[proofplan] The assertion is local on $X$, so we work in a coordinate ball in $\mathbb C^n$. The main analytic input is Hörmander's weighted $L^2$ existence theorem for the $\bar\partial$-operator, applied with a logarithmic auxiliary weight that forces high-order vanishing at a chosen point. This gives a finite-jet approximation statement for germs in the multiplier ideal. The Noetherian property of the local ring of holomorphic germs and Krull's intersection theorem then upgrade finite-jet generation to generation of the full stalk, and the same local argument gives coherence of the sheaf. [/proofplan]

[step:Reduce coherence to a local statement in a coordinate ball]Coherence of an analytic sheaf is local on the base. Fix a point $p \in X$. Choose a holomorphic coordinate chart $(U,\psi)$ with $p \in U$ and \begin{align*} \psi: U &\to \psi(U) \subset \mathbb C^n \end{align*} a biholomorphism onto an [open set](/page/Open%20Set). After replacing $U$ by a smaller coordinate neighbourhood, we may assume that $\psi(p)=0$ and that there are Euclidean balls \begin{align*} B' := B(0,r) \subset\subset B := B(0,R) \subset \psi(U) \end{align*} with $0<r<R$. Let $\mathcal L^{2n}$ denote Lebesgue measure on the real Euclidean space underlying $\mathbb C^n$. Let \begin{align*} \varphi: B &\to [-\infty,\infty),\\ z &\mapsto \phi(\psi^{-1}(z)) \end{align*} denote the plurisubharmonic weight in these coordinates. Since holomorphic coordinate changes have smooth non-vanishing Jacobian determinants, local integrability of $|f|^2 e^{-\phi}$ is equivalent to local integrability of $|f\circ\psi^{-1}|^2 e^{-\varphi}$ with respect to $\mathcal L^{2n}$. Thus it is enough to prove that the multiplier ideal sheaf \begin{align*} \mathcal I(\varphi) \subset \mathcal O_B \end{align*} is coherent near $0$.[/step]

[guided]The theorem is local because coherence is checked in neighbourhoods of each point. We therefore fix $p \in X$ and choose holomorphic coordinates around $p$. The coordinate map is declared explicitly: \begin{align*} \psi: U &\to \psi(U) \subset \mathbb C^n. \end{align*} We shrink $U$ so that the image contains two concentric Euclidean balls \begin{align*} B' := B(0,r) \subset\subset B := B(0,R). \end{align*} Let $\mathcal L^{2n}$ denote Lebesgue measure on the real Euclidean space underlying $\mathbb C^n$. The weight in coordinates is the map \begin{align*} \varphi: B &\to [-\infty,\infty),\\ z &\mapsto \phi(\psi^{-1}(z)). \end{align*} Because $\psi$ is biholomorphic, its real Jacobian determinant is smooth and nowhere zero on compact subsets of $U$. Hence multiplication by the Jacobian factor changes local integrability only by a locally bounded positive factor. Therefore $|f|^2 e^{-\phi}$ is locally integrable near $p$ if and only if $|f\circ \psi^{-1}|^2 e^{-\varphi}$ is locally integrable near $0$. It remains to prove the theorem for a plurisubharmonic function on a ball in $\mathbb C^n$.[/guided]

