_{1}(\hat{f})$ denotes the first Fitting ideal of
$\mathscr{O}_{n+k}$ as an $\mathscr{O}_{n+1+k}$-module via $\hat{f}^{*}$.

Let us denote by $(y,z)$ to the coordinates in $(\mathbb{C}^{n+1+k},0)$, with
$y\in\mathbb{C}^{n+1}$ and $z\in\mathbb{C}^{k}$, and consider the Jacobian
ideal $J_{y}(\hat{g})=\langle\partial\hat{g}/\partial
y_{1},\ldots,\partial\hat{g}/\partial y_{n+1}\rangle$ generated by the
derivatives with respect to the variables $y=(y_{1},\ldots,y_{n+1})$.

######  Definition 3.2.

The restriction of $\hat{f}^{*}$ to $\mathscr{F}_{1}(\hat{f})$ induces an
epimorphism of $\mathscr{O}_{n+1+k}$-modules

$\dfrac{\mathscr{F}_{1}(\hat{f})}{J_{y}(\hat{g})}\rightarrow\dfrac{C(\hat{f})}{J(\hat{g})\cdot\mathscr{O}_{n+k}}.$

We define $N(\hat{g})$ to be the $\mathscr{O}_{n+1+k}$-module given by the
kernel of this morphism, and define

$M(g)=N(\hat{g})\otimes\dfrac{\mathscr{O}_{n+1+k}}{\mathfrak{m}_{k}\cdot\mathscr{O}_{n+1+k}},$

where $\mathfrak{m}_{k}=(z_{1},\ldots,z_{k})$ is generated by the parameters
of the unfolding $\hat{f}$ of $f$. Note that $M(g)$ has a natural
$\mathscr{O}_{n+1}$-module structure inherited from the tensor product.

Hence, $M(g)$ is defined by taking into account that $\hat{f}$ is an unfolding
of $f$, and following the spirit of [4] in the case of mappings with smooth
source. Furthermore, it is relevant to notice that this module $M(g)$
coincides with the given in the smooth case just by taking $\hat{f}=f$ and
$k=0$.

######  Remark 3.3.

Since $\hat{f}$ is defined in a smooth source, it lies naturally in the
context of [4]. However, the module $N(\hat{g})$ that we have defined is not
exactly the same as the module that is defined in [4]. Indeed, the source of
the morphism that defines $N(\hat{g})$ has the Jacobian ideal $J_{y}(\hat{g})$
where only partial derivatives with respect to the parameters
$y_{1},\ldots,y_{n+1}$ are taken into consideration, while the module in the
target does have all the partial derivatives. Although this may seem to be
somewhat whimsical, this will be shown to be exactly what is needed for the
module to specialise properly. This is due to the fact that, in general,
$J(\hat{g})\cdot\mathscr{O}_{n+k}\neq J_{y}(\hat{g})\cdot\mathscr{O}_{n+k}$,
in contrast with what happens in the smooth case. This important fact is what
infuences this definition for $M(g)$, and what makes that another possible
definition may not specialise properly.

######  Remark 3.4.

The module $N(\hat{g})$ is determined by the short exact sequence

$0\rightarrow
N(\hat{g})\rightarrow\dfrac{\mathscr{F}_{1}(\hat{f})}{J_{y}(\hat{g})}\rightarrow\dfrac{C(\hat{f})}{J(\hat{g})\cdot\mathscr{O}_{n+k}}\rightarrow
0.$

######  Proposition 3.5.

The following formula holds:

$N(\hat{g})=\dfrac{(\hat{f}^{*})^{-1}(J(\hat{g})\cdot\mathscr{O}_{n+k})}{J_{y}(\hat{g})}.$

###### Proof.

By definition, it is clear that

$N(\hat{g})=\dfrac{(\hat{f}^{*})^{-1}(J(\hat{g})\cdot\mathscr{O}_{n+k})\cap\mathscr{F}_{1}(\hat{f})}{J_{y}(\hat{g})}.$

Since $J(\hat{g})\cdot\mathscr{O}_{n+k}\subset C(\hat{f})$, then
$(\hat{f}^{*})^{-1}(J(\hat{g})\cdot\mathscr{O}_{n+k})\subset\mathscr{F}_{1}(\hat{f})$,
and the formula follows. ∎

##  4\. A relative version for the generalised Jacobian module

In this section, a relative version of the module for unfoldings of $\hat{f}$
is defined. For the sake of simplicity, we consider unfoldings
$F:(\mathbb{C}^{n+k+r},S\times 0)\rightarrow(\mathbb{C}^{n+1+k+r},0)$ of the
mapping $\hat{f}$ instead of general unfoldings of $f$ with possibly non-
smooth source. Let $G\in\mathscr{O}_{n+1+k+r}$ be an equation for the image of
$F$ such that $G(y,z,0)=\hat{g}(y,z)$, where $(y,z,u)$ are the coordinates of
$(\mathbb{C}^{n+1+k+r},0)$, with $y\in\mathbb{C}^{n+1},z\in\mathbb{C}^{k}$ and
$u\in\mathbb{C}^{r}$. We then have that the diagram

