Let $A$ be an analytic subset of a complex manifold $M$ and $O\_{M}$ be the sheaf of complex analytic functions on $M$. The sheaf of ideals $\mathcal{J}\_{A}$ is defined as the subsheaf of $O\_{M}$ whoes stalk at each point $x\in M$ consists of germs of analytic functions vanishing on $A$. If $x\notin A$, then $\mathcal{J}\_{A,x}=O\_{M,x}$.


On page 99 of Demailly's book "[Complex analytic and differential geometry](https://www-fourier.ujf-grenoble.fr/%7Edemailly/manuscripts/agbook.pdf)", he proved following theorem concerning the coherence of the ideal sheaf $\mathcal{J}\_{A}$:


**Theorem([Cartan-Oka 1950]).** $\mathcal{J}\_{A}$ is coherent.


Since coherece is a local property, we can assume that $A\_{x}$(the germ of $A$ at $x$) is irreducible, i.e. $\mathcal{J}\_{A,x}$ is a prime ideal of $O\_{M,x}$. Then the proof of this theorem relies heavily on the fact that we can choose an appropriate coordinate system centered at $x$ such that $O\_{M,x}/\mathcal{J}\_{A,x}$ is a finite integral extension of $O\_{d,x}:=\mathbb{C}\{z\_{1},\dots,z\_{d}\}$ for some integer $d\leq Dim M$. Indeed, it can be shown that for any prime ideal $\mathcal{J}$, we can find some interger $d$ such that $\mathcal{J}\cap O\_{d,x}=\{0\}$ whence $O\_{d,x}$ naturally embedded into $O\_{M,x}/\mathcal{J}\_{A,x}$. Moreover it is finitely generated as an $O\_{d,x}$-module by the family of monomials $z\_{d+1}^{\alpha\_{d+1}}\dots z\_{n}^{\alpha\_{n}}$.


As $\mathcal{J}\_{A,x}$ is prime, $O\_{M,x}/\mathcal{J}\_{A,x}$ is an entire ring. We denote by $\tilde{f}$ the class of any germ $f\in O\_{M,x}$ in $\mathcal{J}\_{A,x}$, by $M\_{A}$ and $M\_{d}$ the quotient fields of $\mathcal{J}\_{A,x}$ and $O\_{d,x}$ respectively. Then $M\_{A}=M\_{d}[\tilde{z}\_{d+1}\dots\tilde{z}\_{n}]$ is a finite algebraic extension of $M\_{d}$. Let $q=[M\_{A}:M\_{d}]$ be its degree of extension and let $\sigma\_{1},\dots,\sigma\_{q}$ be the $q$ embeddings of $M\_{A}$ over $M\_{d}$ in an algebraic closure $\overline{M\_{A}}$. Since a factorial ring is integrally closed in its quotient field, every element of $M\_{d}$ which is integral over $O\_{d,x}$ lies in fact in $O\_{d,x}$. By the primative element theorem, there exists a linear form $u(z'')=c\_{d+1}z\_{d+1}+\dots+c\_{n}z\_{n}$, such that $M\_{A}=M\_{d}[\tilde{u}]$. As $\tilde{u}$ is integral over the integrally closed ring $O\_{d,x}$, the unitary irreducible polynomial $W\_{u}$ of $\tilde{u}$ over $M\_{d}$ has coefficents in $O\_{d,x}$, i.e. $W\_{u}=W\_{u}(z',T)$ where $z'$ represents the first $d$ complex variables.


Moreover, let $\delta(z')\in O\_{d,x}$ be the discriminant of $W\_{u}(z',T)$. Then it is easy to show that for every element $g$ of $M\_{A}$ which is integral over $O\_{d}$ we have $\delta g\in O\_{d}[\tilde{u}]$.(Demailly proved this claim as lemma 4.15 on page 94). In particular, for $k\geq d+1$, we have $\delta z\_{k}\in O\_{d}[\tilde{u}]$. Or more precisely, there exists an unique polynomial $B\_{k}\in O\_{d}[T]$ with degree less than $q$ such that $\delta z\_{k}=B\_{k}(z',\tilde{u}(z''))$.


**Then he claimed that $\delta$ and the coefficents of the polynomials $W\_{u}(z',T)$ and $B\_{k}(z',T)$ all can be expressed as polynomials in the $c\_{j}$'s with coefficients in $O\_{d,x}$, because all of them can be expressed in terms of the elementary symmetric functions of the $\delta\_{k}\tilde{u}$'s.**


Since $W\_{u}(z',T)$ is the irreducible polynomial of $\tilde{u}$ over the quotient field $M\_{d}$ of $O\_{d}$, his claim is obvious. But for the other polynomial $B\_{k}(z',T)$, I really cannot fill in the gap. The only thing I know is that the coefficients $b\_{j}$'s of the polynomial $B\_{k}$ can be calculated by the cramer rule according to his proof of lemma 4.15. But I really don't know how to show that they are elementary symmetric polynomials of the $\delta\_{k}\tilde{u}$'s with coefficients in $O\_{d,x}$.



A question on Demailly's proof of coherence of ideal sheaf
We need the concept of multi-symmetric polynomial which is a straight forward generalization of symmetric polynomial. The definitions and relevant results concerning it can be found in [this paper](https://%20arxiv.org/pdf/math/0205233.pdf).


Indeed, the coefficients $\delta b\_{j}$ of $B\_{k}$ can be expressed as a multi-symmetric polynomial of $\sigma\_{i}\tilde{u}$ and $\sigma\_{i}\tilde{z}\_{k}$. So it is quite misleading to say that they are just symmetric polynomials of $\sigma\_{i}\tilde{u}$. And it was proved in the paper that the ring of multi-symmetric polynomials $\mathbb{C}[x\_{1},\dots,x\_{n},y\_{1},\dots,y\_{n}]^{S\_{n}}$ can be generated by
$$
\bigg\{ e\_{\alpha\beta}=\sum\_{i=1}^{n}x\_{i}^{\alpha}y\_{i}^{\beta}\Big|\alpha,\beta \in \mathbb{N},\alpha+\beta\leq n
\bigg\}.
$$


And it is easy to see that $e\_{\alpha\beta}\in O\_{d,x}[c\_{k+1},\dots,c\_{n}]$.