[step:State the local finite generation criterion]Let $\mathcal O_{B,0}$ be the local ring of holomorphic germs at $0$, and let $\mathfrak m_0 \subset \mathcal O_{B,0}$ be its maximal ideal. More generally, for each $a \in B$, let $\mathcal O_{B,a}$ denote the local ring of holomorphic germs at $a$, and let $\mathfrak m_a \subset \mathcal O_{B,a}$ denote its maximal ideal of germs vanishing at $a$. Define the stalk ideal \begin{align*} J_0 := \mathcal I(\varphi)_0 = \left\{ f_0 \in \mathcal O_{B,0} : |f|^2 e^{-\varphi} \in L^1_{\mathrm{loc}} \text{ near } 0 \right\}, \end{align*} where $f$ is any holomorphic representative of the germ $f_0$. We shall use the following local finite generation lemma. [claim:Local $L^2$ finite generation lemma] There exist a neighbourhood $V \subset B'$ of $0$ and finitely many holomorphic functions \begin{align*} g_1,\dots,g_N \in \mathcal O_B(B) \end{align*} such that $g_j \in \mathcal I(\varphi)(B')$ for every $1 \le j \le N$, and for every point $a \in V$ the germs $(g_1)_a,\dots,(g_N)_a$ generate $\mathcal I(\varphi)_a$ as an ideal of $\mathcal O_{B,a}$. [/claim] [proof] We prove the lemma from two standard analytic inputs: [Hörmander's weighted $L^2$ existence theorem for $\bar\partial$](/page/Hormander%20L2%20Estimates) on bounded pseudoconvex domains, and the [strong Noetherian theorem for Banach spaces of holomorphic functions](/page/Strong%20Noetherian%20Theorem). The Noetherian input is not Nadel coherence itself; it is a functional-analytic strengthening of the Weierstrass-Noetherian argument for ideals generated by a Banach family of holomorphic functions. The precise form used here is the following: if $E$ is a [Banach space](/page/Banach%20Space) of holomorphic functions on $B$ such that, for every $B''\subset\subset B$, the restriction map $E\to\mathcal O_B(B'')$ is continuous when $\mathcal O_B(B'')$ has the compact-open topology, then the analytic ideal sheaf generated by the germs of all elements of $E$ is locally generated by finitely many elements of $E$. Thus the theorem produces one finite list of generators on a neighbourhood, rather than a different finite list at each stalk. Let $\mathcal L^{2n}$ denote Lebesgue measure on the real Euclidean space underlying $\mathbb C^n$. Define \begin{align*} H(B,\varphi) := \left\{ f \in \mathcal O_B(B) : \int_B |f(z)|^2 e^{-\varphi(z)}\,d\mathcal L^{2n}(z) < \infty \right\}. \end{align*} If $\varphi\equiv -\infty$ on a non-empty open subset of the connected ball $B$, then every element of $H(B,\varphi)$ vanishes on that open subset and hence vanishes on $B$ by the identity theorem; in this case $H(B,\varphi)=\{0\}$, which is a [Hilbert space](/page/Hilbert%20Space). Otherwise $\varphi$ is locally bounded above on $B$, so for every compact set $K\subset\subset B$ there is a finite constant $C_K\in\mathbb R$ with $\varphi\le C_K$ on $K$. Hence \begin{align*} \int_K |f(z)|^2\,d\mathcal L^{2n}(z) \le e^{C_K}\int_B |f(z)|^2 e^{-\varphi(z)}\,d\mathcal L^{2n}(z) \end{align*} for every $f\in H(B,\varphi)$. This estimate shows that convergence in the weighted norm implies local $L^2$ convergence. Equipped with the inner product \begin{align*} (f,h)_{H(B,\varphi)} := \int_B f(z)\overline{h(z)} e^{-\varphi(z)}\,d\mathcal L^{2n}(z), \end{align*} $H(B,\varphi)$ is therefore closed in the weighted $L^2$ space: a weighted-$L^2$ limit of holomorphic functions is a local $L^2$ limit and hence is holomorphic by the distributional Cauchy-Riemann equations and Weyl regularity. Thus $H(B,\varphi)$ is a [Hilbert