Theorem 6.19.2993CE3FB0997

Proof. Coherence is a local property. Fix $x\in U$ and shrink $U$ to a connected neighborhood of $x$. If all $g_ j$ vanish identically there, then $\mathcal J_{p/q}(g)$ is the zero ideal near $x$. Otherwise choose an index $i$ with $g_ i$ not identically zero, and put $U_ i=\{ g_ i\ne 0\} $. By the identity theorem $U_ i$ is dense in $U$, and by lemma 9.2.9 it is connected. Steps 1 and 2 construct the normal space described at the beginning of this section, Steps 3 to 5 show that its functions of weight $p$ are finitely generated over the holomorphic functions of $z$, and Step 6 identifies their coefficients with the germs of $\mathcal J_{p/q}(g)$.

Step 1: a cone over the blowup. Let $B\to U$ be the blowup of the ideal $(g_1,\ldots ,g_ r)$, constructed in proposition 9.6.2. Thus $B\subset U\times \mathbb P^{r-1}$, and a point $(z,[\xi ])$ of $B$ determines the line $\mathbb C\xi \subset \mathbb C^ r$. Let $\Lambda =\{ (z,[\xi ],a)\in B\times \mathbb C^ r:a\in \mathbb C\xi \} $ be the total space of this line bundle, and consider the map $\Lambda \to U\times \mathbb C^ r$, $(z,[\xi ],a)\mapsto (z,a)$. It is proper: the inverse image of a compact set is closed and contained in the product of a compact subset of $B$, namely the inverse image of its projection to $U$, with a bounded set of vectors. By theorem 9.7.2, its image $C\subset U\times \mathbb C^ r$ is analytic, and we regard it as a reduced complex space. It is invariant under the scalar multiplications $a\mapsto \lambda a$. Over $U_ i$ the blowup is the graph of $z\mapsto [g(z)]$, so the part of $C$ over $U_ i$ is the smooth family of lines

This family is dense in $C$. Indeed, the graph over $U_ i$ is dense in $B$, since $U_ i$ is dense in $U\setminus V(g)$ and the graph over $U\setminus V(g)$ is dense in $B$. Hence its part of $\Lambda $ is dense in $\Lambda $: if $(z,[\xi ],\lambda \xi )\in \Lambda $, choose graph points $(z_ k,[\xi _ k])\to (z,[\xi ])$ over $U_ i$ with representatives $\xi _ k\to \xi $, which is possible in the affine charts of $\mathbb P^{r-1}$; then $(z_ k,[\xi _ k],\lambda \xi _ k)\to (z,[\xi ],\lambda \xi )$. The image of a dense set under the continuous surjection $\Lambda \to C$ is dense.

This family is the part of $C$ over the open set $U_ i$, hence open in $C$, and it is the graph of the holomorphic map $(z,a_ i)\mapsto (a_ ig_ l(z)/g_ i(z))_{l\neq i}$ on the connected manifold $U_ i\times \mathbb C$, hence a connected complex manifold. Therefore $C$ is irreducible. Indeed, suppose that $C$ is the union of two analytic subsets different from $C$. Their traces on this connected manifold are analytic subsets covering it, so one of them has nonempty interior in it and hence, by the identity theorem, contains all of it. Since this manifold is dense in $C$ and analytic subsets are closed, that analytic subset contains $C$, a contradiction. Taking $\tau =0$, we see that $C$ contains $U_ i\times \{ 0\} $, hence $U\times \{ 0\} $, since $C$ is closed and $U_ i$ is dense in $U$.