${{(\mathbb{C}^{n+k+r},S\times
0)}}$${{(\mathbb{C}^{n+1+k+r},0)}}$${{(\mathbb{C}^{n+k},S)}}$${{(\mathbb{C}^{n+1+k},0)}}$${{(\mathbb{C},0)}}$${{(X,S)}}$${{(\mathbb{C}^{n+1},0)}}$$\scriptstyle{\hat{f}}$$\scriptstyle{f}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{\hat{i}}$$\scriptstyle{\hat{j}}$$\scriptstyle{F}$$\scriptstyle{g}$$\scriptstyle{\hat{g}}$$\scriptstyle{G}$

commutes, where $\hat{i},\hat{j}$ are the natural immersions. Let us denote by

$J_{y}(G)=\Big{\langle}\dfrac{\partial G}{\partial
y_{1}},\ldots,\dfrac{\partial G}{\partial
y_{n+1}}\Big{\rangle},\,\,J_{z}(G)=\Big{\langle}\dfrac{\partial G}{\partial
z_{1}},\ldots,\dfrac{\partial G}{\partial z_{k}}\Big{\rangle},$

and $J_{y,z}(G)=J_{y}(G)+J_{z}(G)$. We therefore define $M_{\textup{rel}}(G)$
as the kernel of the module epimorphism

$\dfrac{\mathscr{F}_{1}(F)}{J_{y}(G)}\rightarrow\dfrac{C(F)}{J_{y,z}(G)\cdot\mathscr{O}_{n+k+r}}.$

Hence, $M_{\textup{rel}}(G)$ fits into the short exact sequence

$0\rightarrow
M_{\textup{rel}}(G)\rightarrow\dfrac{\mathscr{F}_{1}(F)}{J_{y}(G)}\rightarrow\dfrac{C(F)}{J_{y,z}(G)\cdot\mathscr{O}_{n+k+r}}\rightarrow
0.$

Furthermore, it is straightforward to verify that

######  Proposition 4.1.

The following formula holds:

$M_{\textup{rel}}(G)=\dfrac{(F^{*})^{-1}(J_{y,z}(G)\cdot\mathscr{O}_{n+k+r})}{J_{y}(G)}.$

In contrast with the case of $M(g)$, we have that
$J_{y,z}(G)\cdot\mathscr{O}_{n+k+r}=J(G)\cdot\mathscr{O}_{n+k+r}$ in virtue of
lemma 4.5 of [4]. Hence, they can be interchanged indistinctively in this
formula.

Notice, in addition, that the relative module in [4] is denoted by $M_{y}(G)$
with the intention to highlight that the module is relative with respect to
the parameters $y=(y_{1},\ldots,y_{n+1})$. Since this interpretation in the
case of mappings with non-smooth source is less clear, due to the fact that
one has both parameters $y,z$, the authors have decided to consider the
notation $M_{\textup{rel}}(G)$.

In what follows, we show that the given definition of the
$\mathscr{O}_{n+1+k+r}$-module $M_{\textup{rel}}(G)$ properly specialises to
$M(g)$ seen as an $\mathscr{O}_{n+1}$-module. Before giving a proof of this
result, let us comment on the specialisation method that should be considered.

######  Remark 4.2.

It is relevant to notice that the specialisation process that is performed in
[4] for mappings with smooth source restricts the relative module
$M_{y}(G)=M_{\textup{rel}}(G)$ to be an $\mathscr{O}_{r}$-module, and then
$M_{\textup{rel}}(G)$ is tensored with $\mathscr{O}_{r}/\mathfrak{m}_{r}$.
Hence, one naturally obtains a $\mathbb{C}$-vector space that is isomorphic to
$M(g)$ (see Theorem 4.6 of [4]). However, this specialisation process ignores
the fact that $M(g)$ is an $\mathscr{O}_{n+1}$-module. In order to take this
into account, one should perform the specialisation process in a slightly
different way. Since $M_{\textup{rel}}(G)$ is an $\mathscr{O}_{n+1+r}$-module,
the tensor product