space](/page/Hilbert%20Space). Moreover, the same local $L^2$ estimate together with the mean-value inequality for holomorphic functions gives, for every $B''\subset\subset B$, a constant $C_{B''}>0$ such that \begin{align*} \sup_{z\in B''}|f(z)|\le C_{B''}\|f\|_{H(B,\varphi)}. \end{align*} Consequently the restriction map $H(B,\varphi)\to\mathcal O_B(B'')$ is continuous for the compact-open topology. For each $a \in B'$, let $\mathcal J_a \subset \mathcal O_{B,a}$ be the ideal generated by the germs at $a$ of all elements of $H(B,\varphi)$. Since every element of $H(B,\varphi)$ is locally square-integrable with weight $e^{-\varphi}$, we have $\mathcal J_a \subset \mathcal I(\varphi)_a$. We prove the reverse inclusion by finite-jet approximation. Fix $a \in B'$ and an integer $k \ge 1$. If $f_a \in \mathcal I(\varphi)_a$ is represented by a holomorphic map $f: W \to \mathbb C$ on a ball $W \subset\subset B$ containing $a$, choose a smooth cut-off map \begin{align*} \eta: B &\to [0,1] \end{align*} with compact support in $W$ and equal to $1$ on a smaller ball around $a$. Define \begin{align*} u: B &\to \mathbb C,\\ z &\mapsto \eta(z)f(z), \end{align*} and define the smooth compactly supported $(0,1)$-form \begin{align*} \alpha := \bar\partial u. \end{align*} The support of $\alpha$ is contained in the annulus where $f$ is holomorphic and $\eta$ is not constant. Since $f_a \in \mathcal I(\varphi)_a$, we choose $W$ so that \begin{align*} \int_W |f(z)|^2 e^{-\varphi(z)}\,d\mathcal L^{2n}(z) < \infty. \end{align*} The coefficients of $\bar\partial\eta$ are smooth and bounded, and $\operatorname{supp}\alpha \subset W$, so \begin{align*} \int_B |\alpha(z)|^2 e^{-\varphi(z)}\,d\mathcal L^{2n}(z) < \infty. \end{align*} This verifies the integrability hypothesis needed for Hörmander's theorem; no integrability away from the chosen representative's domain is being assumed. Choose a real number $A>2(n+k)$ and define the auxiliary plurisubharmonic weight \begin{align*} \Phi_{a,k}: B &\to [-\infty,\infty),\\ z &\mapsto \varphi(z) + A\log|z-a|. \end{align*} The function $z \mapsto \log|z-a|$ is plurisubharmonic on $B$, with value $-\infty$ at $a$, and hence $\Phi_{a,k}$ is plurisubharmonic. On $\operatorname{supp}\alpha$ the function $|z-a|^{-A}$ is bounded because the support is separated from $a$; therefore \begin{align*} \int_B |\alpha(z)|^2 e^{-\Phi_{a,k}(z)}\,d\mathcal L^{2n}(z) < \infty. \end{align*} We use the following global weighted consequence of [Hörmander's weighted $L^2$ existence theorem for $\bar\partial$](/page/Hormander%20L2%20Estimates): on a bounded pseudoconvex domain $\Omega \subset \mathbb C^n$, for every plurisubharmonic weight $\Psi$ and every smooth compactly supported $\bar\partial$-closed $(0,1)$-form $\beta$ with \begin{align*} \int_\Omega |\beta(z)|^2 e^{-\Psi(z)}\,d\mathcal L^{2n}(z) < \infty, \end{align*} there exists $w \in L^2_{\mathrm{loc}}(\Omega)$ satisfying $\bar\partial w=\beta$ in distributions and \begin{align*} \int_\Omega |w(z)|^2 e^{-\Psi(z)}\,d\mathcal L^{2n}(z) \le C_{\Omega,\operatorname{supp}\beta}\int_\Omega |\beta(z)|^2 e^{-\Psi(z)}\,d\mathcal L^{2n}(z) \end{align*} with a finite constant $C_{\Omega,\operatorname{supp}\beta}>0$. This is obtained from the standard Hörmander