Let $\mathcal{Z}$ be an analytic subset of a domain $\Omega$ in $\mathbb{C}^{m}$. For a point $p\in\mathcal{Z}$, define the dimension of $\mathcal{Z}$ at $p$ as $dim_{p}\mathcal{Z}:=\overline{lim}_{z\to\infty,~z\in reg{\mathcal{Z}}}dim_{z}\mathcal{Z}$, where $reg{\mathcal{Z}}$ is the collection of regular points of $\mathcal{Z}$. Now, the dimension of $\mathcal{Z}$ will be defined as $dim{\mathcal{Z}}:=max_{z\in\mathcal{Z}}dim_{z}\mathcal{Z}=max_{z\in reg\mathcal{Z}}dim_{z}\mathcal{Z}$ and we'll define its codimension as $codim\mathcal{Z}:=m - dim\mathcal{Z}$. If every point of $\mathcal{Z}$ has dimension $d$, then we'd call it a pure $d$-dimensional analytic subset of $\Omega$. Similarly, one has analytic subset of pure codimension $d$.
Now, suppose $\mathcal{Z}$ has pure codimension $1$. Then $\mathcal{Z}$ will be an analytic hypersurface (See Corollary 1, P-26, Complex Analytic Sets by E.M. Chirka). So, for each point $p\in\mathcal{Z}$ there exists an open neighbourhood $U_{p}$ of $p$ in $\Omega$, $\phi^{U_{p}}\in\mathcal{O}(U_{p})$ s.t. $\mathcal{Z}\cap U_{p}=Z(\phi^{U_{p}})$ and for any $f^{U_{p}}\in\mathcal{O}(U_{p})$ which vanishes on $\mathcal{Z}\cap U_{p}$, we'd have $\phi^{U_{p}}$ divides $f^{U_{p}}$. In particular, $f^{U_{p}}\in<\phi^{U_{p}}>$ which is the ideal generated by $\phi^{U_{p}}$ in the ring $\mathcal{O}(U_{p})$. Such a $\phi^{U_{p}}$ is often called a local defining function of $\mathcal{Z}$.
My question is, does there exists an analogue of the above phenomenon in the pure codimension $d$ case$?$ More precisely, what I think is the following:
Let $\mathcal{Z}$ be an analytic subset of pure codimension $d$. For each point $p\in\mathcal{Z}$ there exists an open neighbourhood $U_{p}$ of $p$ in $\Omega$, $\phi_{1}^{U_{p}},\ldots\phi_{d}^{U_{p}}\in\mathcal{O}(U_{p})$ s.t. i). $\mathcal{Z}\cap U_{p}=Z(\phi_{1}^{U_{p}})\cap\ldots\cap Z(\phi_{d}^{U_{p}})$ ii). for any $f^{U_{p}}\in\mathcal{O}(U_{p})$ which vanishes on $\mathcal{Z}\cap U_{p}, f^{U_{p}}\in <\phi_{1}^{U_{p}},\ldots,\phi_{d}^{U_{p}}>$.
Note that if $\mathcal{Z}$ becomes a submanifold, then one can show the above through a power series argument. But whether or not it(or any variant of it) holds in the general case is unclear to me. Any answer/suggestion/reference is appreciated.