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On page 99 of Demailly's book "Complex analytic and differential geometry", he proved following theorem:

Theorem([Cartan-Oka 1950]). For any analytic set $A\subset M$, the sheaf of ideals $\mathcal{J}_{A}$ is a coherent analytic sheaf where $\mathcal{J}_{A}$ is a prime deal of the ring $O_{n}$ that is the germs of complex analytic functions at orgin.

The proof of this theorem is closely related to the fact that $O_{n}/\mathcal{J}_{A}$ is a finite integral extension of $O_{d}:=\mathbb{C}\{z_{1},\dots,z_{d}\}$. Indeed, Demailly has showed that for any prime ideal $\mathcal{J}$, we can find some interger $d$ such that $\mathcal{J}\cap O_{d}=\{0\}$ whence $O_{d}$ naturally embedded into $O_{n}/\mathcal{J}_{A}$ which is finitely generated as an $O_{d}$-module by the family of monomials $z_{d+1}^{\alpha_{d+1}}\dots z_{n}^{\alpha_{n}}$.

As $\mathcal{J}_{A}$ is prime, $O_{n}/\mathcal{J}_{A}$ is an entire ring. We denote by $\tilde{f}$ the class of any germ $f\in O_{n}$ in $\mathcal{J}_{A}$, by $M_{A}$ and $M_{d}$ the quotient fields of $\mathcal{J}_{A}$ and $O_{d}$ 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 $O_{d}$ lies in fact in $O_{d}$. 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 $O_{d}$, the unitary irreducible polynomial $W_{u}$ of $\tilde{u}$ over $M_{d}$ has coefficents in $O_{d}$, i.e. $W_{u}=W_{u}(z',T)$ where $z'$ represents the first $d$ complex variables.

Moreover, let $\delta(z')\in O_{d}$ 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 states that the coefficents of the polynomials $W_{u}(z',T)$ and $B_{k}(z',T)$ is a polynomial in the $c_{j}$'s with coefficients in $O_{d}$ because they can be expressed in terms of the elementary symmetric functions of the $\delta_{k}\tilde{u}$'s.(how come?)

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 in fact elementary symmetric functions of the $\delta_{k}\tilde{u}$'s.

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", 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 $O_{d,x}$.

On page 99 of Demailly's book "Complex analytic and differential geometry", he proved following theorem:

Theorem([Cartan 1950]). For any analytic set $A\subset M$, the sheaf of ideals $\mathcal{J}_{A}$ is a coherent analytic sheaf where $\mathcal{J}_{A}$ is a prime deal of the ring $O_{n}$ that is the germs of complex analytic functions at orgin.

The proof of this theorem is closely related to the fact that $O_{n}/\mathcal{J}_{A}$ is a finite integral extension of $O_{d}:=\mathbb{C}\{z_{1},\dots,z_{d}\}$. Indeed, Demailly has showed that for any prime ideal $\mathcal{J}$, we can find some interger $d$ such that $\mathcal{J}\cap O_{d}=\{0\}$ whence $O_{d}$ naturally embedded into $O_{n}/\mathcal{J}_{A}$ which is finitely generated as an $O_{d}$-module by the family of monomials $z_{d+1}^{\alpha_{d+1}}\dots z_{n}^{\alpha_{n}}$.

As $\mathcal{J}_{A}$ is prime, $O_{n}/\mathcal{J}_{A}$ is an entire ring. We denote by $\tilde{f}$ the class of any germ $f\in O_{n}$ in $\mathcal{J}_{A}$, by $M_{A}$ and $M_{d}$ the quotient fields of $\mathcal{J}_{A}$ and $O_{d}$ 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 $O_{d}$ lies in fact in $O_{d}$. 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 $O_{d}$, the unitary irreducible polynomial $W_{u}$ of $\tilde{u}$ over $M_{d}$ has coefficents in $O_{d}$, i.e. $W_{u}=W_{u}(z',T)$ where $z'$ represents the first $d$ complex variables.

Moreover, let $\delta(z')\in O_{d}$ 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 states that the coefficents of the polynomials $W_{u}(z',T)$ and $B_{k}(z',T)$ is a polynomial in the $c_{j}$'s with coefficients in $O_{d}$ because they can be expressed in terms of the elementary symmetric functions of the $\delta_{k}\tilde{u}$'s.(how come?)

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 in fact elementary symmetric functions of the $\delta_{k}\tilde{u}$'s.

On page 99 of Demailly's book "Complex analytic and differential geometry", he proved following theorem:

Theorem([Cartan-Oka 1950]). For any analytic set $A\subset M$, the sheaf of ideals $\mathcal{J}_{A}$ is a coherent analytic sheaf where $\mathcal{J}_{A}$ is a prime deal of the ring $O_{n}$ that is the germs of complex analytic functions at orgin.

The proof of this theorem is closely related to the fact that $O_{n}/\mathcal{J}_{A}$ is a finite integral extension of $O_{d}:=\mathbb{C}\{z_{1},\dots,z_{d}\}$. Indeed, Demailly has showed that for any prime ideal $\mathcal{J}$, we can find some interger $d$ such that $\mathcal{J}\cap O_{d}=\{0\}$ whence $O_{d}$ naturally embedded into $O_{n}/\mathcal{J}_{A}$ which is finitely generated as an $O_{d}$-module by the family of monomials $z_{d+1}^{\alpha_{d+1}}\dots z_{n}^{\alpha_{n}}$.

