Proof. If $(\phi ,\,f):(\mathbb {C}^{n},\,0)\to (\mathbb {C}^{2},\,0)$ defines an ICIS then $\mu _{BR}^{-}(f,\,X)$ is finite because

\[ V(\textrm{d}f(\Theta_{X}^{-}))\subset V(J(f,\phi)+I_{X})\subset\{0\}. \]

For the converse, if $\mu _{BR}^{-}(f,\,X)<\infty$ then the restriction of $f$ to each logarithmic stratum, excluding $\mathbb {C}^{n}\setminus X$ is non-singular. The proof is now the same of Proposition 2.8 in [Reference Biviá-Ausina and Ruas2].

Proof. We see $A$ and $A'$ as homomorphims of modules over $R:=\mathcal {O}_n$:

\[ A\colon R^{n}\longrightarrow R^{2},\quad A'\colon R^{n+1}\longrightarrow R^{2}. \]

We consider the $R$-module $R^{2}/M=\operatorname {coker}(A)$, which has support $V(I_2(A))=V(J(f,\,g))$. Therefore, $\operatorname {dim}(R^{2}/M)=1=n-(n-2+1)$ and hence it is Cohen–Macaulay (see [Reference Buchsbaum and Rim5]). In particular, it is unmixed. Now, $M'/M$ is a submodule of $R^{2}/M$, so the associated primes $\operatorname {Ass} (M'/M)$ are included in $\operatorname {Ass}(R^{2}/M)$. If $M'/M\ne 0$ then $\operatorname {Ass} (M'/M)\ne \emptyset$ and it follows that $\operatorname {dim}(M'/M)=1$.

Let $U$ be a neighbourhood of $0$ in $\mathbb {C}^{n}$ such that $0$ is the only critical point of $f$. For all $x\in U\setminus \{0\}$, there exist $i_{0}\in \{1,\,\ldots ,\,n\}$, such that $\partial f/\partial x_{i_{0}}(x)\neq 0$. We may suppose $i_{0}=1$. Making elementary column operations in the matrices $A$ and $A'$, we obtain

\[ B=\begin{pmatrix} 1 & 0 & \ldots & 0 \\ c_1 & c_{2} & \ldots & c_n \end{pmatrix}, \quad B'=\begin{pmatrix} \mu & 1 & 0 & \ldots & 0 \\ \lambda & c_1 & c_{2} & \ldots & c_n \end{pmatrix} \]

such that

\[ I_{2}(A)=I_{2}(B),\quad I_{2}(A')=I_{2}(B'),\ \operatorname{Im}(A)=\operatorname{Im}(B) \text{ and }\operatorname{Im}(A')=\operatorname{Im}(B'). \]

By hypothesis $I_{2}(A)=I_{2}(A')$ and consequently $\langle c_{2},\,\ldots ,\,c_{n}\rangle =\langle \mu c_{1}-\lambda ,\,c_{2},\,\ldots ,\,c_{n}\rangle .$ This implies $\lambda =\mu c_{1}+\alpha _{2}c_{2}+\cdots +\alpha _{n}c_{n}$, for some $\alpha _{2},\,\cdots ,\,\alpha _{n}\in R$. Thus,

\[ \left(\begin{array}{@{}c@{}} \mu \\ \lambda\end{array}\right)=\mu \left(\begin{array}{@{}c@{}} 1 \\ c_1\end{array}\right)+ \alpha_2 \left(\begin{array}{@{}c@{}} 0 \\ c_2\end{array}\right)+\cdots+ \alpha_n \left(\begin{array}{@{}c@{}} 0 \\ c_n\end{array}\right). \]

and hence $(M'/M)_{x}=0$. This shows that $\operatorname {Supp}(M'/M)\subset \{0\}$ and hence, $M'=M$.

Proof.

