Theorem 9.2.7 (Oka's coherence theorem)F28642698B29

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. $\square$