[proofplan] We prove the local syzygy statement by induction on the number $n$ of complex variables. Near a point, the morphism $\phi$ is represented by a row of holomorphic functions; if one coefficient is nonzero, a linear change of coordinates and the Weierstrass Preparation Theorem make that coefficient a monic polynomial $h$ in the last variable. Weierstrass division then decomposes every syzygy into elementary syzygies involving $h$ and a finite-dimensional remainder problem over the base variables $z' \in \mathbb{C}^{n-1}$. The induction hypothesis makes that lower-dimensional remainder kernel locally finitely generated, and the lifted generators generate the original kernel. [/proofplan]

[step:Reduce the theorem to local syzygies of a row of holomorphic functions] For $m \geq 0$, let $\mathsf{Coh}(m)$ denote the following assertion: for every open set $X \subset \mathbb{C}^m$, every integer $r \geq 1$, and every $\mathcal{O}_X$-module morphism $\psi: \mathcal{O}_X^r \to \mathcal{O}_X$, the sheaf $\ker \psi$ is locally finitely generated. Let $U \subset \mathbb{C}^n$ be open, let $r \geq 1$, and let $\phi: \mathcal{O}_U^r \to \mathcal{O}_U$ be an $\mathcal{O}_U$-module morphism. For $1 \leq i \leq r$, let $e_i \in \mathcal{O}_U^r(U)$ denote the $i$-th standard basis section and define the holomorphic coefficient \begin{align*} f_i: U &\to \mathbb{C} \\ x &\mapsto \phi(e_i)(x). \end{align*} For every open set $V \subset U$ and every section $G=(G_1,\ldots,G_r) \in \mathcal{O}_U^r(V)$, $\mathcal{O}_U$-linearity gives \begin{align*} G \in \ker \phi(V) \quad \Longleftrightarrow \quad \sum_{i=1}^r f_i|_V G_i = 0 \text{ in } \mathcal{O}_U(V). \end{align*} Thus $\ker \phi$ is the sheaf of syzygies among the row $(f_1,\ldots,f_r)$. Fix a point $p \in U$. If $f_{i,p}=0$ in the stalk $\mathcal{O}_{U,p}$ for every $i$, then for each $i$ there is a neighbourhood $V_i$ of $p$ on which $f_i=0$. On $V=\bigcap_{i=1}^r V_i$, the morphism $\phi|_V$ is the zero morphism, so $\ker \phi|_V=\mathcal{O}_V^r$, generated by $e_1,\ldots,e_r$. Hence only the case where at least one coefficient germ is nonzero needs proof. [/step]

[step:Extend the induction hypothesis to finite free targets] [claim:Kernels with finite free targets follow from kernels with rank one target] Assume $\mathsf{Coh}(m)$. Let $X \subset \mathbb{C}^m$ be open, let $N \geq 1$, let $s \geq 1$, and let $\Psi: \mathcal{O}_X^N \to \mathcal{O}_X^s$ be an $\mathcal{O}_X$-module morphism. Then $\ker \Psi$ is locally finitely generated. [/claim] [proof] We prove the claim by induction on $s$. The case $s=1$ is exactly $\mathsf{Coh}(m)$. Assume the statement known for target rank $s-1$, where $s \geq 2$. Write \begin{align*} \Psi=(\Psi_1,\Psi_2): \mathcal{O}_X^N \to \mathcal{O}_X \oplus \mathcal{O}_X^{s-1}, \end{align*} where $\Psi_1$ is the first component and $\Psi_2$ is the remaining $(s-1)$-tuple of components. Fix $x \in X$. By $\mathsf{Coh}(m)$ applied to $\Psi_1$, there is a neighbourhood $V \subset X$ of $x$ and sections $\sigma_1,\ldots,\sigma_M \in \mathcal{O}_X^N(V)$ generating $\ker \Psi_1|_V$; if $M=0$, then $\ker \Psi_1|_V=0$ and therefore $\ker \Psi|_V=0$. Assume $M \geq 1$. Define the morphism \begin{align*} B: \mathcal{O}_V^M &\to \mathcal{O}_V^{s-1} \\ (c_1,\ldots,c_M) &\mapsto \Psi_2\left(\sum_{j=1}^M c_j \sigma_j\right). \end{align*} By the induction hypothesis on the target rank, after shrinking $V$ to a neighbourhood $V_0$ of $x$, the sheaf $\ker B$ is generated by finitely many sections $\tau_1,\ldots,\tau_L \in \mathcal{O}_V^M(V_0)$. For each $1 \leq \ell \leq L$, define \begin{align*} \eta_\ell := \sum_{j=1}^M (\tau_\ell)_j \sigma_j \in \mathcal{O}_X^N(V_0). \end{align*} If $G \in \ker \Psi(V')$ for an open set $V' \subset V_0$, then $G \in \ker \Psi_1(V')$, so $G=\sum_{j=1}^M c_j\sigma_j$ for some $c_j \in \mathcal{O}_V(V')$. The condition $\Psi_2(G)=0$ says $(c_1,\ldots,c_M)\in \ker B(V')$, hence $(c_1,\ldots,c_M)$ is an $\mathcal{O}_V(V')$-linear combination of the $\tau_\ell$. Therefore $G$ is an $\mathcal{O}_V(V')$-linear combination of the $\eta_\ell$. Thus $\ker \Psi$ is locally finitely generated. [/proof] [/step]