$M_{\textup{rel}}(G)\otimes\dfrac{\mathscr{O}_{n+1+r}}{\mathfrak{m}_{r}\cdot\mathscr{O}_{n+1+r}}$

provides naturally an $\mathscr{O}_{n+1}$-module, due to the fact that
$\mathscr{O}_{n+1+r}/(\mathfrak{m}_{r}\cdot\mathscr{O}_{n+1+r})\cong\mathscr{O}_{n+1}$,
where $\mathfrak{m}_{r}=(u_{1},\ldots,u_{r})$ is the maximal ideal of
$\mathscr{O}_{r}$ generated by the parameters of the unfolding $F$. In fact,
this specialisation process both captures the spirit of forcing the parameters
of the unfolding to be equal to 0 and keeps the natural
$\mathscr{O}_{n+1}$-module structure of $M(g)$, as it can be easily verified
with minor modifications in the proofs of section 4 of [4]. Hence, it can be
easily checked that the module $M_{\textup{rel}}(G)$ of [4] in the smooth case
satisfies that

$M_{\textup{rel}}(G)\otimes\dfrac{\mathscr{O}_{n+1+r}}{\mathfrak{m}_{r}\cdot\mathscr{O}_{n+1+r}}\cong
M(g),$

where $\cong$ denotes isomorphism of $\mathscr{O}_{n+1}$-modules. Although
this is not relevant in the smooth case, we have already seen in the
definition 3.2 that specialisations should be taken through this method in
order to preserve the module structure.

Taking this into account, we are now able to check that the module
$M_{\textup{rel}}(G)$ in this setting properly specialises to $M(g)$:

######  Theorem 4.3.

If $n\geq 2$, then

$M_{\textup{rel}}(G)\otimes\dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_{k+r}\cdot\mathscr{O}_{n+1+k+r}}\cong
M(g)$

as $\mathscr{O}_{n+1}$-modules.

###### Proof.

The same proof of the smooth case (Theorem 4.6 of [4] and the previous remark)
shows that

$M_{\textup{rel}}(G)\otimes\dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_{r}\cdot\mathscr{O}_{n+1+k+r}}\cong
N(\hat{g}),$

since this first specialisation restricts the parameters that were added in
the unfolding $F$ of $\hat{f}$, both with smooth domain. Indeed, the only
difference is that the definition of $N(\hat{g})$ has a term $J_{y}(\hat{g})$,
but it holds that $\hat{j}(J_{y}(G))=J_{y}(\hat{g})$ and, as in the smooth
case, $\hat{j}(J_{y,z}(G))=J(\hat{g})$. Hence,

$\displaystyle
M_{\textup{rel}}(G)\otimes\dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_{k+r}\cdot\mathscr{O}_{n+1+k+r}}$
$\displaystyle=\left(M_{\textup{rel}}(G)\otimes\dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_{r}\cdot\mathscr{O}_{n+1+k+r}}\right)\otimes\dfrac{\mathscr{O}_{n+1+k}}{\mathfrak{m}_{k}\cdot\mathscr{O}_{n+1+k}}$
$\displaystyle=N(\hat{g})\otimes\dfrac{\mathscr{O}_{n+1+k}}{\mathfrak{m}_{k}\cdot\mathscr{O}_{n+1+k}}=M(g),$

where the last equality holds by definition of $M(g)$. ∎

######  Remark 4.4.

As we have commented on before in Remark 4.2, the specialisation performed is
different than the one appearing in [4] for mappings with smooth source.
Although this different approach lets us transfer the module structure, it is
important for applications to analyse the analogue process. It turns out that,
when we restrict $M_{\textup{rel}}(G)$ to be an $\mathscr{O}_{k+r}$-module via
the natural inclusion $\mathscr{O}_{k+r}\rightarrow\mathscr{O}_{n+1+k+r}$
induced by the projection
$\mathbb{C}^{n+1}\times\mathbb{C}^{k+r}\rightarrow\mathbb{C}^{k+r}$, then it
is straightforward to verify that

$M_{\textup{rel}}(G)\otimes\dfrac{\mathscr{O}_{k+r}}{\mathfrak{m}_{k+r}}\cong
M(g)$

as $\mathbb{C}$-vector spaces, just by noticing that

$\displaystyle
M_{\textup{rel}}(G)\otimes\dfrac{\mathscr{O}_{k+r}}{\mathfrak{m}_{k+r}}$
$\displaystyle\cong\dfrac{M_{\textup{rel}}(G)}{\mathfrak{m}_{k+r}\cdot
M_{\textup{rel}}(G)}=\dfrac{M_{\textup{rel}}(G)}{(\mathfrak{m}_{k+r}\cdot\mathscr{O}_{n+1+k+r})\cdot
M_{\textup{rel}}(G)}\cong$ $\displaystyle\cong
M_{\textup{rel}}(G)\otimes\dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_{k+r}\cdot\mathscr{O}_{n+1+k+r}}\cong
M(g).$

In a nutshell, the specialisation process can be performed either restricting
first $M_{\textup{rel}}(G)$ to be an $\mathscr{O}_{k+r}$-module, or rather
keeping its whole $\mathscr{O}_{n+1+k+r}$-module structure. The obtained
modules are, respectively,