estimate by using a bounded strictly plurisubharmonic exhaustion weight on $\Omega$; since $\operatorname{supp}\beta\subset\subset\Omega$, the auxiliary factors are bounded above and below on the support of $\beta$, and the resulting solution has the displayed global weighted integrability. The norm of a $(0,1)$-form is the Euclidean coefficient norm induced by the standard Hermitian metric on $\mathbb C^n$. In the present application, $B$ is a bounded pseudoconvex ball, $\Phi_{a,k}$ is plurisubharmonic, $\alpha$ is smooth compactly supported, $\bar\partial\alpha=\bar\partial^2u=0$, and the weighted integral above is finite. Hence there exists a measurable function \begin{align*} v_{a,k}: B &\to \mathbb C \end{align*} with $\bar\partial v_{a,k}=\alpha$ in the sense of distributions and \begin{align*} \int_B |v_{a,k}(z)|^2 e^{-\Phi_{a,k}(z)}\,d\mathcal L^{2n}(z) < \infty. \end{align*} Define the measurable function \begin{align*} F_{a,k}: B &\to \mathbb C,\\ z &\mapsto u(z)-v_{a,k}(z). \end{align*} Since $\bar\partial F_{a,k}=0$ in the sense of distributions, [Weyl regularity for the $\bar\partial$-operator](/page/Weyl%20Regularity%20for%20dbar) implies that $F_{a,k}$ agrees almost everywhere with a [holomorphic function](/page/Holomorphic%20Function) on $B$. Replacing $F_{a,k}$ by that holomorphic representative, we have $F_{a,k} \in H(B,\varphi)$, and \begin{align*} (F_{a,k})_a-f_a \in \mathfrak m_a^{k+1}. \end{align*} Indeed, holomorphicity follows from the distributional identity $\bar\partial u=\bar\partial v_{a,k}$ and Weyl regularity. To prove membership in $H(B,\varphi)$, set $M_{a,A}:=\max\{1,(R+r)^A\}$. Since $a\in B'=B(0,r)$ and $z\in B=B(0,R)$ imply $|z-a|\le R+r$, we have \begin{align*} e^{-\varphi(z)} \le M_{a,A}|z-a|^{-A}e^{-\varphi(z)} \end{align*} for $z\in B\setminus\{a\}$, and the point $a$ is negligible for $\mathcal L^{2n}$. The weighted estimate for $v_{a,k}$ and the defining weighted integrability of $u$ therefore give $F_{a,k}\in H(B,\varphi)$. The last assertion follows because the integrability of $|v_{a,k}|^2 |z-a|^{-A}e^{-\varphi}$ and the local upper boundedness of the plurisubharmonic function $\varphi$ away from the identically $-\infty$ case force every holomorphic term of degree at most $k$ in the germ of $v_{a,k}$ at $a$ to vanish. If $\varphi\equiv -\infty$ on a neighbourhood of $a$, then $\mathcal I(\varphi)_a=0$ and the generation statement at $a$ is empty. We now pass from finite-jet approximation to exact generation. Fix $a \in B'$. For every integer $k\ge 1$ and every $f_a\in \mathcal I(\varphi)_a$, the preceding construction gives $F_{a,k}\in H(B,\varphi)$ with \begin{align*} f_a-F_{a,k,a}\in \mathfrak m_a^{k+1}. \end{align*} Thus \begin{align*} \mathcal I(\varphi)_a\subset \mathcal J_a+\mathfrak m_a^{k+1}\mathcal O_{B,a} \end{align*} for every $k\ge 1$. Since $\mathcal O_{B,a}$ is Noetherian, $\mathcal O_{B,a}/\mathcal J_a$ is a finitely generated local ring module over itself, and [Krull's intersection theorem](/page/Krull%20Intersection%20Theorem) gives \begin{align*} \bigcap_{k=1}^{\infty}\mathfrak m_a^{k+1}(\mathcal O_{B,a}/\mathcal J_a)=0. \end{align*} Therefore $\mathcal I(\varphi)_a\subset \mathcal J_a$. Combined with $\mathcal J_a\subset\mathcal