As $\mathcal{J}_{A}$ is prime, $O_{n}/\mathcal{J}_{A}$ is an entire ring. We denote by $\tilde{f}$ the class of any germ $f\in O_{n}$ in $\mathcal{J}_{A}$, by $M_{A}$ and $M_{d}$ the quotient fields of $\mathcal{J}_{A}$ and $O_{d}$ 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 $O_{d}$ lies in fact in $O_{d}$. 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 $O_{d}$, the unitary irreducible polynomial $W_{u}$ of $\tilde{u}$ over $M_{d}$ has coefficents in $O_{d}$, i.e. $W_{u}=W_{u}(z',T)$ where $z'$ represents the first $d$ complex variables.

Moreover, let $\delta(z')\in O_{d}$ 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 states that the coefficents of the polynomials $W_{u}(z',T)$ and $B_{k}(z',T)$ is a polynomial in the $c_{j}$'s with coefficients in $O_{d}$ because they can be expressed in terms of the elementary symmetric functions of the $\delta_{k}\tilde{u}$'s.(how come?)

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 in fact elementary symmetric functions of the $\delta_{k}\tilde{u}$'s.

On page 99 of Demailly's book "Complex analytic and differential geometry", he proved following theorem:

Theorem([Cartan 1950]). For any analytic set $A\subset M$, the sheaf of ideals $\mathcal{J}_{A}$ is a coherent analytic sheaf where $\mathcal{J}_{A}$ is a prime deal of the ring $O_{n}$ that is the germs of complex analytic functions at orgin.

The proof of this theorem is closely related to the fact that $O_{n}/\mathcal{J}_{A}$ is a finite integral extension of $O_{d}:=\mathbb{C}\{z_{1},\dots,z_{d}\}$. Indeed, Demailly has showed that for any prime ideal $\mathcal{J}$, we can find some interger $d$ such that $\mathcal{J}\cap O_{d}=\{0\}$ whence $O_{d}$ naturally embedded into $O_{n}/\mathcal{J}_{A}$ which is finitely generated as an $O_{d}$-module by the family of monomials $z_{d+1}^{\alpha_{d+1}}\dots z_{n}^{\alpha_{n}}$.

As $\mathcal{J}_{A}$ is prime, $O_{n}/\mathcal{J}_{A}$ is an entire ring. We denote by $\tilde{f}$ the class of any germ $f\in O_{n}$ in $\mathcal{J}_{A}$, by $M_{A}$ and $M_{d}$ the quotient fields of $\mathcal{J}_{A}$ and $O_{d}$ 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. Hence every element of $M_{d}$ which is integral over $O_{d}$ lies in fact in $O_{d}$. 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 $O_{d}$, the unitary irreducible polynomial $W_{u}$ of $\tilde{u}$ over $M_{d}$ has coefficents in $O_{d}$, i.e. $W_{u}=W_{u}(z',T)$ where $z'$ represents the first $d$ complex variables.

Moreover, let $\delta(z')\in O_{d}$ 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 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 states that the coefficents of the polynomials $W_{u}(z',T)$ and $B_{k}(z',T)$ is a polynomial in the $c_{j}$'s with coefficients in $O_{d}$ because they can be expressed in terms of the elementary symmetric functions of the $\delta_{k}\tilde{u}$'s.(how come?)

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 $b_{j}$ 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 in fact elementary symmetric functions of the $\delta_{k}\tilde{u}$'s.

On page 99 of Demailly's book "Complex analytic and differential geometry", he proved following theorem:

Theorem([Cartan 1950]). For any analytic set $A\subset M$, the sheaf of ideals $\mathcal{J}_{A}$ is a coherent analytic sheaf where $\mathcal{J}_{A}$ is a prime deal of the ring $O_{n}$ that is the germs of complex analytic functions at orgin.

The proof of this theorem is closely related to the fact that $O_{n}/\mathcal{J}_{A}$ is a finite integral extension of $O_{d}:=\mathbb{C}\{z_{1},\dots,z_{d}\}$. Indeed, Demailly has showed that for any prime ideal $\mathcal{J}$, we can find some interger $d$ such that $\mathcal{J}\cap O_{d}=\{0\}$ whence $O_{d}$ naturally embedded into $O_{n}/\mathcal{J}_{A}$ which is finitely generated as an $O_{d}$-module by the family of monomials $z_{d+1}^{\alpha_{d+1}}\dots z_{n}^{\alpha_{n}}$.

As $\mathcal{J}_{A}$ is prime, $O_{n}/\mathcal{J}_{A}$ is an entire ring. We denote by $\tilde{f}$ the class of any germ $f\in O_{n}$ in $\mathcal{J}_{A}$, by $M_{A}$ and $M_{d}$ the quotient fields of $\mathcal{J}_{A}$ and $O_{d}$ 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 $O_{d}$ lies in fact in $O_{d}$. 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 $O_{d}$, the unitary irreducible polynomial $W_{u}$ of $\tilde{u}$ over $M_{d}$ has coefficents in $O_{d}$, i.e. $W_{u}=W_{u}(z',T)$ where $z'$ represents the first $d$ complex variables.

Moreover, let $\delta(z')\in O_{d}$ 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 states that the coefficents of the polynomials $W_{u}(z',T)$ and $B_{k}(z',T)$ is a polynomial in the $c_{j}$'s with coefficients in $O_{d}$ because they can be expressed in terms of the elementary symmetric functions of the $\delta_{k}\tilde{u}$'s.(how come?)

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 in fact elementary symmetric functions of the $\delta_{k}\tilde{u}$'s.