$M_{\textup{rel}}(G)\otimes(\mathscr{O}_{n+1+k+r}/\mathfrak{m}_{k+r}\cdot\mathscr{O}_{n+1+k+r})\text{
and }M_{\textup{rel}}(G)\otimes(\mathscr{O}_{k+r}/\mathfrak{m}_{k+r}).$

In both scenarios one recovers $M(g)$ with some structure: in the former case,
the result is an $\mathscr{O}_{n+1}$-module isomorphic to $M(g)$, and, in the
latter, a $\mathbb{C}$-module isomorphic to $M(g)$. Hence, the specialisation
procces that one should perform depends on whether one needs to keep the
module structure or not. For most of the cases, the only information that
needs to be keeped is the complex dimension as a vector space, and hence both
methods are valid.

Lastly, an important result regarding the form of the module
$M_{\textup{rel}}(G)$ when $F$ is stable is the following:

######  Proposition 4.5.

Let $F$ be a stable unfolding of $\hat{f}$ and $G$ an equation such that $G\in
J(G)$. Then,

$M_{\textup{rel}}(G)=\dfrac{J(G)}{J_{y}(G)}.$

###### Proof.

Since $F$ is stable and $G\in J(G)$, the smooth version of the module $M(G)$
from [4] satisfies that $M(G)=0$. In this case, the formula of proposition 5.1
of this article yields that

$M(G)=\dfrac{(F^{*})^{-1}(J(G)\cdot\mathscr{O}_{n+k+r})}{J(G)},$

and hence $(F^{*})^{-1}(J(G)\cdot\mathscr{O}_{n+k+r})=J(G)$. Furthermore, in
the smooth case, it follows that
$J(G)\cdot\mathscr{O}_{n+k+r}=J_{y,z}(G)\cdot\mathscr{O}_{n+k+r}$. Hence,
proposition 4.1 yields

$M_{\textup{rel}}(G)=\dfrac{(F^{*})^{-1}(J_{y,z}(G)\cdot\mathscr{O}_{n+k+r})}{J_{y}(G)}=\dfrac{J(G)}{J_{y}(G)},$

and the claim follows. ∎

Note that, since $\hat{f}$ is a finite mapping, the existence of a stable
unfolding $F$ of $\hat{f}$ is granted. Furthermore, as it is commented in [4],
there is always a stable unfolding $F$ which admits an equation $G$ with the
property that $G\in J(G)$. Such an equation is referred to as a good defining
equation. Indeed, if $F(u,x)=(u,f_{u}(x))$ is a stable unfolding of $\hat{f}$,
then let $F^{\prime}$ be the $1$-parameter stable unfolding of $F$ given by
$F^{\prime}(t,u,x)=(t,u,f_{u}(x))$. Let $G=0$ be a reduced equation of the
image of $F$ and consider $G^{\prime}(t,u,y)=e^{t}G(u,y)$. It therefore
follows that $G^{\prime}=0$ is a reduced equation defining the image of
$F^{\prime}$ with the property that $G^{\prime}=\partial_{t}G^{\prime}\in
J(G^{\prime})$.

##  5\. Relation between $\dim_{\mathbb{C}}M(g)$ and
$\textup{codim}_{\mathscr{A}_{e}}(X,f)$

Let $F$ be a stable unfolding of $\hat{f}$, and consider an equation $G$ of
the image of $F$, namely, $(Z,0)$, that satisfies $G\in J(G)$. Then, the last
result of the previous section showed that
$M_{\textup{rel}}(G)=J(G)/J_{y}(G)$. Let us relate this with the codimension
of $(X,f)$, which can be determined through the formula

$\textup{codim}_{\mathscr{A}_{e}}(X,f)=\dim_{\mathbb{C}}\dfrac{\theta(i)}{ti(\theta_{n+1})+i^{*}(\textup{Derlog}\,Z)},$

where $i:(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{C}^{n+1+k+r},0)$ denotes the
natural immersion (see [10] for more details).