Step 2: a root cover and its normalization. In $C\times \mathbb C_ b$ consider the analytic set defined by

and let $C'$ be the closure of its part over $U_ i$. The set $C'$ is analytic by lemma 9.3.7, applied in $U\times \mathbb C^ r\times \mathbb C_ b$ to this analytic set and to the analytic set $\{ g_ i(z)=0\} $; near each of its points it is the union of the branches of that set not contained in $\{ g_ i(z)=0\} $. The projection $C'\to C$ is finite: the monic equation bounds $b$ over compact subsets of $C$ and allows at most $q$ values of $b$. Over $U_ i$ we have $a=\tau g(z)$ and $b^ q=\tau g_ i(z)^ q$, so $b=tg_ i(z)$ with $t^ q=\tau $. Hence the part of $C'$ over $U_ i$ is the image of the injective immersion

whose inverse is $t=b/g_ i(z)$; it is a dense open connected submanifold of $C'$, so $C'$ is irreducible by the argument just given for $C$.

Let $Y\to C'$ be the normalization (theorem 9.4.6), and let $\nu \colon Y\to C$ be its composition with the projection; $\nu $ is finite. Over $U_ i$ the normalization is an isomorphism, and we identify the corresponding open subset of $Y$ with $U_ i\times \mathbb C_ t$; we call it the dense open family. Its complement in $Y$ is the zero set of $g_ i$ composed with $\nu $, an analytic subset, and it is nowhere dense. Indeed, by theorem 9.4.6 the map $Y\to C'$ is bimeromorphic: it restricts to a homeomorphism from the complement of a nowhere dense analytic subset of $Y$ onto the complement of a nowhere dense analytic subset of $C'$. The part of $C'$ over $U_ i$ is open and dense in $C'$, so its part in the second complement is dense there, and the inverse image of this part is dense in the first complement, hence in $Y$.

Step 3: finite generation over $C$. For an open set $W\subset C$ let $\mathcal F(W)$ be the ring of holomorphic functions on $\nu ^{-1}(W)$; holomorphic functions on $W$ act on it by multiplication after composition with $\nu $. We show that every point $w_0\in C$ has a neighborhood $W$ in $C$ with the following property: there are finitely many $s_1,\ldots ,s_ N\in \mathcal F(W)$ such that for every $w\in W$ and every holomorphic function $s$ on a neighborhood of $\nu ^{-1}(w)$ there are holomorphic functions $k_1,\ldots ,k_ N$ of $(z,a)$ on a neighborhood of $w$ in $U\times \mathbb C^ r$ with $s=\sum _ j(k_ j\circ \nu )s_ j$ near $\nu ^{-1}(w)$.

The tool is division by monic polynomials whose roots need not coincide. Let $\Phi (\xi ,u)=u^\mu +c_1(\xi )u^{\mu -1}+\cdots +c_\mu (\xi )$, with coefficients holomorphic on an open subset of a complex vector space, let $\xi _1$ be a point of this set, let $\Gamma $ be the union of small disjoint circles about the distinct roots of $\Phi (\xi _1,\cdot )$, and let $\Delta $ be the union of the open discs they bound. Choose a relatively compact neighborhood $A_1$ of $\xi _1$ such that $\Phi \neq 0$ on a neighborhood of $\overline{A_1}\times \Gamma $. Let $H$ be holomorphic on a neighborhood of $\overline{A_1}\times \overline\Delta $. The two integrals (9.2.3) of proposition 9.2.2, with $f$, $P$ and $w$ replaced by $H$, $\Phi $ and $u$, and taken over $\Gamma $ instead of a single circle,

define, by differentiation under the integral sign, a holomorphic function $Q$ on $A_1\times \Delta $ and, since $(\Phi (\xi ,\zeta )-\Phi (\xi ,u))/(\zeta -u)$ is a polynomial in $u$ of degree less than $\mu $ whose coefficients are polynomials in $\zeta $ and the $c_ k(\xi )$, a polynomial $R=\sum _{k<\mu }r_ k(\xi )u^ k$ with coefficients holomorphic on a neighborhood of $\overline{A_1}$. They satisfy $H=\Phi Q+R$ on $A_1\times \Delta $: for $u\in \Delta $, Cauchy's formula on the disc containing $u$ gives $H(\xi ,u)=\frac1{2\pi \mathrm i}\int _\Gamma H(\xi ,\zeta )(\zeta -u)^{-1}\, \mathrm d\zeta $, since the circles that do not surround $u$ contribute nothing, and writing $1=\bigl (\Phi (\xi ,u)+(\Phi (\xi ,\zeta )-\Phi (\xi ,u))\bigr )/\Phi (\xi ,\zeta )$ in this integral gives $\Phi (\xi ,u)Q(\xi ,u)+R(\xi ,u)$. If $H$ depends holomorphically on further variables, so do $Q$ and the $r_ k$.

