9.2 Division and finite analytic modulesD4E6B7455291

Proof. Polydiscs as in the statement exist with arbitrarily small radii, because the roots of $P(z',\cdot )$ tend to $0$ as $z'\to 0$, by (9.2.1).

Let $\Delta '\times \Delta $ be such a polydisc and $f$ a holomorphic function on it. For $\rho <s<r$ and $|w|<s$ define $q$ and $R$ by (9.2.3). For fixed $z'$ and $w$, both integrands are holomorphic in $\zeta $ on the annulus $\{ \rho <|\zeta |<r\} $, except for the pole of the first one at $\zeta =w$; by Cauchy's theorem the integrals do not depend on $s$ as long as $|w|<s$, and the second one does not depend on $s\in (\rho ,r)$ at all. Hence $q$ and $R$ are defined on $\Delta '\times \Delta $, and they are holomorphic by differentiation under the integral sign. The difference quotient $(P(z',\zeta )-P(z',w))/(\zeta -w)$ is a polynomial in $w$ of degree $m-1$ whose coefficients are polynomials in $\zeta $ and in the $a_ j(z')$; hence $R$ is a polynomial in $w$ of degree less than $m$ with holomorphic coefficients on $\Delta '$. Adding the two integrals, we obtain $Pq+R=(2\pi \mathrm i)^{-1}\int _{|\zeta |=s}f(z',\zeta )(\zeta -w)^{-1}\, \mathrm d\zeta =f$ by Cauchy's formula.

For the estimate, let $f$ be bounded, and fix $s_0=(\rho +r)/2$. For $|\zeta |\geq s_0$ we have $|P(z',\zeta )|=\prod _ l|\zeta -\lambda _ l(z')|\geq (s_0-\rho )^ m$, where $\lambda _ l(z')$ are the roots, and the coefficients of the difference quotient are bounded for $z'\in \Delta '$, $|\zeta |=s_0$ and $|w|<r$, since the $a_ j$ are bounded on $\Delta '$. The integral over $|\zeta |=s_0$ therefore gives $|R|\leq C_1\sup |f|$ on $\Delta '\times \Delta $. For $s_0\leq |w|<r$ we have $|q|=|f-R|/|P|\leq (1+C_1)(s_0-\rho )^{-m}\sup |f|$, and the maximum principle for the holomorphic functions $q(z',\cdot )$ on the discs $\{ |w|\leq s\} $, $s_0\leq s<r$, extends this bound to all of $\Delta '\times \Delta $.

For uniqueness, suppose that $Pq+R=0$ on a polydisc as above, with $R$ a polynomial in $w$ of degree less than $m$. For each $z'$, the polynomial $R(z',\cdot )=-P(z',\cdot )q(z',\cdot )$ vanishes at the $m$ roots of $P(z',\cdot )$, counted with multiplicity, all of which lie in $\Delta $. Its degree is less than $m$, so $R(z',\cdot )=0$. Then $P(z',\cdot )q(z',\cdot )=0$, and since $P(z',\cdot )$ has only finitely many zeros, the continuous function $q(z',\cdot )$ vanishes on a dense set, hence everywhere. Finally, every germ is represented by a bounded holomorphic function on some polydisc as above, and an identity between germs holds on some such polydisc; so the assertions for germs follow. $\square$

Proof. We first recall three algebraic facts about a commutative ring $\Lambda $ all of whose ideals are finitely generated. Every submodule $N$ of $\Lambda ^ r$ is finitely generated, by induction on $r$: the projection of $N$ to the last coordinate is an ideal; choose elements of $N$ whose last coordinates generate it, and add generators of $N\cap (\Lambda ^{r-1}\times \{ 0\} )$, which exist by the induction hypothesis. A finitely generated module is a quotient of some $\Lambda ^ r$, and the inverse image of a submodule is a submodule of $\Lambda ^ r$; hence submodules of finitely generated modules are finitely generated. Finally, the union of an increasing sequence of submodules is a submodule; its finitely many generators already lie in one member of the sequence, which is therefore stationary from then on.