Recall that
$\textup{Derlog}\,Z=\\{\xi\in\theta_{n+1+k+r}:\xi(G)=\lambda(G)\\}$, and
$\textup{Derlog}\,G=\\{\xi\in\theta_{n+1+k+r}:\xi(G)=0\\}$. Notice that $G\in
J(G)$, so that $G=\sum_{s}a_{s}\partial_{s}G$, where $\partial_{s}G$ denotes
the partial derivatives with respect to all the variables in
$(y,z,u)\in\mathbb{C}^{n+1+k+r}$. Hence, the vector field
$\epsilon=\sum_{s}a_{s}\partial_{s}$ satisfies that $\epsilon(G)=G$, where
$\partial_{s}$ denotes the coordinate vector field associated with the $s$-th
coordinate, where $s\in\\{1,\ldots,n+1+k+r\\}$. Furthermore,
$\textup{Derlog}\,Z=\textup{Derlog}\,G\oplus\langle\epsilon\rangle$. We
therefore have that

$\textup{codim}_{\mathscr{A}_{e}}(X,f)=\dim_{\mathbb{C}}\dfrac{\theta(i)}{ti(\theta_{n+1})+i^{*}(\textup{Derlog}\,G)+i^{*}(\epsilon)}.$

Notice that the evaluation mapping $\text{ev}:\theta_{n+1+k+r}\rightarrow
J(G)$ given by $\xi\mapsto\xi(G)$ is a surjective mapping with kernel
$\textup{Derlog}\,G$. Hence, it induces an isomorphism

$\dfrac{\theta_{n+1+k+r}}{\textup{Derlog}\,G}\cong J(G).$

Thus,

$\dfrac{\theta_{n+1+k+r}}{\langle\frac{\partial}{\partial
y_{1}},\ldots,\frac{\partial}{\partial
y_{n+1}}\rangle+\textup{Derlog}\,G}\cong\dfrac{J(G)}{J_{y}(G)}=M_{\textup{rel}}(G).$

Tensoring with $\mathscr{O}_{k+r}/\mathfrak{m}_{k+r}$ yields that

$\dfrac{\theta(i)}{ti(\theta_{n+1})+i^{*}\textup{Derlog}\,G}\cong M(g).$

Now, notice that the evaluation map acting on $\epsilon$ gives
$\epsilon(G)=G$, and hence $i^{*}(\text{ev}(\epsilon))=i^{*}(G)=g$. Therefore,
if $K(g)=(J(g)+(g))/J(g)$, then

$0\rightarrow
K(g)\rightarrow\dfrac{\theta(i)}{ti(\theta_{n+1})+i^{*}\textup{Derlog}\,G}\rightarrow\dfrac{\theta(i)}{ti(\theta_{n+1})+i^{*}\textup{Derlog}\,G+i^{*}(\epsilon)}\rightarrow
0$

is a short exact sequence. Indeed, the evaluation map satisfies that
$\text{ev}(ti(\theta_{n+1}))=J(g)$ and that
$\text{ev}(i^{*}\textup{Derlog}\,G)=0$. Hence, the evaluation map yields an
isomorphism

$\dfrac{ti(\theta_{n+1})+i^{*}\textup{Derlog}\,G+i^{*}(\epsilon)}{ti(\theta_{n+1})+i^{*}\textup{Derlog}\,G}\cong\dfrac{J(g)+(g)}{J(g)}=K(g).$

After taking lengths in the exact sequence, and taking into account the
previous assertions, it follows that

######  Theorem 5.1.

In the above conditions,

$\dim_{\mathbb{C}}M(g)=\dim_{\mathbb{C}}K(g)+\textup{codim}_{\mathscr{A}_{e}}(X,f).$

In particular,
$\dim_{\mathbb{C}}M(g)\geq\textup{codim}_{\mathscr{A}_{e}}(X,f)$, with
equality in case that $g$ is weighted homogeneous.

Furthermore, this formula shows that $\dim_{\mathbb{C}}M(g)$ only depends on
the isomorphism class of $g$, and neither depends on the mapping $f$ nor on
the chosen extension $\hat{f}$.

When either $(n,n+1)$ are nice dimensions or in corank one, all stable
singularities are weighted homogeneous (see [11]). This gives the following
direct consequence of Theorem 5.1:

######  Corollary 5.2.

Assume that either $(n,n+1)$ are nice dimensions or $(X,f)$ has corank one.
Then, $M(g)=0$ if and only if $(X,f)$ is stable.

##  6\. The Jacobian module and the Mond conjecture

This section is the centerpiece of this paper. We present one of the main
results we aim to establish, namely, a formula for the image Milnor number
expressed in terms of the Samuel multiplicity of the module
$M_{\textup{rel}}(G)$. Additionally, we prove that the generalised Mond
conjecture holds provided the module $M_{\textup{rel}}(G)$ is Cohen-Macaulay.

This theorem is the key ingredient to show the main result of the article [4],
which is Theorem 6.1, and whose proof can be easily adapted to this new
setting:

######  Theorem 6.1.

Assume that either $(n,n+1)$ are nice dimensions or $(X,f)$ has corank one.
Let $F$ be a stable unfolding of $\hat{f}$. With the notations of Section 3,

$\mu_{I}(X,f)=e(\mathfrak{m}_{k+r},M_{\textup{rel}}(G)),$

that is, $\mu_{I}(X,f)$ equals the Samuel multiplicity of the
$\mathscr{O}_{k+r}$-module $M_{\textup{rel}}(G)$ with respect to
$\mathfrak{m}_{k+r}$.