Applying this successively to variables $u_1,\ldots ,u_ m$ and monic polynomials $\Phi _1(\xi ,u_1),\ldots ,\Phi _ m(\xi ,u_ m)$, with the corresponding sets $\Delta _1,\ldots ,\Delta _ m$ and a common $A_1$, we find: a function holomorphic on a neighborhood of $\overline{A_1}\times \overline{\Delta _1}\times \cdots \times \overline{\Delta _ m}$ agrees, at the points of $A_1\times \Delta _1\times \cdots \times \Delta _ m$ where all $\Phi _ l(\xi ,u_ l)$ vanish, with a polynomial $\sum _\beta r_\beta (\xi )u^\beta $, $\beta _ l<\deg \Phi _ l$, whose coefficients are holomorphic on $A_1$.

Next we embed $Y$ locally by functions that are integral over the functions of $(z,a)$. Let $y\in Y$, let $y'$ be its image in $C'$, and let $w=\nu (y)$. By corollary 9.4.7, applied to the analytic subset $C'$ of the domain $U\times \mathbb C^ r\times \mathbb C_ b$ and to the point $y'$, a neighborhood of $y$ in $Y$ is identified, compatibly with the maps to $C'$, with the closure of the graph of finitely many weakly holomorphic functions $f_1,\ldots ,f_ m$ over the regular part of the branch of $(C',y')$ that corresponds to $y$. This closure is an analytic subset of $\Omega '\times \mathbb C^ m$ for a neighborhood $\Omega '$ of $y'$ in $U\times \mathbb C^ r\times \mathbb C_ b$, embedded by the coordinates $(z,a,b)$ and by the coordinates $f_1,\ldots ,f_ m$ of $\mathbb C^ m$. Each $f_ j$ satisfies on it a monic equation whose coefficients are holomorphic functions of $(z,a,b)$ on $\Omega '$, and every holomorphic function on $Y$ near a point of this neighborhood of $y$ is the restriction of a holomorphic function of $(z,a,b,f_1,\ldots ,f_ m)$.

We first reduce the coefficients of the monic equations of the $f_ j$ to polynomials in $b$. Divide each of these coefficients, extended by zero near the roots of $b^ q-a_ ig_ i(z)^{q-1}$ different from the $b$-coordinate of $y'$, by this monic polynomial in $b$. Since the polynomial vanishes on $C'$, each coefficient agrees on $Y$ near $y$ with a polynomial in $b$ of degree less than $q$ whose coefficients are holomorphic functions of $(z,a)$ near $w$.

Let $\mathcal O_ w$ be the ring of germs at $w$ of holomorphic functions of $(z,a)$, acting through $\nu $. Reducing powers of $b$ by $b^ q=a_ ig_ i^{q-1}$ and the top power of $f_ j$ by its equation, we see that the $\mathcal O_ w$-module spanned by the germs at $y$ of the products $b^ kf_ j^ l$, with $k<q$ and $l$ less than the degree of the equation of $f_ j$, contains $1$ and is stable under multiplication by $f_ j$. Thus, if $v_1,\ldots ,v_ M$ are these products, then $f_ jv_\kappa =\sum _\lambda e_{\kappa \lambda }v_\lambda $ with $e_{\kappa \lambda }\in \mathcal O_ w$; in other words, the column vector $v=(v_\lambda )$ satisfies $(f_ jI-(e_{\kappa \lambda }))v=0$. Multiplying by the adjugate matrix of $f_ jI-(e_{\kappa \lambda })$, whose product with this matrix is $\det (f_ jI-(e_{\kappa \lambda }))\, I$ over every commutative ring, we find that $\det (f_ jI-(e_{\kappa \lambda }))$ annihilates every $v_\kappa $, in particular $1$. Expanding this determinant gives a monic equation for $f_ j$ with coefficients in $\mathcal O_ w$. Consequently, after shrinking the neighborhood of $y$, each coordinate of $u=(b,f_1,\ldots ,f_ m)$ satisfies a monic equation $\Phi _ l(z,a;u_ l)=0$ whose coefficients are holomorphic near $w$ in $U\times \mathbb C^ r$.