1. (i) The homomorphism $\Psi :\Theta _{X}\to \textrm {d}f(\Theta _{X})+I_{X}$ defined by $\Psi (\xi )=\textrm {d}f(\xi )$ induces the isomorphism

   \[ \overline{\Psi}:\frac{\Theta_{X}}{\Theta_{X}^{T}}\to\frac{\textrm{d}f(\Theta_{X})+I_{X}}{\textrm{d}f(\Theta_{X}^{T})+I_{X}}. \]
   In fact, it is enough to show that $\Psi ^{-1}(\textrm {d}f(\Theta _{X}^{T})+I_{X})\subset \Theta _{X}^{T}.$ Let $\xi \in \Psi ^{-1}(\textrm {d}f(\Theta _{X}^{T})+I_{X})$ then $\Psi (\xi )\in \textrm {d}f(\Theta _{X}^{T})+I_{X}$, that is, there exist $\eta \in \Theta _{X}^{T}$ and $\mu ,\, \lambda \in \mathcal {O}_{n}$, such that
   \[ \begin{cases} \textrm{d}f(\xi-\eta)=\mu\phi\\ \textrm{d}\phi(\xi-\eta)=\lambda\phi \end{cases}, \]
   then
   \[ \left(\begin{array}{@{}c@{}} \mu\phi\\ \lambda\phi \end{array}\right)\in \left\langle \left(\begin{array}{@{}c@{}} \dfrac{\partial f}{\partial x_{i}}\\ \dfrac{\partial \phi}{\partial x_{i}} \end{array}\right) \quad i=1,\ldots,n \right\rangle \]
   and
   \[ I_{2}\begin{pmatrix}\mu\phi & \dfrac{\partial f}{\partial x_{1}} & \cdots & \dfrac{\partial f}{\partial x_{n}}\\ \lambda\phi & \dfrac{\partial \phi}{\partial x_{1}} & \cdots & \dfrac{\partial \phi}{\partial x_{n}} \end{pmatrix}=I_{2}\begin{pmatrix}\dfrac{\partial f}{\partial x_{1}} & \cdots & \dfrac{\partial f}{\partial x_{n}}\\ \dfrac{\partial \phi}{\partial x_{1}} & \cdots & \dfrac{\partial \phi}{\partial x_{n}} \end{pmatrix}=J(f,\phi). \]
   Therefore
   \[ \left|\begin{array}{cc} \mu & \dfrac{\partial f}{\partial x_{i}} \\ \lambda & \dfrac{\partial \phi}{\partial x_{i}} \end{array}\right|\phi\in J(f,\phi) \]
   and since $\phi$ is regular in $\frac {\mathcal {O}_{n}}{J(f,\,\phi )}$ then
   \[ \left|\begin{array}{cc} \mu & \dfrac{\partial f}{\partial x_{i}}\\ \lambda & \dfrac{\partial \phi}{\partial x_{i}} \end{array}\right|\in J(f,\phi),\quad i=1,\ldots,n. \]
   By Lemma 2.2, $\lambda \in J\phi$ and using [Reference Nuño-Ballesteros, Oréfice-Okamoto, Lima-Pereira and Tomazella9, Lemma 3.1], $\xi \in \Theta _{X}^{T}.$

2. (ii) This equality also was proved in [Reference Nuño-Ballesteros, Oréfice-Okamoto, Lima-Pereira and Tomazella9] with the additional hypothesis that $f$ is $\mathcal {R}_{X}$-finitely determined.

   The epimorphism $\psi :\Theta _{X}\to \textrm {d}f(\Theta _{X})$ defined by $\psi (\xi )=\textrm {d}f(\xi )$ induces the isomorphism

   \[ \overline{\psi}:\frac{\Theta_{X}}{\Theta_{X}^{T}}\to \frac{\textrm{d}f(\Theta_{X})}{\textrm{d}f(\Theta_{X}^{T})}. \]
   In fact, let $\xi \in \ker (\psi )$, then there exist $\lambda \in \mathcal {O}_{n}$, such that
   \[ \begin{cases} \textrm{d}f(\xi)=0\\ \textrm{d}\phi(\xi)=\lambda\phi \end{cases} \]
   The rest is similar to the proof of (i).