###### Proof.

Take a representative of $F$ and let $w\in\mathbb{C}^{k}\times\mathbb{C}^{r}$
be a generic value. The conservation of multiplicity implies that

$e\Big{(}\mathfrak{m}_{k+r},M_{\textup{rel}}(G)\Big{)}=\sum_{p\in
B_{\epsilon}}e\Big{(}\mathfrak{m}_{k+r,w},M_{\textup{rel}}(G)_{(p,w)}\Big{)}.$

In order to compute the previous multiplicity, let us take first the points
$p\in Y_{w}\cap B_{\epsilon}$, where $Y_{w}$ is the image of
$f_{w}:X_{w}\rightarrow\mathbb{C}^{n+1}$. Since the module
$M_{\textup{rel}}(G)$ specialises to $M(g)$, it follows that

$M_{\textup{rel}}(G)_{(p,w)}\otimes\dfrac{\mathscr{O}_{k+r,w}}{\mathfrak{m}_{k+r,w}}\cong
M(g_{w})_{p}$

as $\mathbb{C}$-vector spaces in virtue of Remark 4.4. If $F$ is a stable
unfolding, it follows that, provided $w$ is generic, $f_{w}$ is a stable
mapping, and hence $M(g_{w})_{p}=0$ for each $p\in Y_{w}$, by Corollary 5.2.
Hence, the points $p\in Y_{w}\cap B_{\epsilon}$ do not contribute to the term
$e(\mathfrak{m}_{k+r},M_{\textup{rel}}(G))$.

On the other hand, if $p\in B_{\epsilon}/Y_{w}$, then

$M_{\textup{rel}}(G)_{(p,w)}=\dfrac{\mathscr{O}_{\mathbb{C}^{k+r}\times
B_{\epsilon},(p,w)}}{J_{y}(G)}$

is a module with dimension $\geq k+r$. Indeed, this follows from the exact
sequence that defines $M_{\textup{rel}}(G)$, since the localisation of
$C(F)/(J_{y,z}(G)\cdot\mathscr{O}_{n+k+r})$ is zero if $p\notin Y_{w}$, and
$\mathscr{F}_{1}(F)$ localises to $\mathscr{O}_{\mathbb{C}^{k+r}\times
B_{\epsilon},(p,w)}$. Moreover,

$M_{\textup{rel}}(G)_{(p,w)}\otimes\dfrac{\mathscr{O}_{k+r,w}}{\mathfrak{m}_{k+r,w}}\cong\dfrac{\mathscr{O}_{B_{\epsilon},p}}{J(g_{w})}$

is a module with dimension $0$, since it has finite length due to the fact
that $g_{u}$ has isolated singularity. This implies that the dimension of
$M_{\textup{rel}}(G)$ is $\leq k+r$. Hence, $M_{\textup{rel}}(G)_{(p,u)}$ is a
complete intersection ring, and in particular a Cohen-Macaulay
$\mathscr{O}_{k+r}$-module of dimension $k+r$. Hence,

$\displaystyle
e\Big{(}\mathfrak{m}_{k+r,w},M_{\textup{rel}}(G)_{(p,w)}\Big{)}$
$\displaystyle=\dim_{\mathbb{C}}\left(M_{\textup{rel}}(G)_{(p,w)}\otimes\dfrac{\mathscr{O}_{k+r,w}}{\mathfrak{m}_{k+r,w}}\right)$
$\displaystyle=\dim_{\mathbb{C}}\dfrac{\mathscr{O}_{B_{\epsilon},p}}{J(g_{w})}=\mu(g_{w},p).$

By Siersma’s Theorem 2.8, it follows that $\sum_{p\in
B_{\epsilon}/Y_{w}}\mu(g_{w},p)=\mu_{I}(X,f)$. ∎

Once this result has been proven, the following theorem is now an immediate
consequence:

######  Theorem 6.2.

In the above conditions, the following statements are equivalent and imply the
generalised Mond conjecture for $(X,f)$:

  1. (1)

$\dim_{\mathbb{C}}M(g)=\mu_{I}(X,f)$,

  2. (2)

$M_{\textup{rel}}(G)$ is a Cohen-Macaulay $\mathscr{O}_{k+r}$-module of
dimension $k+r$.

Furthermore, if $g$ is weighted homogeneous and satisfies the generalised Mond
conjecture, then the above assertions hold.

