Theorem 9.3.10 (Cartan's coherence theorem)3675C20D7E18

Proof. The assertion is local. Near a point of $A$, the set $A$ is the union of representatives $A_1,\ldots ,A_ p$ of the branches of its germ, and $\mathcal I_ A=\bigcap _ i\mathcal I_{A_ i}$. The intersection of two coherent subsheaves $\mathcal F,\mathcal G$ of $\mathcal O_ M$ is the kernel of the morphism $\mathcal F\oplus \mathcal G\to \mathcal O_ M$, $(a,b)\mapsto a-b$, and is coherent by theorem 9.2.7, lemma 9.2.6. It therefore suffices to prove that $\mathcal I_ A$ is coherent near $0$ when $(A,0)\subset (\mathbb C^ n,0)$ is irreducible. We use the parametrization of theorem 9.3.4, with its primitive linear form $u$, its Weierstrass polynomial $W$ and $\delta =\operatorname {Disc}(W)$, and we put $m=\max \{ q,(n-d)(q-1)\} $. Since $A$ may have several branches at points near $0$, we describe all stalks near $0$ at once: we construct holomorphic functions $G_1,\ldots ,G_ N$ vanishing on $A$ and a holomorphic function $h\not\equiv 0$ of $z'$ such that, at every point $x$ near $0$, the stalk $\mathcal I_{A,x}$ consists of the germs $f$ with $hf\in (G_{1,x},\ldots ,G_{N,x})$; such a sheaf is coherent by Oka's theorem.

Step 1: dependence on the primitive form. For $c=(c_{d+1},\ldots ,c_ n)\in \mathbb C^{n-d}$ put $u_ c=\sum _ kc_ kz_ k$. For $z'\in \Delta '\setminus \{ \delta =0\} $ let $z^{(1)},\ldots ,z^{(q)}$ be the points of $A$ over $z'$, and define

where $D_ c$ is the Vandermonde determinant $\det (u_ c(z^{(i)})^ j)_{i,j}$ and $D_{kcj}$ is obtained from it by replacing the $j$th column with the coordinates $z_ k^{(i)}$. These expressions do not change under permutations of the points $z^{(i)}$, since $D_ c$ and $D_{kcj}$ both change sign under a transposition. Hence their coefficients in $T$ are polynomials in $c$ whose coefficients are bounded holomorphic functions on $\Delta '\setminus \{ \delta =0\} $, and these extend holomorphically to $\Delta '$ by lemma 9.2.9. On $A_\delta $ we have $W_ c(z',u_ c(z''))=0$ and, by Cramer's rule in the form $\sum _ jD_{kcj}u_ c(z^{(i)})^ j=D_ cz_ k^{(i)}$, also $\delta _ c(z')z_ k=B_{kc}(z',u_ c(z''))$, for every $c$. Let $G_1,\ldots ,G_ N$ be the coefficients, as polynomials in $c$, of the functions $W_ c(z',u_ c)$ and $\delta _ cz_ k-B_{kc}(z',u_ c)$, $d<k\leq n$. They are holomorphic on $\Delta '\times \Delta ''$ and vanish on $A_\delta $, hence on $A$, which is the closure of $A_\delta $. For the coefficient vector $c$ of the primitive linear form $u$, the values $u(z^{(i)})$ are the roots of $W(z',\cdot )$, by theorem 9.3.4, so $\delta _ c=\operatorname {Disc}(W)=\delta $ by (9.3.2), and $\delta _ c\neq 0$ off $\{ \delta =0\} $; so the polynomial $\delta _ c^ m=\sum _\alpha h_\alpha (z')c^\alpha $ in $c$ is not identically zero, and we choose a coefficient $h=h_\alpha \not\equiv 0$.

Step 2: a description of the stalks. We claim that on a small polydisc around $0$,

for every point $x$. If $hf\in (G_{1,x},\ldots ,G_{N,x})$, then $hf$ vanishes on $A$ near $x$, so $f$ vanishes on $A\setminus \{ h=0\} $ near $x$. This set is dense in $A$: the covering locus $A_\delta $ is dense in $A$, and since $\pi $ is a local biholomorphism on $A_\delta $, the set where $h\circ \pi $ vanishes is nowhere dense in $A_\delta $. Hence $f\in \mathcal I_{A,x}$, which proves the inclusion from right to left.