I(\varphi)_a$, this proves \begin{align*} \mathcal J_a=\mathcal I(\varphi)_a \end{align*} for every $a\in B'$. It remains only to replace the possibly infinite generating family $H(B,\varphi)$ by a finite one near $0$. The [strong Noetherian theorem for Banach spaces of holomorphic functions](/page/Strong%20Noetherian%20Theorem) stated at the beginning of the proof applies because $H(B,\varphi)$ is a [Hilbert space](/page/Hilbert%20Space) of holomorphic functions on $B$ and local $L^2$ estimates for holomorphic functions make point evaluation and restriction to smaller balls continuous. Hence there exist a neighbourhood $V\subset B'$ of $0$ and elements \begin{align*} g_1,\dots,g_N\in H(B,\varphi) \end{align*} such that the germs $(g_1)_a,\dots,(g_N)_a$ generate $\mathcal J_a$ for every $a\in V$. Since $\mathcal J_a=\mathcal I(\varphi)_a$, these same germs generate $\mathcal I(\varphi)_a$ for every $a\in V$. Also $g_j\in H(B,\varphi)$ implies $g_j\in\mathcal I(\varphi)(B')$ by restriction of the defining weighted integral from $B$ to $B'$. This proves the lemma. [/proof][/step]

[guided]The goal is to avoid assuming the local coherence theorem we are proving. We use Hörmander's theorem only to manufacture global weighted-$L^2$ holomorphic functions with prescribed finite jets, and then use Noetherian algebra to turn finite-jet approximation into exact generation. Let $\mathcal L^{2n}$ denote Lebesgue measure on the real Euclidean space underlying $\mathbb C^n$, and define \begin{align*} H(B,\varphi) := \left\{ f \in \mathcal O_B(B) : \int_B |f(z)|^2 e^{-\varphi(z)}\,d\mathcal L^{2n}(z) < \infty \right\}. \end{align*} For $a\in B'$, let $\mathcal J_a\subset\mathcal O_{B,a}$ be the ideal generated by germs of functions in $H(B,\varphi)$. The inclusion $\mathcal J_a\subset\mathcal I(\varphi)_a$ follows directly from the definition, because a function square-integrable on $B$ with weight $e^{-\varphi}$ is square-integrable on a neighbourhood of $a$. We prove the reverse inclusion. Take $f_a\in\mathcal I(\varphi)_a$, represented by a holomorphic map $f:W\to\mathbb C$ on a ball $W\subset\subset B$ containing $a$. Choose a smooth compactly supported cut-off map $\eta:B\to[0,1]$ that is identically $1$ near $a$, set $u=\eta f$, and set $\alpha=\bar\partial u$. The form $\alpha$ is smooth, compactly supported, and $\bar\partial$-closed because $\bar\partial^2u=0$. Since $f_a\in\mathcal I(\varphi)_a$, after shrinking $W$ we have \begin{align*} \int_W |f(z)|^2e^{-\varphi(z)}\,d\mathcal L^{2n}(z)<\infty. \end{align*} The coefficients of $\bar\partial\eta$ are bounded and supported in $W$, so \begin{align*} \int_B |\alpha(z)|^2e^{-\varphi(z)}\,d\mathcal L^{2n}(z)<\infty. \end{align*} Choose $A>2(n+k)$ and define \begin{align*} \Phi_{a,k}:B&\to[-\infty,\infty),\\ z&\mapsto\varphi(z)+A\log|z-a|. \end{align*} The function $z\mapsto\log|z-a|$ is plurisubharmonic, hence $\Phi_{a,k}$ is plurisubharmonic. The support of $\alpha$ is separated from $a$, so the factor $|z-a|^{-A}$ is bounded on $\operatorname{supp}\alpha$, and therefore \begin{align*} \int_B |\alpha(z)|^2e^{-\Phi_{a,k}(z)}\,d\mathcal L^{2n}(z)<\infty. \end{align*} [Hörmander's weighted $L^2$ theorem](/page/Hormander%20L2%20Estimates) applies because $B$ is a