###### Proof.

Recall that the Samuel multiplicity of an $R$-module $M$ is generally smaller
than the length of $M/(\mathfrak{m}\cdot M)$, with equality if and only if $M$
is a Cohen-Macaulay module with the same dimension as $R$. Hence,
$M_{\textup{rel}}(G)$ is Cohen-Macaulay of dimension $k+r$ if and only if

$\displaystyle\mu_{I}(X,f)$
$\displaystyle=e\Big{(}\mathfrak{m}_{k+r},M_{\textup{rel}}(G)\Big{)}=\dim_{\mathbb{C}}\dfrac{M_{\textup{rel}}(G)}{\mathfrak{m}_{k+r}M_{\textup{rel}}(G)}$
$\displaystyle=\dim_{\mathbb{C}}M_{\textup{rel}}(G)\otimes\dfrac{\mathscr{O}_{k+r}}{\mathfrak{m}_{k+r}}=\dim_{\mathbb{C}}M(g).$

Therefore, if one of the items hold, it then follows that

$\mu_{I}(X,f)=\dim_{\mathbb{C}}M(g)=\dim_{\mathbb{C}}K(g)+\textup{codim}_{\mathscr{A}_{e}}(X,f)\geq\textup{codim}_{\mathscr{A}_{e}}(X,f),$

and hence, the generalised Mond conjecture holds for $(X,f)$. Moreover, if $g$
is weighted homogeneous, then $K(g)=0$. Thus, if the generalised Mond
conjecture holds for $(X,f)$, then

$\mu_{I}(X,f)=\textup{codim}_{\mathscr{A}_{e}}(X,f)=\dim_{\mathbb{C}}M(g)-\dim_{\mathbb{C}}K(g)=\dim_{\mathbb{C}}M(g).$

Hence, the above assertions hold. ∎

######  Remark 6.3.

Recent work by Nuño-Ballesteros and Giménez Conejero [7] has shown that the
requirement for $M_{\textup{rel}}(G)$ to be $k+r$-dimensional can be
eliminated from the second condition. Thus, we have that
$\dim_{\mathbb{C}}M(g)=\mu_{I}(X,f)$ if and only if $M_{\textup{rel}}(G)$ is a
Cohen-Macaulay module. Indeed, if the dimension of $M_{\textup{rel}}(G)$ is
strictly less than $k+r$, then $\mu_{I}(X,f)$ is zero, which, according to
[7], implies that $(X,f)$ is a stable map-germ, and hence
$\textup{codim}_{\mathscr{A}_{e}}(X,f)=0$.

This result shows that in order to establish the validity of the Mond
conjecture for $f$, it suffices to check the Cohen-Macaulay property of the
module $M_{\textup{rel}}(G)$. While it has been observed that this module is
Cohen-Macaulay in every computed example, it remains an open question to
provide a rigorous proof of this statement.

##  7\. Proof of the generalised Mond conjecture for n = 2

In this section, we achieve our main objective of the article, which is to
show the validity of the generalised Mond conjecture for $n=2$ by employing
the results from the previous section.

Building on the main result of the previous section, to establish the
generalised Mond conjecture for mappings
$f:(X,0)\rightarrow(\mathbb{C}^{3},0)$ where $(X,0)$ is a 2-dimensional icis,
it suffices to check that $M_{\textup{rel}}(G)$ is a Cohen-Macaulay module of
dimension $k+r$. Notice first that the dimension of $M_{\textup{rel}}(G)$ is
$\leq k+r$, due to the fact that

$\dim_{\mathbb{C}}M_{\textup{rel}}(G)\otimes\dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_{k+r}\cdot\mathscr{O}_{n+1+k+r}}=\dim_{\mathbb{C}}M(g)<+\infty.$

Therefore, it is enough to show that $0ptM_{\textup{rel}}(G)\geq k+r$. To
achieve this, we apply the depth lemma to the exact sequence

$0\rightarrow
M_{\textup{rel}}(G)\rightarrow\dfrac{\mathscr{F}_{1}(F)}{J_{y}(G)}\rightarrow\dfrac{C(F)}{J_{y,z}(G)\cdot\mathscr{O}_{n+k+r}}\rightarrow
0$

to obtain that

$0ptM_{\textup{rel}}(G)\geq\min\left\\{0pt\dfrac{\mathscr{F}_{1}(F)}{J_{y}(G)},0pt\dfrac{C(F)}{J_{y,z}(G)\cdot\mathscr{O}_{n+k+r}}+1\right\\}.$

Hence, both terms of the minimum have to be shown to be greater than or equal
to $k+r$. For the module $C(F)/J_{y,z}(G)\cdot\mathscr{O}_{n+k+r}$, it has
been checked in Remark 3.9 of [4] that this module is Cohen-Macaulay of
dimension $n+k+r-2$ provided $n\geq 2$, due to the fact that it is isomorphic
to the determinantal ring $\mathscr{O}_{n+k+r}/R(F)$, where $R(F)$ denotes the
ramification ideal of $F$. In particular, it follows that