[step:Start the induction in zero variables] We prove $\mathsf{Coh}(n)$ by induction on $n$. For $n=0$, the space $\mathbb{C}^0$ is a point. On the one-point space, $\mathcal{O}_{\mathbb{C}^0}$ is the constant sheaf with value $\mathbb{C}$. Any morphism $\psi:\mathcal{O}_{\mathbb{C}^0}^r \to \mathcal{O}_{\mathbb{C}^0}$ is a linear map $\mathbb{C}^r \to \mathbb{C}$, and its kernel is a finite-dimensional $\mathbb{C}$-vector space. Choosing a finite $\mathbb{C}$-basis of that kernel gives finitely many global generators. Hence $\mathsf{Coh}(0)$ holds. [/step]

[step:Prepare one nonzero coefficient as a monic polynomial in the last variable]Assume $n \geq 1$ and assume $\mathsf{Coh}(n-1)$. Fix $p \in U$ and suppose at least one coefficient germ is nonzero. Reindex so that $f_{r,p}\neq 0$ in $\mathcal{O}_{U,p}$. If $f_r(p)\neq 0$, shrink to a neighbourhood $V$ of $p$ on which $f_r$ has no zeros. Then $f_r|_V$ is a unit in $\mathcal{O}_V(V)$, and $\ker \phi|_V$ is generated by the $r-1$ sections \begin{align*} e_i - \frac{f_i}{f_r}e_r, \qquad 1 \leq i \leq r-1. \end{align*} Thus assume $f_r(p)=0$. Local finite generation is invariant under biholomorphic changes of coordinates because pullback by a biholomorphism identifies the structure sheaves and carries kernels to kernels. Translate $p$ to $0$. Let $d \geq 1$ be the order of vanishing of the germ $f_{r,0}$. Choose a complex linear coordinate system $z=(z',z_n)\in \mathbb{C}^{n-1}\times \mathbb{C}$ such that the restriction $z_n \mapsto f_r(0',z_n)$ has a zero of order $d$ at $0$. By the Weierstrass Preparation Theorem, after shrinking to a product neighbourhood $P=P'\times D \subset U$ of $0$, there are holomorphic maps \begin{align*} u: P &\to \mathbb{C}^{\times} \end{align*} and \begin{align*} h: P &\to \mathbb{C} \\ (z',z_n) &\mapsto z_n^d+\sum_{k=0}^{d-1} a_k(z')z_n^k, \end{align*} where $\mathbb{C}^{\times}=\mathbb{C}\setminus\{0\}$ and each $a_k:P'\to\mathbb{C}$ is holomorphic with $a_k(0')=0$, such that \begin{align*} f_r|_P = u h. \end{align*} Multiplication of the target $\mathcal{O}_P$ by the unit $u^{-1}$ is an automorphism, so it does not change the kernel. Define \begin{align*} F_i: P &\to \mathbb{C} \\ z &\mapsto u(z)^{-1} f_i(z), \qquad 1 \leq i \leq r-1. \end{align*} It is enough to prove local finite generation of the kernel of \begin{align*} \psi: \mathcal{O}_P^r &\to \mathcal{O}_P \\ (G_1,\ldots,G_r) &\mapsto \sum_{i=1}^{r-1} F_iG_i+hG_r. \end{align*}[/step]