3. (iii) Let $\xi \in \Theta _{X}$ be such that $\textrm {d}f(\xi )\in I_{X}$, then there exist $\mu ,\,\lambda \in \mathcal {O}_{n}$, such that

   \[ \begin{cases} \textrm{d}f(\xi)=\mu\phi\\ \textrm{d}\phi(\xi)=\lambda\phi \end{cases} \]
   Using the same techniques of the proof of (i), we have
   \[ \textrm{d}f(\Theta_{X})\cap I_{X}\subset Jf I_{X}. \]
   The other inclusion is immediate.

4. (iv) It follows from the isomorphisms

   \[ \frac{\textrm{d}f(\Theta_{X}^{-})}{\textrm{d}f(\Theta_{X})}=\frac{\textrm{d}f(\Theta_{X})+I_{X}}{\textrm{d}f(\Theta_{X})}\approx \frac{I_{X}}{\textrm{d}f(\Theta_{X})\cap I_{X}}\stackrel{(iii)}{=}\frac{I_{X}}{JfI_{X} }\approx\frac{\mathcal{O}_{n}}{Jf}. \]

5. (v) It follows from (iii).

6. (vi) It follows from (v) and $Jf\subset \textrm {d}f(\Theta _{X}^{T}):I_{X}$.

Proof. We consider the exact sequence

\[ 0\longrightarrow \frac{\textrm{d}f(\Theta_{X}^{-})}{\textrm{d}f(\Theta_{X}^{T})+I_{X}}\stackrel{i}{\longrightarrow}\frac{\mathcal{O}_{n}}{\textrm{d}f(\Theta_{X}^{T})+I_{X}}\stackrel{\pi}{\longrightarrow}\frac{\mathcal{O}_{n}}{\textrm{d}f(\Theta_{X}^{-})}\longrightarrow 0. \]

Since $(X,\,0)$ is an IHS

\[ \textrm{d}f(\Theta_{X}^{T})=J(f,\phi)+JfI_{X}, \]

hence

\begin{align*} mu_{BR}^{-}(f,X)& =\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{n}}{J(f,\phi)+I_{X}}-\operatorname{dim}_{\mathbb{C}}\frac{\textrm{d}f(\Theta_{X})+I_{X}}{\textrm{d}f(\Theta_{X}^{T})+I_{X}}\\ & =\mu(f^{{-}1}(0)\cap X,0)+\mu(X,0)-\tau(X,0). \end{align*}

The last equality is a consequence of the Lê-Greuel formula [Reference Brieskorn and Greuel3] and Theorem 2.3 (i).

Proof. We consider $(0,\,p)\in LC(X)^{-}$, then $(0,\,p)\in LC(X)$ and there exists $f\in \mathcal {O}_{n}$ such that $\textrm {d}f(0)=p$. In [Reference Nuño-Ballesteros, Oréfice-Okamoto, Lima-Pereira and Tomazella9], we proved that $LC(X)$ is Cohen–Macaulay. Therefore, by [Reference Bruce and Roberts4, Proposition 5.8],

\[ \mu_{BR}(f,X)=\sum_{i=0}^{k+1}m_{i}n_{i}=m_{0}n_{0}+\sum_{i=1}^{k+1}m_{i}n_{i}=\mu(f)+\sum_{i=1}^{k+1}m_{i}n_{i}. \]

where $n_{i}$ is the number of critical points of a Morsification of $f$ in $X_{i}$ and $m_{i}$ is the multiplicity of irreducible component $Y_{i}$. Thus,

\[ \mu_{BR}^{-}(f,X)=\mu_{BR}(f,X)-\operatorname{dim}_{\mathbb{C}}\frac{\textrm{d}f(\Theta_{X}^{-})}{\textrm{d}f(\Theta_{X})}=\mu_{BR}(f,X)-\mu(f)=\sum_{i=1}^{k+1}m_{i}n_{i}. \]

and by [Reference Bruce and Roberts4, Proposition 5.11], we obtain that $LC(X)^{-}$ is Cohen–Macaulay.