It remains to prove that the ideals of $\mathcal O_ n$ are finitely generated, which we do by induction on $n$. The ring $\mathcal O_0=\mathbb C$ is a field. Let $n\geq 1$ and let $I\subset \mathcal O_ n$ be an ideal. If $I=0$ or $I$ contains a unit, it is generated by one element. Otherwise choose a nonzero germ in $I$, and coordinates in which it is distinguished in $w$. By the preparation theorem it is a unit times a Weierstrass polynomial $P$ of some degree $m\geq 1$, and $P\in I$. Divide every $f\in I$ by $P$ using proposition 9.2.2; the remainder $R_ f=f-Pq$ again belongs to $I$. The remainders of degree less than $m$ that lie in $I$ form an $\mathcal O_{n-1}$-submodule of $\mathcal O_{n-1}[w]$, which we identify, through their coefficients, with a submodule of $\mathcal O_{n-1}^ m$. By the induction hypothesis and the first paragraph, it is generated by finitely many remainders $R_1,\ldots ,R_ k$. Then $P,R_1,\ldots ,R_ k$ generate $I$. $\square$

Proof. In the coordinates $(z'-x',w-w_0)$ centered at $x$, the function $w\mapsto P(x',w)$ has a zero of order $\mu '$ at $w_0$. By the Weierstrass preparation theorem (theorem 5.5.2), $P$ is, near $x$, a nowhere vanishing holomorphic function $U$ times a Weierstrass polynomial $P_ x$ of degree $\mu '$ in $w-w_0$. Since $P_ x$ is monic in $w$, Euclidean division of the polynomial $P$ by $P_ x$, with coefficients holomorphic near $x'$, gives $P=P_ xQ+R$, where $Q$ is a monic polynomial of degree $\mu -\mu '$ in $w$ and $R$ is a polynomial in $w$, equivalently in $w-w_0$, of degree less than $\mu '$. The identities $P=P_ xQ+R$ and $P=P_ xU+0$ are two divisions of the germ of $P$ at $x$ by $P_ x$; by the uniqueness in proposition 9.2.2, applied in the coordinates centered at $x$, we get $Q=U$ near $x$ and $R=0$. Hence $Q_ x=Q$ has the required properties, with $Q_ x(x)=U(x)\neq 0$. $\square$

Proof. A locally finitely generated submodule of a coherent sheaf is coherent, because its sections have the same relations when they are regarded as sections of the larger sheaf. The image of a morphism from a coherent sheaf is locally generated by the images of local generators of the source, so the image of a morphism between coherent sheaves is coherent. For the kernel of a morphism $\phi \colon \mathcal F\to \mathcal G$ between coherent sheaves, choose local generators $F_1,\ldots ,F_ p$ of $\mathcal F$ and local generators $R_ j=(R_{j1},\ldots ,R_{jp})$ of the relations of $\phi (F_1),\ldots ,\phi (F_ p)$. If $\sum _ ia_ iF_ i$ lies in the kernel, then $(a_ i)$ is a relation of the $\phi (F_ i)$, hence a combination of the $R_ j$; so the sections $\sum _ iR_{ji}F_ i$ generate the kernel, which is therefore coherent.

Let $\mathcal F\subset \mathcal S$ be coherent sheaves and consider $\mathcal G=\mathcal S/\mathcal F$; this covers cokernels, because images are coherent. Local generators of $\mathcal S$ map to local generators of $\mathcal G$. Given sections $G_1,\ldots ,G_ p$ of $\mathcal G$, lift them locally to sections $S_1,\ldots ,S_ p$ of $\mathcal S$, and choose local generators $F_1,\ldots ,F_ k$ of $\mathcal F$. A tuple $(a_ i)$ is a relation of the $G_ i$ exactly when $\sum _ ia_ iS_ i\in \mathcal F$, that is, when $(a,b)$ is a relation of $(S_1,\ldots ,S_ p,F_1,\ldots ,F_ k)$ in $\mathcal S$ for some $b$. Hence the projections to the first $p$ coordinates of local generators of the latter relations generate the relations of the $G_ i$, and $\mathcal G$ is coherent.

Now let $\mathcal F$ and $\mathcal G$ be coherent. Local lifts of generators of $\mathcal G$, together with local generators of $\mathcal F$, generate $\mathcal S$. Given sections $S_1,\ldots ,S_ p$ of $\mathcal S$, let $R_1,\ldots ,R_ l$ generate the relations of their images in $\mathcal G$. The combinations $H_ j=\sum _ iR_{ji}S_ i$ lie in $\mathcal F$; let $Q_1,\ldots ,Q_ k$ generate their relations. Every relation $(a_ i)$ of the $S_ i$ is first a relation of their images, so $a=\sum _ jc_ jR_ j$, and then $\sum _ jc_ jH_ j=\sum _ ia_ iS_ i=0$, so $c$ is a combination of $Q_1,\ldots ,Q_ k$. Hence the tuples $\sum _ jQ_{\nu j}R_ j$, $1\leq \nu \leq k$, which are relations of the $S_ i$, generate all of them, and $\mathcal S$ is coherent. The remaining case of the exact sequence, with $\mathcal S$ and $\mathcal G$ coherent, is the kernel case. A finite direct sum is an extension $0\to \mathcal F\to \mathcal F\oplus \mathcal G\to \mathcal G\to 0$. Finally, if $\mathcal A$ is coherent, so is $\mathcal A^ r$, and the last assertion follows from the first sentence of the proof. $\square$