$0pt\dfrac{C(F)}{J_{y}(G)\cdot\mathscr{O}_{n+k+r}}+1=n+k+r-1\geq k+r.$

Therefore, it is enough to verify that $0pt\mathscr{F}_{1}(F)/J_{y}(G)\geq
k+r$. Notice that, up to this point, the assumption $n=2$ has not been used
yet. In general, the module $\mathscr{F}_{1}(F)/J_{y}(G)$ has dimension

$\dim\dfrac{\mathscr{F}_{1}(F)}{J_{y}(G)}=\max\left\\{\dim
M_{\textup{rel}}(G),\dim\dfrac{C(F)}{J_{y}(G)\cdot\mathscr{O}_{n+k+r,0}}\right\\}=n+k+r-2,$

due to the fact that $\dim M_{\textup{rel}}(G)\leq k+r\leq n+k+r-2$ provided
$n\geq 2$. Therefore, it is not expected that $M_{\textup{rel}}(G)$ will be
Cohen-Macaulay for every $n\geq 2$. The only case in which we can expect this
is when $n=2$, since, in this case, its dimension is precisely $k+r$. To
verify this claim, we make use of Pellikaan’s Theorem:

######  Theorem 7.1 (Pellikaan, Section 3 of [13]).

If $J\subset F\subset R$ are ideals of $R$ where $J$ is generated by $m$
elements, $\textup{grade}\,(F/J)\geq m$ and $\textup{pd}\,(R/F)=2$, then $F/J$
is a perfect module and grade$\,(F/J)=m$.

This result plays a crucial role to show that the module is indeed Cohen-
Macaulay, as it is proven in the following proposition:

######  Theorem 7.2.

If $n=2$, then $\mathscr{F}_{1}(F)/J_{y}(G)$ is a Cohen-Macaulay module.

###### Proof.

We follow the notation of the previous result, taking
$R=\mathscr{O}_{n+1+k+r}$, $F=\mathscr{F}_{1}(F)$ and $J=J_{y}(G)$. It follows
from the proof of Theorem 3.4 of [12] that
$R/F=\mathscr{O}_{n+1+k+r}/\mathscr{F}_{1}(F)$ is a determinantal ring of
dimension $n+k+r-1$, and hence Cohen-Macaulay. Hence, by the Auslander-
Buchsbaum formula,

$\displaystyle\textup{pd}\,(R/F)$ $\displaystyle=0ptR-0ptR/F=\dim R-\dim R/F$
$\displaystyle=n+1+k+r-(n+k+r-1)=2.$

Moreover, $J=J_{y}(G)$ is clearly generated by $n+1=3$ elements, namely the
partial derivatives of $G$ with respect to the variables
$y_{1},\ldots,y_{n+1}$. Furthermore,

$\displaystyle\textup{grade}\,(F/J)$
$\displaystyle=0pt(\textup{Ann}\,(F/J),\mathscr{O}_{n+1+r})=0pt\textup{Ann}\,(F/J)=$
$\displaystyle=\dim\mathscr{O}_{n+1+r}-\dim F/J=n+1+r-(n+r-2)=3.$

Thus, Pellikaan’s Theorem states that $F/J$ is a perfect
$\mathscr{O}_{n+1+k+r}$-module. Furthermore, since $\mathscr{O}_{n+1+k+r}$ is
a local Cohen-Macaulay ring, then $F/J$ is Cohen-Macaulay. ∎

Now the main result of this article follows easily as an application of the
results presented in the previous section.

###### Proof of Theorem 1.2.

Theorem 7.2 implies that $M_{\textup{rel}}(G)$ is a Cohen-Macaulay module.
Hence, Theorem 6.2 implies that the generalised Mond conjecture holds for
$(X,f)$. ∎

In this setting, the image Milnor number therefore satisfies that
$\mu_{I}(X,f)=\dim_{\mathbb{C}}M(g)$. This gives an operative method to
compute this number, as the following example shows:

######  Example 7.3.

Let $(X,0)\subset(\mathbb{C}^{3},0)$ be the hypersurface defined by
$x^{3}+y^{3}-z^{2}=0$ and let $f:(X,0)\rightarrow(\mathbb{C}^{3},0)$ be the
$\mathscr{A}$-finite mapping $f(x,y,z)=(x,y,z^{3}+xz+y^{2})$. In this case,
$\hat{f}(x,y,z)=(x,y,z^{3}+xz+y^{2},x^{3}+y^{3}-z^{2})$ turns out to be a
stable mapping. With Singular [3], one easily obtains that
$\textup{codim}_{\mathscr{A}_{e}}(X,f)=6$ and
$\mu_{I}(X,f)=\dim_{\mathbb{C}}M(g)=6$.

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