Proof. Let $(0,\,p)\in LC(X)$, then there exists an $\mathcal {R}_{X}$-finitely determined function germ $f\in \mathcal {O}_{n},$ such that $\textrm {d}f(0)=p$. Let $F:(\mathbb {C}^{n}\times \mathbb {C})\to (\mathbb {C},\,0)$, $F(x,\,t)=f_{t}(x)$,

be a Morsification of $f$. By hypothesis $\mathcal {O}_{n+1}/\textrm {d}f_{t}(\Theta _{X})$ is Cohen–Macaulay of dimension 1, then by the principle of conservation of number

\[ \mu_{BR}(f,X)=\sum_{i=0}^{k+1}\sum_{x\in\Sigma f_{t}\cap X_{i}}\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{n,x}}{\textrm{d}f_{t}(\Theta_{X,x})}=\sum_{i=0}^{k+1}\sum_{x\in\Sigma f_{t}\cap X_{i}}m_{i}=\sum_{i=0}^{k+1}n_{i}m_{i} \]

because if $x\in X_{i}$ is a Morse critical point of $f_{t}$, then $\mu _{BR}(f_{t},\,X)_{x}=m_{i}$, and by [Reference Bruce and Roberts4, Proposition 5.8], $LC(X)$ is Cohen–Macaulay.

Proof. Let $Z_{i}=Y_{i}\setminus \bigcup _{j\neq i}Y_{j}$ where $Y_{i}$ are the irreducible components of $LC(X)$. We know from [Reference Bruce and Roberts4, Proposition 5.12] that $LC(X)$ is Cohen–Macaulay at points in $Z_{i}$, $i=1,\,\ldots ,\,k+1$. We see that $LC(X)^{-}$ coincides locally with $LC(X)$ and hence, $LC(X)^{-}$ is also Cohen–Macaulay at points in $Z_{i}$, $i=1,\,\ldots ,\,k+1$.

In fact, let $(0,\,p)\in Z_{i}$ with $i\neq 0$, then $(x,\,p)\not \in Y_{0}$. Let $V:= T^{*}\mathbb {C}^{n}\setminus Y_{0}$, which is an open neighbourhood of $(x,\,p)$. Obviously, we have the equality of sets

\[ LC(X)\cap V=LC(X)^{-}\cap V. \]

Moreover, let $I,\,\;I^{-}$ and $I_{j}$ be the ideals which define $LC(X)$, $LC(X)^{-}$ and $Y_{j}$, $j=0,\,\ldots ,\,k+1$, respectively. Then,

\[ I=I_{0}\cap I_{1}\cap\cdots\cap I_{k+1}, \quad I^{-}=I_{1}\cap\cdots\cap I_{k+1}\text{ and }I_{0}=\langle p_{1},\ldots,p_{n}\rangle. \]

Since $p\neq 0$, $I_{0}$ is the total ring at $(x,\,p)$, so we have an equality between germs of complex spaces.

Finally, we have

\[ \mu_{BR}(f,X)_{x}^{-}\stackrel{(*)}=\sum_{i=1}^{k+1}m_{i}n_{i}\stackrel{(**)}=m_{\alpha}. \]

The equalities $(*)$ and $(**)$ are consequences of [Reference Bruce and Roberts4, Propositions 5.11 and 5.2], respectively.