Proof. The sheaf $\mathcal O_ M$ is generated by the constant $1$, and the statement is local, so we must show that the relations of finitely many holomorphic functions $F_1,\ldots ,F_ q$ near a point of $\mathbb C^ n$, which we take to be the origin, are locally finitely generated. We argue by induction on $n$: we reduce to relations that are polynomials in one variable of bounded degree, and apply the induction hypothesis to their coefficients. For $n=0$ the assertion is finite-dimensional linear algebra. Let $n\geq 1$. If the germ of some $F_ j$ at $0$ is zero, then $F_ j$ vanishes on a connected neighborhood, and the relations of the $F_ i$ are those of the remaining functions, with an arbitrary $j$th coordinate; so we may assume that no germ $F_ j$ is zero (if all are zero, every tuple is a relation). Choose coordinates $(z',w)$ in which all $F_ j$ are distinguished in $w$. By the preparation theorem, on a small polydisc each $F_ j$ is a holomorphic unit $u_ j$ times a Weierstrass polynomial; since $(g_ j)$ is a relation of the $F_ j$ exactly when $(u_ jg_ j)$ is a relation of the Weierstrass polynomials, we may assume that each $F_ j$ is a Weierstrass polynomial in $w$ whose coefficients are holomorphic on a polydisc $\Delta '\subset \mathbb C^{n-1}$. Renumbering, let $F_ q$ have the maximal degree $\mu $.

We claim that at every point $x=(x',w_0)$ of the polydisc, the stalk of relations is generated by relations whose coordinates are polynomials in $w$ of degree at most $\mu $ with coefficients holomorphic near $x'$. If $F_ q(x)\neq 0$, then for a relation $g=(g_1,\ldots ,g_ q)$ we have $g_ q=-\sum _{j<q}g_ jF_ j/F_ q$, so $g=\sum _{j<q}(g_ j/F_ q)(F_ qe_ j-F_ je_ q)$ is a combination of the elementary relations $F_ qe_ j-F_ je_ q$, which are polynomial of degree at most $\mu $. Suppose that $F_ q(x)=0$, and let $\mu '\geq 1$ be the order of the root $w_0$ of $F_ q(x',\cdot )$. By lemma 9.2.5, near $x$ the monic polynomial $F_ q$ is the product of a Weierstrass polynomial $P_ x$ of degree $\mu '$ in $w-w_0$ and a monic polynomial $Q_ x$ in $w$ of degree $\mu -\mu '$:

Let $g$ be a relation at $x$. Dividing each $g_ j$, $j<q$, by $P_ x$ (proposition 9.2.2) and writing $P_ x=F_ qQ_ x^{-1}$, we obtain $g_ j=F_ qt_ j+r_ j$ with $\deg _ wr_ j<\mu '$. Subtracting $\sum _{j<q}t_ j(F_ qe_ j-F_ je_ q)$ from $g$, we obtain a relation whose first $q-1$ coordinates are $r_1,\ldots ,r_{q-1}$; we call its last coordinate $r_ q$. Then

The right-hand side is a polynomial in $w$; dividing it by $P_ x$ as a polynomial and comparing with the uniqueness of division at $x$, we find that $Q_ xr_ q$ is a polynomial of degree less than $\mu $. Each $Q_ xr_ j$, $j<q$, is also a polynomial of degree at most $(\mu -\mu ')+(\mu '-1)<\mu $. Thus $(Q_ xr_1,\ldots ,Q_ xr_ q)$ is a polynomial relation of degree less than $\mu $, and $(r_ j)$ is its product with the unit $Q_ x^{-1}$. This proves the claim.

A tuple of polynomials $g_ j=\sum _{k=0}^{\mu }c_{jk}(z')w^ k$ is a relation exactly when the $2\mu +1$ coefficients of the polynomial $\sum _ jF_ jg_ j$ vanish. These coefficients are given by a holomorphic matrix acting on the coefficient vectors $(c_{jk})$, with entries formed from the coefficients of the $F_ j$. By the induction hypothesis and lemma 9.2.6, the kernel of this matrix is coherent, so it has finitely many generators on a smaller polydisc $\Delta '_1$, generating its stalk at every point of $\Delta '_1$. The corresponding polynomial tuples are relations of the $F_ j$, and they generate, over the rings of germs on the base, all polynomial relations of degree at most $\mu $ at every point over $\Delta '_1$. By the claim, they generate all relations at every point of the polydisc over $\Delta '_1$. This proves that $\mathcal O_ M$ is coherent. The last assertion follows from lemma 9.2.6, since the kernel of a morphism between the coherent sheaves $\mathcal O_ U^ q$ and $\mathcal O_ U^ p$ is coherent. $\square$

