Lemma 6.8.4 (Strong Noetherian property)4C0C502048AA

Proof. The assertion on compact sets follows from the local one by covering a compact set with finitely many of the neighborhoods $U$ and taking the largest of the corresponding indices. To prove the local assertion at a point $x$, note that $\mathcal F$, being locally finitely generated, is near $x$ the image of a surjective morphism $\phi \colon \mathcal O^ r\to \mathcal F$ defined by finitely many generators. The inverse images $\phi ^{-1}(\mathcal F_ j)$ form an increasing sequence of subsheaves of $\mathcal O^ r$, and they are coherent by lemma 9.2.6, being the kernels of the morphisms $\mathcal O^ r\to \mathcal F/\mathcal F_ j$ between coherent sheaves; here $\mathcal O^ r$ is coherent by theorem 9.2.7. Since $\mathcal F_ j=\phi (\phi ^{-1}(\mathcal F_ j))$, the sequence $(\mathcal F_ j)$ is stationary wherever $(\phi ^{-1}(\mathcal F_ j))$ is.

We may therefore assume that $\mathcal F=\mathcal O^ r$, and we reduce to the case of ideals, $r=1$, by induction on $r$. Let $p\colon \mathcal O^ r\to \mathcal O$ be the projection to the last coordinate. The images $p(\mathcal F_ j)$ form an increasing sequence of coherent ideals, and the kernels $\mathcal F_ j\cap (\mathcal O^{r-1}\oplus 0)$ of $p|_{\mathcal F_ j}$ form an increasing sequence of coherent subsheaves of $\mathcal O^{r-1}$, by lemma 9.2.6. Suppose that both sequences are constant for $j\geq j_0$ on a neighborhood $U$ of $x$. If $\sigma $ is a germ of $\mathcal F_ j$ at a point of $U$, with $j\geq j_0$, choose a germ $\sigma '$ of $\mathcal F_{j_0}$ with $p(\sigma ')=p(\sigma )$; then $\sigma -\sigma '$ lies in the kernel for the index $j$, which equals the kernel for $j_0$, so $\sigma \in \mathcal F_{j_0}$. Thus $\mathcal F_ j=\mathcal F_{j_0}$ on $U$.

For ideals $\mathcal I_1\subset \mathcal I_2\subset \cdots $ we argue by induction on the dimension $n$ of $X$ at $x$; by the reduction just made, the induction hypothesis applies to coherent subsheaves of $\mathcal O^\mu $ in dimension $n-1$. In dimension zero, $x$ is an open point of $X$ and the only ideals of $\mathcal O_{X,x}=\mathbb C$ are $0$ and $\mathbb C$, so the increasing sequence $(\mathcal I_{j,x})$ is eventually constant, and $U=\{ x\} $ has the required property.

Let $n\geq 1$, and let $W$ be a connected coordinate neighborhood of $x$. If all stalks $\mathcal I_{j,x}$ vanish, then all $\mathcal I_ j$ vanish on $W$ by lemma 6.8.2, and there is nothing to prove. Otherwise some $\mathcal I_{j_1}$ has a nonzero germ $g$ at $x$. After a linear change of coordinates $z=(z',w)$ centered at $x$, in which $g$ is distinguished in $w$ (section 9.2), the Weierstrass preparation theorem (theorem 5.5.2) shows that $g$ is a unit times a Weierstrass polynomial $P(z',w)$ of some degree $\mu $, so that the germ of $P$ belongs to $\mathcal I_{j_1,x}$. If $\mu =0$, then $g$ is a unit, the section $1=g\cdot g^{-1}$ belongs to $\mathcal I_{j_1}$ near $x$, and $\mathcal I_ j$ is the unit ideal near $x$ for all $j\geq j_1$.

Let $\mu \geq 1$. By (9.2.1) the roots of $P(z',\cdot )$ tend to $0$ as $z'\to 0$, so there is a product $D=W'\times \Delta \subset W$ of a polydisc $W'\subset \mathbb C^{n-1}$ centered at $0$ and a disc $\Delta =\{ |w|<r\} $ such that the coefficients of $P$ are holomorphic on $W'$, $P$ is a section of $\mathcal I_{j_1}$ over $D$, and all roots of $P(z',\cdot )$, $z'\in W'$, lie in $\{ |w|\leq \rho \} $ for some $\rho <r$. For $j\geq j_1$ the ideals $\mathcal I_ j|_ D$ contain $P$, and by lemma 6.8.3(3) the sheaves $\mathcal M(\mathcal I_ j|_ D)$ form an increasing sequence of coherent subsheaves of $\mathcal O_{W'}^\mu $. By the induction hypothesis, they are constant for $j\geq j_0$ on a neighborhood $V'$ of $0$, for some $j_0\geq j_1$, and lemma 6.8.3(3) gives $\mathcal I_ j=\mathcal I_{j_0}$ on the neighborhood $V'\times \Delta $ of $x$ for all $j\geq j_0$. This proves the local assertion.

Finally, let $h_1,h_2,\ldots $ be holomorphic functions on $X$. The ideals $\mathcal J_ j$ generated by $h_1,\ldots ,h_ j$ are locally finitely generated subsheaves of $\mathcal O$, hence coherent by theorem 9.2.7, lemma 9.2.6, and the stalk at $y$ of the ideal $\mathcal J$ generated by all the $h_ l$ is the union of the stalks $\mathcal J_{j,y}$. By the local assertion, every point has a neighborhood $U$ and an index $j_0$ with $\mathcal J_ j=\mathcal J_{j_0}$ on $U$ for $j\geq j_0$; then $\mathcal J=\mathcal J_{j_0}$ on $U$. Hence $\mathcal J$ is coherent, and near each point it is generated by $h_1,\ldots ,h_{j_0}$. $\square$