Proof. Let $(0,\,p)\in LC(X)^{-}$, then there exists an $\mathcal {R}_{X}$-finitely determined function germ $f\in \mathcal {O}_{n},$ such that $\textrm {d}f(0)=p$. Let $F:(\mathbb {C}^{n}\times \mathbb {C},\,0)\to (\mathbb {C},\,0)$ be a Morsification of $f$ and set $f_{t}(x)=F(x,\,t)$.

By hypothesis $\mathcal {O}_{n+1}/\textrm {d}f_{t}(\Theta _{X}^{-})$ is Cohen–Macaulay of dimension 1. By the principle of the conservation of the multiplicity,

\[ \operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{n}}{\textrm{d}f(\Theta_{X}^{-})}=\sum_{i=1}^{k+1}\sum_{x\in\Sigma f\cap X_{i}}\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{n,x}}{\textrm{d}f_{t}(\Theta_{X,x}^{-})}=\sum_{i=1}^{k+1}\sum_{x\in\Sigma f\cap X_{i}}m_{i}=\sum_{i=1}^{k+1}n_{i}m_{i}, \]

because if $x\in X_{i}$ is a Morse critical point of $f_{t}$, then $\mu _{BR}(f_{t},\,X)^{-}_{x}=m_{i}$ by Lemma 4.4. By [Reference Bruce and Roberts4, Proposition 5.11], $LC(X)^{-}$ is Cohen–Macaulay.

Proof. The equivalence follows from

\[ V(I')=V(I)\times\mathbb{C}^{n-t},\quad V(J')=\mathbb{C}^{t}\times V(J) \text{ and } V(I'+J')=V(I)\times V(J). \]

For the equality, by hypothesis there exist positive integer numbers $k',\,\;k_{i}$ and $k_{j}$ such that

\[ \mathcal{M}_{n}^{k'}\subset I'+J',\quad \mathcal{M}_{k}^{k_{i}}\subset I,\ ;\mathcal{M}_{n-k}^{k_{j}}\subset J, \]

where $\mathcal {M}_\ell$ is the maximal ideal of $\mathcal {O}_\ell$. Let $r=\max \{k',\,\ k_{i},\, \ k_{j}\}$, then

\[ \frac{\mathcal{O}_{n}}{I'+J'}\approx\frac{\frac{\mathcal{O}_{n}}{\mathcal{M}_{n}^{r}}}{\frac{I'+J'}{\mathcal{M}_{n}^{r}}}=\frac{\frac{\mathbb{C}[z_1,z_2]}{\mathcal{M}_{n}^{r}}}{\frac{I''+J''}{\mathcal{M}_{n}^{r}}}=\frac{\mathbb{C}[z_1,z_2]}{I''+J''}, \]

where $z_1=(x_1,\,\ldots ,\,x_k)$, $z_2=(x_{k+1},\,\ldots ,\,x_n)$ and $I''$ and $J''$ are the ideals in $\mathbb {C}[z_1,\,z_2]$ generated by the $r-1$-jets of the generators of $I$ and $J$, respectively. Analogously,

\[ \frac{\mathcal{O}_{k}}{I}\approx\frac{\mathbb{C}[z_1]}{I'''} \text{ and } \frac{\mathcal{O}_{n-t}}{J} \approx\frac{\mathbb{C}[z_{2}]}{J'''}, \]

where $I'''$ and $J'''$ are the ideals in $\mathbb {C}[z_1]$ and $\mathbb {C}[z_2]$ generated by the $r-1$-jets of the generators of $I$ and $J$, respectively. Finally, the equality follows from

\[ \frac{\mathbb{C}[z_1]}{I'''}\otimes_{\mathbb{C}}\frac{\mathbb{C}[z_2]}{J'''}=\frac{\mathbb{C}[z_1,z_2]}{I''+J''}, \]

where $\otimes _{\mathbb {C}}$ denotes the tensor product, see [Reference Greuel and Pfister7, Proposition 2.7.13].

Proof. It is a consequence of the characterization of $\Theta _{\tilde {X}}$ and the previous theorem.