Proof. The interior of $A$ is open, and we show that it is closed in $M$. Let $a\in M$ be a limit of interior points of $A$, and choose a connected coordinate neighborhood $U$ of $a$ and functions $g_1,\ldots ,g_ k$ with $A\cap U=\{ g_1=\cdots =g_ k=0\} $. Each $g_ i$ vanishes on a nonempty open subset of $U$, hence on $U$ by the identity theorem; so $U\subset A$, and $a$ is an interior point. Since $M$ is connected and $A\neq M$, the interior of $A$ is empty. In the second assertion, if all $g_ i$ vanished identically on $U$, the nonempty open set $U$ would lie in $A$, contradicting the first. For the third, take for $U$ a connected coordinate ball on which $A$ is the common zero set of finitely many holomorphic functions, and for $g$ one of them that is not identically zero, which exists by the second assertion. $\square$

Proof. Let $f$ be holomorphic on $M\setminus A$ and locally bounded near $A$. If $\dim M=0$, then $A=\varnothing $ and there is nothing to prove. Fix $a\in A$. By lemma 9.2.8 there is a holomorphic function $g\not\equiv 0$ on a connected neighborhood of $a$ vanishing on $A$; choose linear coordinates $(z',w)$ centered at $a$ in which $g$ is distinguished in $w$. By the Weierstrass preparation theorem and (9.2.1), there is a polydisc $\Delta '\times \Delta $ centered at $a$, with $\Delta =\{ |w|<r\} $, on which $g$ is a unit times a Weierstrass polynomial $W$ and $f$ is bounded off $A$, such that all roots of $W(z',\cdot )$, $z'\in \Delta '$, have modulus at most some $\rho <r$. For fixed $z'\in \Delta '$, the function $f(z',\cdot )$ is holomorphic on $\Delta $ outside the finitely many points of $A$ in the fiber, which are roots of $W(z',\cdot )$, and it is bounded near them; by the removable singularity theorem in one variable it extends holomorphically to $\Delta $. For $\rho <s<r$ the function

is holomorphic, by differentiation under the integral sign, since $f$ is holomorphic near $\Delta '\times \{ |\zeta |=s\} $, a set disjoint from $A$. By Cauchy's formula for the extended functions $f(z',\cdot )$, it agrees with $f$ outside $A$. Letting $s\uparrow r$, we obtain a holomorphic extension of $f$ to $\Delta '\times \Delta $. Since $A$ is nowhere dense, two holomorphic extensions on overlapping open sets agree, being continuous functions that coincide on a dense subset; so the local extensions glue to a holomorphic function on $M$, and this extension is unique.

If $M\setminus A$ were disconnected, the function equal to $0$ on one nonempty union of its connected components and to $1$ on the others would be bounded and holomorphic on $M\setminus A$. Its holomorphic extension is continuous and takes values in $\{ 0,1\} $ on the dense set $M\setminus A$, hence everywhere; since $M$ is connected, it is constant, a contradiction.

Now let $b,c\in \mathcal O_ n$, $c\neq 0$, with $|b|\leq C|c|$ on a connected neighborhood $U$ of the origin on which both are defined. The set $\{ c=0\} $ is a proper analytic subset of $U$, and the holomorphic function $b/c$ on $U\setminus \{ c=0\} $ is bounded by $C$. By the first part it extends to a holomorphic function $h$ on $U$, and $ch=b$ on the dense set $U\setminus \{ c=0\} $, hence on $U$; so $b\in c\, \mathcal O_ n$. Finally, let $b/c\in \operatorname {Frac}(\mathcal O_ n)$, with $c\neq 0$, satisfy a monic equation $(b/c)^ m+a_1(b/c)^{m-1}+\cdots +a_ m=0$ with $a_ j\in \mathcal O_ n$. Multiplying by $c^ m$, we obtain the identity $b^ m+a_1b^{m-1}c+\cdots +a_ mc^ m=0$ between germs, which holds on a connected neighborhood of $0$ on which the $a_ j$ are bounded. At the points of this neighborhood where $c\neq 0$, the number $b/c$ is a root of a monic polynomial with bounded coefficients, so $|b/c|$ is bounded there by (9.2.1); at the points where $c=0$ the identity gives $b^ m=0$. Hence $|b|\leq C|c|$ on this neighborhood for some constant $C$, and by what we have just proved, $b/c\in \mathcal O_ n$, which proves that $\mathcal O_ n$ is integrally closed. $\square$