bounded pseudoconvex ball, $\Phi_{a,k}$ is plurisubharmonic, $\alpha$ is smooth compactly supported and $\bar\partial$-closed, and the Euclidean coefficient norm of $\alpha$ is weighted-square-integrable. Thus there is a measurable map $v_{a,k}:B\to\mathbb C$ such that $\bar\partial v_{a,k}=\alpha$ in distributions and \begin{align*} \int_B |v_{a,k}(z)|^2e^{-\Phi_{a,k}(z)}\,d\mathcal L^{2n}(z)<\infty. \end{align*} Set $F_{a,k}=u-v_{a,k}$. Since $\bar\partial F_{a,k}=0$ distributionally, [Weyl regularity for $\bar\partial$](/page/Weyl%20Regularity%20for%20dbar) gives a holomorphic representative on $B$. To see that $F_{a,k}\in H(B,\varphi)$, set $M_{a,A}:=\max\{1,(R+r)^A\}$. For $a\in B'=B(0,r)$ and $z\in B=B(0,R)$, the triangle inequality gives $|z-a|\le R+r$, hence \begin{align*} e^{-\varphi(z)}\le M_{a,A}|z-a|^{-A}e^{-\varphi(z)} \end{align*} away from the single point $a$, which has $\mathcal L^{2n}$-measure zero. The weighted estimate for $v_{a,k}$ and the weighted integrability of $u=\eta f$ therefore imply $F_{a,k}\in H(B,\varphi)$. Why does $F_{a,k}$ match the $k$-jet of $f$ at $a$? Near $a$ the cut-off is $1$, so $F_{a,k}-f=-v_{a,k}$. The weighted estimate says $|v_{a,k}|^2|z-a|^{-A}e^{-\varphi}$ is integrable. Since a plurisubharmonic function is locally bounded above unless it is identically $-\infty$ on the component under consideration, this forces every holomorphic term of degree at most $k$ in the germ of $v_{a,k}$ at $a$ to vanish when $A>2(n+k)$. If $\varphi\equiv-\infty$ near $a$, then $\mathcal I(\varphi)_a=0$ and the assertion is immediate. Hence \begin{align*} f_a-F_{a,k,a}\in\mathfrak m_a^{k+1}. \end{align*} Because $F_{a,k,a}\in\mathcal J_a$, this gives \begin{align*} \mathcal I(\varphi)_a\subset\mathcal J_a+\mathfrak m_a^{k+1}\mathcal O_{B,a} \end{align*} for every $k\ge1$. Now apply [Krull's intersection theorem](/page/Krull%20Intersection%20Theorem) to the Noetherian local ring $\mathcal O_{B,a}$ and the quotient by $\mathcal J_a$. Intersecting over all $k$ gives \begin{align*} \mathcal I(\varphi)_a\subset\mathcal J_a. \end{align*} Together with the obvious reverse inclusion, this proves $\mathcal J_a=\mathcal I(\varphi)_a$. Finally, the [strong Noetherian theorem for Banach spaces of holomorphic functions](/page/Strong%20Noetherian%20Theorem) turns the infinite family $H(B,\varphi)$ into finitely many generators on one neighbourhood of $0$. This is an external Banach-family Noetherian theorem, not the coherence theorem currently being proved. Its hypotheses have been verified above: $H(B,\varphi)$ is a [Hilbert space](/page/Hilbert%20Space), including the possible case where $\varphi\equiv-\infty$ on an open component, and for each $B''\subset\subset B$ the local upper bound for $\varphi$ plus the mean-value inequality gives a compact-open estimate \begin{align*} \sup_{z\in B''}|f(z)|\le C_{B''}\|f\|_{H(B,\varphi)}. \end{align*} Therefore there are $V\subset B'$ and $g_1,\dots,g_N\in H(B,\varphi)$ whose germs generate $\mathcal J_a$, hence $\mathcal I(\varphi)_a$, for every $a\in V$.[/guided]