Step 3: the inclusion from left to right at points of $A$. Fix $x\in A$ and $f\in \mathcal I_{A,x}$. Call $c$ admissible if $\delta _ c\not\equiv 0$ and $u_ c$ takes distinct values at the points of the finite fiber of $A$ over $x'$; the admissible $c$ form a dense open subset of $\mathbb C^{n-d}$, the complement of an algebraic subset and finitely many proper linear subspaces. Fix an admissible $c$. Since $W_ c(z',u_ c(z''))$ vanishes on $A$ by Step 1, the number $u_ c(x'')$ is a root of $W_ c(x',\cdot )$. By lemma 9.2.5, $W_ c=W_ c^ xQ_ c^ x$ near $(x',u_ c(x''))$, where $W^ x_ c$ is a Weierstrass polynomial in $T-u_ c(x'')$ and $Q^ x_ c$ is a polynomial in $T$ with $Q^ x_ c(x',u_ c(x''))\neq 0$.

We call $z'$ generic if $\delta (z')\delta _ c(z')\neq 0$. For generic $z'$ near $x'$, the roots of $W^ x_ c(z',\cdot )$ are distinct and are exactly the values of $u_ c$ at the points of $A$ over $z'$ that are close to $x$; let $q_ x$ be their number, which is the degree of $W^ x_ c$. Indeed, the roots of $W_ c(z',\cdot )$ are the $q$ distinct values of $u_ c$ at the points of $A$ over $z'$, and by properness these points are close to the points of the fiber over $x'$. The values of $u_ c$ at the points close to $x$ are close to $u_ c(x'')$, where $Q^ x_ c(z',\cdot )$ does not vanish, so they are roots of $W^ x_ c(z',\cdot )$. The values at the points close to the other points of the fiber are close to values of $u_ c$ different from $u_ c(x'')$, since $c$ is admissible, whereas the roots of the Weierstrass polynomial $W^ x_ c(z',\cdot )$ in $T-u_ c(x'')$ tend to $u_ c(x'')$ as $z'\to x'$, by (9.2.1).

Let $\mathfrak g_ c\subset \mathcal O_{M,x}$ be the ideal generated by the germs at $x$ of $W_ c(z',u_ c)$ and $\delta _ cz_ k-B_{kc}(z',u_ c)$, $d<k\leq n$. These functions vanish on $A$, and they are complex linear combinations of $G_1,\ldots ,G_ N$, so $\mathfrak g_ c\subset (G_{1,x},\ldots ,G_{N,x})$. The germ of $W^ x_ c(z',u_ c)$ belongs to $\mathfrak g_ c$, because it equals $Q^ x_ c(z',u_ c)^{-1}W_ c(z',u_ c)$ and $Q^ x_ c(z',u_ c)$ is a unit at $x$. We use the following divisibility property: if $H(z',T)$ is a polynomial in $T$ whose coefficients are germs at $x'$, and $H(y',u_ c(y''))=0$ at all points $y\in A$ close to $x$ with $y'$ generic, then the germ of $H(z',u_ c)$ belongs to $\mathfrak g_ c$. Indeed, the remainder of the division of $H$ by the monic polynomial $W^ x_ c$ has degree less than $q_ x$, and for generic $z'$ near $x'$ it vanishes at the $q_ x$ distinct roots of $W^ x_ c(z',\cdot )$; so it is zero, and $H(z',u_ c)$ is a multiple of $W^ x_ c(z',u_ c)$.