Now fix $w_0\in C$, and let $y_1,\ldots ,y_\ell $ be the points of the finite fiber $\nu ^{-1}(w_0)$. Choose embeddings as above on pairwise disjoint neighborhoods $N_1,\ldots ,N_\ell $ of these points, with coordinates $u^{(\iota )}$ on $N_\iota $ and monic equations $\Phi _{\iota ,l}$. Since $\nu $ is proper, there is a neighborhood $W$ of $w_0$ in $C$ with $\nu ^{-1}(W)\subset N_1\cup \cdots \cup N_\ell $, and we may assume that $W=C\cap A$ for an open set $A\subset U\times \mathbb C^ r$ on which the coefficients of all the $\Phi _{\iota ,l}$ are holomorphic. For each $\iota $ and each multi-index $\beta $ with $\beta _ l<\deg \Phi _{\iota ,l}$, let $s_{\iota ,\beta }\in \mathcal F(W)$ be the function equal to the monomial $(u^{(\iota )})^\beta $ on $N_\iota \cap \nu ^{-1}(W)$ and to zero on the other sets $N_{\iota '}\cap \nu ^{-1}(W)$.

We show that these functions have the required property. Let $w\in W$, let $s$ be holomorphic near $\nu ^{-1}(w)$, and fix $\iota $. The coordinates $u^{(\iota )}$ of the points of $\nu ^{-1}(w)\cap N_\iota $ are distinct points of the finite product of the sets of roots of the polynomials $\Phi _{\iota ,l}(w,\cdot )$, and near each of them $s$ is the restriction of a holomorphic function of $(z,a,u^{(\iota )})$. Let $H$ be the function, defined on a neighborhood of $\{ w\} $ times this finite product, that equals such an extension near the coordinates of the points of $\nu ^{-1}(w)\cap N_\iota $ and zero near the other points of the product. By the iterated division, $H$ agrees with a polynomial $\sum _\beta r_{\iota ,\beta }(z,a)(u^{(\iota )})^\beta $, with coefficients holomorphic near $w$, on the common zero set of the $\Phi _{\iota ,l}$ near these points. This zero set contains $Y\cap N_\iota $ near $\nu ^{-1}(w)\cap N_\iota $, so $s=\sum _\beta (r_{\iota ,\beta }\circ \nu )s_{\iota ,\beta }$ there. Summing over $\iota $, we obtain the claim with the finitely many functions $s_{\iota ,\beta }$.

Step 4: homogeneous components. For $x'\in U$ and $\varepsilon >0$ with $\{ |z-x'|\leq \varepsilon \} \subset U$ put

These sets form a basis of neighborhoods of $(x',0)$ in $C$, and since $C$ is closed in $U\times \mathbb C^ r$, the closure of $W_{\varepsilon '}(x')$ is a compact subset of $W_\varepsilon (x')$ for $\varepsilon '<\varepsilon $. The part $G$ of $\nu ^{-1}(W_\varepsilon (x'))$ in the dense open family consists of the points $(z,t)$ with $z\in U_ i$, $|z-x'|<\varepsilon $ and $|t^ qg_ l(z)|<\varepsilon $ for all $l$. For fixed $z$ these $t$ form a disc about $0$, and $G$ is invariant under the rotations $t\mapsto \mathrm e^{\mathrm i\vartheta }t$. Let $s\in \mathcal F(W_\varepsilon (x'))$. On $G$ we expand $s(z,t)=\sum _{e\geq 0}c_ e(z)t^ e$, where $c_ e(z)=\partial _ t^ es(z,0)/e!$ is holomorphic on $U_ i\cap \{ |z-x'|<\varepsilon \} $, and for an integer $d$ we define