[step:Apply the local finite generation lemma to obtain coherence near the chosen point]By the local $L^2$ finite generation lemma, after shrinking to a neighbourhood $V \subset B'$ of $0$, there are sections \begin{align*} g_1,\dots,g_N \in \mathcal I(\varphi)(V) \end{align*} such that for every $a \in V$ the germs $(g_1)_a,\dots,(g_N)_a$ generate $\mathcal I(\varphi)_a$ as an ideal of $\mathcal O_{B,a}$. Define the morphism of analytic sheaves \begin{align*} \Theta: \mathcal O_V^N &\to \mathcal O_V,\\ (h_1,\dots,h_N) &\mapsto \sum_{j=1}^N h_j g_j. \end{align*} Its image sheaf is exactly $\mathcal I(\varphi)|_V$: the inclusion $\operatorname{im}\Theta\subset\mathcal I(\varphi)|_V$ follows from $g_j\in\mathcal I(\varphi)(V)$ and the ideal property, while equality on stalks follows because the germs $(g_1)_a,\dots,(g_N)_a$ generate $\mathcal I(\varphi)_a$ for every $a\in V$. By [Oka's coherence theorem for the structure sheaf](/page/Oka%20Coherence%20Theorem), $\mathcal O_V$ is coherent. Hence $\mathcal O_V^N$ and $\mathcal O_V$ are coherent, and the kernel $\ker\Theta$ of a morphism between coherent analytic sheaves is coherent. Therefore \begin{align*} \mathcal I(\varphi)|_V=\operatorname{im}\Theta\cong \mathcal O_V^N/\ker\Theta \end{align*} is a quotient of a coherent analytic sheaf by a coherent subsheaf, and is coherent.[/step]

[guided]The lemma gives actual sections on a neighbourhood, not merely generators of one stalk. This distinction matters: coherence is a sheaf-theoretic local property, so we need generators that work throughout some neighbourhood. We have holomorphic functions \begin{align*} g_1,\dots,g_N \in \mathcal I(\varphi)(V) \end{align*} with the property that, for each $a \in V$, every germ in $\mathcal I(\varphi)_a$ is an $\mathcal O_{B,a}$-linear combination of the germs $(g_j)_a$. We package this generation statement into a morphism landing in the structure sheaf: \begin{align*} \Theta: \mathcal O_V^N &\to \mathcal O_V,\\ (h_1,\dots,h_N) &\mapsto \sum_{j=1}^N h_j g_j. \end{align*} The image is contained in $\mathcal I(\varphi)|_V$ because each $g_j$ is a section of the ideal sheaf. Conversely, at every point $a\in V$, the stalk of $\operatorname{im}\Theta$ is the ideal of $\mathcal O_{B,a}$ generated by $(g_1)_a,\dots,(g_N)_a$, which is $\mathcal I(\varphi)_a$ by the lemma. Since equality of subsheaves can be checked on stalks, $\operatorname{im}\Theta=\mathcal I(\varphi)|_V$. The final sheaf-theoretic input is [Oka coherence](/page/Oka%20Coherence%20Theorem): the structure sheaf $\mathcal O_V$ of a complex manifold is coherent. Therefore $\mathcal O_V^N$ and $\mathcal O_V$ are coherent, and the kernel of the morphism $\Theta:\mathcal O_V^N\to\mathcal O_V$ is coherent. The image is then \begin{align*} \operatorname{im}\Theta\cong \mathcal O_V^N/\ker\Theta, \end{align*} so it is coherent as a quotient of a coherent analytic sheaf by a coherent subsheaf. Since $\operatorname{im}\Theta=\mathcal I(\varphi)|_V$, the multiplier ideal sheaf is coherent on $V$.[/guided]

[step:Return from coordinates to the manifold] The preceding argument proves that for every point $p \in X$ there exists a coordinate neighbourhood $U_p \subset X$ such that \begin{align*} \mathcal I(\phi)|_{U_p} \end{align*} is coherent. Since coherence is local on $X$, the multiplier ideal sheaf $\mathcal I(\phi)$ is coherent on all of $X$. This proves the Nadel Coherence Theorem. [/step]