For $d<k\leq n$ and the unit vector $e_ k$ we have $u_{e_ k}=z_ k$, so by Step 1 the coordinate $z_ k$ of each point of $A$ is a root of the monic polynomial $W_{e_ k}(z',\cdot )$ of degree $q$. Let $W^ x_ k$ be the Weierstrass polynomial in $T-x_ k$ given by lemma 9.2.5 for $W_{e_ k}$ at the root $x_ k$ of $W_{e_ k}(x',\cdot )$, so that $W_{e_ k}=W^ x_ kQ^ x_ k$ near $(x',x_ k)$ with $Q^ x_ k(x',x_ k)\neq 0$. Then $W^ x_ k(z',z_ k)$ vanishes at the points of $A$ close to $x$, and its degree $q_ k$ is at most $q$. Substituting $\delta _ cz_ k\equiv B_{kc}(z',u_ c)$ modulo $\mathfrak g_ c$, we get $\delta _ c^{q_ k}W^ x_ k(z',z_ k)\equiv H_ k(z',u_ c)$ modulo $\mathfrak g_ c$, where

is a polynomial in $T$. At a point $y\in A$ close to $x$ with $y'$ generic, Step 1 gives $B_{kc}(y',u_ c(y''))=\delta _ c(y')y_ k$, so $H_ k(y',u_ c(y''))=\delta _ c(y')^{q_ k}W^ x_ k(y',y_ k)=0$. By the divisibility property, $H_ k(z',u_ c)\in \mathfrak g_ c$; hence $\delta _ c^ qW^ x_ k(z',z_ k)\in \mathfrak g_ c$.

Now divide $f$ successively by the Weierstrass polynomials $W^ x_ k(z',z_ k)$ at $x$, for $k=n,\ldots ,d+1$ (proposition 9.2.2, in coordinates centered at $x$). This writes $f$ as a polynomial $\sum _\alpha e_\alpha (z')z''^\alpha $, with germs $e_\alpha $ at $x'$ and $\alpha _ k<q_ k\leq q$ for all $k$, plus multiples of the $W^ x_ k(z',z_ k)$. After multiplication by $\delta _ c^ m$, these multiples lie in $\mathfrak g_ c$, because $m\geq q$. Substituting $\delta _ cz_ k\equiv B_{kc}(z',u_ c)$, which is possible because $|\alpha |\leq (n-d)(q-1)\leq m$, we obtain $\delta _ c^ mf\equiv H(z',u_ c)$ modulo $\mathfrak g_ c$ for a polynomial $H(z',T)$ whose coefficients are germs at $x'$. Since $f$ and the elements of $\mathfrak g_ c$ vanish on $A$ near $x$, the divisibility property applies to $H$, and we conclude that

The polynomials in $c$ of degree at most $\deg _ c\delta _ c^ m$ form a finite-dimensional space, and the evaluations at points of the dense open set of admissible $c$ span its dual, since a polynomial vanishing on a nonempty open set is zero. Hence each coefficient $h_\alpha $ of $\delta _ c^ m$ is a complex linear combination of finitely many values $\delta _{c^{(l)}}^ m$ with admissible $c^{(l)}$, and $h_\alpha f\in (G_{1,x},\ldots ,G_{N,x})$. This proves (9.3.11) at points of $A$.

Step 4: points outside $A$, and conclusion. Let $x\notin A$. Choose $c$ such that $u_ c(x'')$ differs from the values of $u_ c$ at the finitely many points of $A$ over $x'$. The roots of $W_ c(x',\cdot )$ are values of $u_ c$ at points of $A$ over $x'$. Indeed, choose $z'_ j\to x'$ with $\delta (z'_ j)\neq 0$; by properness, after passing to a subsequence, the $q$ points $z^{(1)}_ j,\ldots ,z^{(q)}_ j$ of $A$ over $z'_ j$ converge to points $y^{(1)},\ldots ,y^{(q)}$ of $A$ over $x'$, so that $W_ c(x',T)=\lim _ j\prod _ i(T-u_ c(z^{(i)}_ j))=\prod _ i(T-u_ c(y^{(i)}))$. Hence $W_ c(x',u_ c(x''))\neq 0$, so one of the coefficients $G_ l$ does not vanish at $x$, and both sides of (9.3.11) are the unit ideal. The right side of (9.3.11) is the projection to the first coordinate of the kernel of the row matrix $(h,-G_1,\ldots ,-G_ N)$, which is coherent by theorem 9.2.7, lemma 9.2.6. Hence $\mathcal I_ A$ is coherent. $\square$