integrating the power series term by term. For $\varepsilon '<\varepsilon $ the circle through a point of $G\cap \nu ^{-1}(W_{\varepsilon '}(x'))$ stays in this set, so $|P_ ds|$ is bounded there by the maximum of $|s|$ on the compact set $\nu ^{-1}(\overline{W_{\varepsilon '}(x')})$. Since every point of $\nu ^{-1}(W_\varepsilon (x'))$ lies in such a set $\nu ^{-1}(W_{\varepsilon '}(x'))$, the holomorphic function $P_ ds$ on $G$ is locally bounded near every point of $\nu ^{-1}(W_\varepsilon (x'))$. Since $Y$ is normal and the complement of $G$ in $\nu ^{-1}(W_\varepsilon (x'))$ is a nowhere dense analytic subset, lemma 9.4.4, applied on each of the open sets $\nu ^{-1}(W_{\varepsilon '}(x'))$, extends $P_ ds$ holomorphically to $\nu ^{-1}(W_\varepsilon (x'))$; the extensions on these sets agree on their overlaps by density of $G$. We denote the extension again by $P_ ds$.

We say that $s$ is homogeneous of weight $d$ if $s(z,\mathrm e^{\mathrm i\vartheta }t)=\mathrm e^{\mathrm id\vartheta }s(z,t)$ on $G$, that is, comparing the power series in $t$, if $c_ e=0$ for every $e\neq d$; then $P_ ds=s$ on $G$, hence on $\nu ^{-1}(W_\varepsilon (x'))$ by density of $G$. In particular, $P_ ds$ is homogeneous of weight $d$, a function homogeneous of negative weight vanishes, and since $a=t^ qg(z)$ on $G$, the function $a^\alpha $ is homogeneous of weight $q|\alpha |$. The operators $P_ d$ commute with restriction to smaller sets $W_{\varepsilon _1}(x'')\subset W_\varepsilon (x')$.

Let $k$ be the restriction to $W_\varepsilon (x')$ of a holomorphic function $K$ on $\{ |z-x'|<\varepsilon \} \times \{ |a_ l|<\varepsilon \text{ for all }l\} $, with Taylor expansion $K(z,a)=\sum _\alpha K_\alpha (z)a^\alpha $ in $a$. At a point of $G$ we have $k(\nu (z,\mathrm e^{\mathrm i\vartheta }t))=\sum _\alpha K_\alpha (z)\mathrm e^{\mathrm iq|\alpha |\vartheta }a^\alpha $ with $a=t^ qg(z)$, and $\sum _\alpha |K_\alpha (z)a^\alpha |<\infty $, while $s$ is bounded on the circle. Integrating term by term, we obtain, first on $G$ and then on $\nu ^{-1}(W_\varepsilon (x'))$ by density,

for every integer $p$, where $K_\alpha (z)$ and $a^\alpha $ are composed with $\nu $; the terms with $q|\alpha |>p$ vanish because $P_ es=0$ for $e<0$.

Step 5: functions of weight $p$. Let $W$ and $s_1,\ldots ,s_ N$ be as in Step 3 for $w_0=(x,0)$, and choose $\varepsilon >0$ with $\{ |z-x|\leq \varepsilon \} \subset U$ and $W_\varepsilon (x)\subset W$. Let $|x'-x|<\varepsilon /2$, let $0<\varepsilon _1\leq \varepsilon /2$, and let $h\in \mathcal F(W_{\varepsilon _1}(x'))$ be homogeneous of weight $p$. Then $W_{\varepsilon _1}(x')\subset W_\varepsilon (x)$. By Step 3, $h=\sum _ j(k_ j\circ \nu )s_ j$ near $\nu ^{-1}(x',0)$, with $k_ j$ holomorphic near $(x',0)$ in $U\times \mathbb C^ r$. Since $\nu $ is proper, after shrinking $\varepsilon _1$ this identity holds on $\nu ^{-1}(W_{\varepsilon _1}(x'))$, and each $k_ j$ is holomorphic on $\{ |z-x'|<\varepsilon _1\} \times \{ |a_ l|<\varepsilon _1\text{ for all }l\} $. We apply $P_ p$ to the identity. Since $P_ ph=h$, Step 4 gives

where the $k_{j,\alpha }$ are the Taylor coefficients of $k_ j$ in $a$. Thus every function of weight $p$ near $\nu ^{-1}(x',0)$ is a combination, with coefficients holomorphic in $z$ near $x'$, of the finitely many functions

which are holomorphic on $\nu ^{-1}(W_\varepsilon (x))$ and homogeneous of weight $p$.

Step 6: identification with $\mathcal J_{p/q}(g)$. By Step 4, each function $a^\alpha P_ es_ j$ of Step 5 equals $f_{j,\alpha }(z)t^ p$ on the dense open family, with $f_{j,\alpha }$ holomorphic on $U_ i\cap \{ |z-x|<\varepsilon \} $. Let $M$ be the maximum of the moduli of these finitely many functions on the compact set $\nu ^{-1}(L)$, where $L=\{ (z,a)\in C:|z-x|\leq \varepsilon /2,\ |a_ l|\leq \varepsilon /2\text{ for all }l\} \subset W_\varepsilon (x)$. For $z$ with $|z-x|<\varepsilon /2$ and $g_ i(z)\neq 0$, choose $t$ with

Then $|t^ qg_ l(z)|\leq \varepsilon /2$ for all $l$, so $(z,t)\in \nu ^{-1}(L)$, and $|f_{j,\alpha }(z)||t|^ p\leq M$, that is,

In particular $f_{j,\alpha }$ is bounded near $\{ g_ i=0\} $, so it extends holomorphically to the ball $\{ |z-x|<\varepsilon /2\} $ by lemma 9.2.9, and the inequality persists on $\{ g_ i=0\} $, since both sides are continuous and $U_ i$ is dense. Hence the finitely many functions $f_{j,\alpha }$ belong to $\mathcal J_{p/q}(g)$ on this ball.

Conversely, let $f$ be a germ of $\mathcal J_{p/q}(g)$ at a point $x'$ with $|x'-x|<\varepsilon /2$, say $|f|\leq C_ f|g|^{p/q}$ on $\{ |z-x'|<\rho \} $, and let $0<\varepsilon _1\leq \min (\rho ,\varepsilon /2)$. On the part of $\nu ^{-1}(W_{\varepsilon _1}(x'))$ in the dense open family, the function $f(z)t^ p$ satisfies

It is therefore bounded, and since $Y$ is normal it extends holomorphically to $\nu ^{-1}(W_{\varepsilon _1}(x'))$ by lemma 9.4.4; the extension $h$ is homogeneous of weight $p$. By Step 5, after shrinking $\varepsilon _1$, $h$ is a combination of the functions $a^\alpha P_ es_ j$ with coefficients $k_{j,\alpha }(z)$ holomorphic near $x'$. Comparing the coefficients of $t^ p$ on the dense open family gives $f=\sum k_{j,\alpha }f_{j,\alpha }$ on $U_ i$ near $x'$, hence near $x'$, since both sides are continuous and $U_ i$ is dense. Thus the functions $f_{j,\alpha }$ generate $\mathcal J_{p/q}(g)$ on $\{ |z-x|<\varepsilon /2\} $, and $\mathcal J_{p/q}(g)$ is coherent by theorem 9.2.7, lemma 9.2.6.

Finally, if $v=(p/q)\log \sum _ j|g_ j|^2+O(1)$ is a local presentation of $v$, then $\mathcal I_\infty (v)=\mathcal J_{p/q}(g)$ on its domain. Since coherence is a local property, $\mathcal I_\infty (v)$ is coherent. (If $g(x)\neq 0$, both ideals are the unit ideal near $x$; in dimension zero every finite weight gives the unit ideal.) $\square$