[guided]Assume $n \geq 1$ and assume the theorem has already been proved in $n-1$ variables. We fix a point $p \in U$ and look only near $p$, because local finite generation is a local property. After the reduction in the first step, at least one coefficient germ is nonzero. Reindex the basis of $\mathcal{O}_U^r$ so that this nonzero germ is $f_{r,p}$. There are two cases. If $f_r(p)\neq 0$, then $f_r$ is nonvanishing on some neighbourhood $V$ of $p$, so $f_r|_V$ is a unit. In that case the equation \begin{align*} \sum_{i=1}^r f_iG_i=0 \end{align*} can be solved for the last component: \begin{align*} G_r=-\sum_{i=1}^{r-1}\frac{f_i}{f_r}G_i. \end{align*} Therefore $\ker \phi|_V$ is generated by \begin{align*} e_i-\frac{f_i}{f_r}e_r, \qquad 1\leq i\leq r-1. \end{align*} The only remaining case is $f_r(p)=0$ but $f_{r,p}$ is not the zero germ. Translate $p$ to the origin. Pullback by a biholomorphic coordinate change identifies the sheaves of holomorphic functions and preserves finite generation of kernels, so we may choose convenient linear coordinates. Let $d\geq 1$ be the order of vanishing of $f_{r,0}$. Choosing a direction on which the first nonzero homogeneous Taylor term of $f_r$ does not vanish gives coordinates $z=(z',z_n)\in\mathbb{C}^{n-1}\times\mathbb{C}$ such that $z_n\mapsto f_r(0',z_n)$ has a zero of order $d$ at $0$. Now the Weierstrass Preparation Theorem applies to $f_r$ in the variable $z_n$. After shrinking to a product neighbourhood $P=P'\times D$, it gives a unit \begin{align*} u:P&\to\mathbb{C}^{\times} \end{align*} and a monic polynomial in $z_n$ \begin{align*} h:P&\to\mathbb{C}\\ (z',z_n)&\mapsto z_n^d+\sum_{k=0}^{d-1}a_k(z')z_n^k, \end{align*} where each coefficient $a_k:P'\to\mathbb{C}$ is holomorphic and satisfies $a_k(0')=0$, such that \begin{align*} f_r|_P=uh. \end{align*} Multiplying the target by the unit $u^{-1}$ is an isomorphism of $\mathcal{O}_P$-modules, so it preserves the kernel. Thus we replace the original row by \begin{align*} (F_1,\ldots,F_{r-1},h), \qquad F_i=u^{-1}f_i|_P, \end{align*} and prove local finite generation for the morphism \begin{align*} \psi:\mathcal{O}_P^r&\to\mathcal{O}_P\\ (G_1,\ldots,G_r)&\mapsto \sum_{i=1}^{r-1}F_iG_i+hG_r. \end{align*} The point of this preparation is that division by $h$ has finite remainders of degree $<d$ in the last variable, so the unresolved part of the syzygy problem lives over $P'\subset\mathbb{C}^{n-1}$.[/guided]

[step:Reduce the remaining coefficients modulo the prepared polynomial] For each $1 \leq i \leq r-1$, the Weierstrass Division Theorem applied to $F_i$ and the Weierstrass polynomial $h$ gives unique holomorphic maps \begin{align*} Q_i:P&\to\mathbb{C} \end{align*} and \begin{align*} \rho_i:P&\to\mathbb{C}\\ (z',z_n)&\mapsto \sum_{k=0}^{d-1} b_{i,k}(z')z_n^k, \end{align*} where each $b_{i,k}:P'\to\mathbb{C}$ is holomorphic, such that \begin{align*} F_i=Q_i h+\rho_i. \end{align*} Define \begin{align*} \Psi:\mathcal{O}_P^r&\to\mathcal{O}_P\\ (G_1,\ldots,G_r)&\mapsto \sum_{i=1}^{r-1}\rho_iG_i+hG_r \end{align*} and \begin{align*} T:\mathcal{O}_P^r&\to\mathcal{O}_P^r\\ (G_1,\ldots,G_r)&\mapsto \left(G_1,\ldots,G_{r-1},G_r+\sum_{i=1}^{r-1}Q_iG_i\right). \end{align*} The inverse of $T$ subtracts $\sum_{i=1}^{r-1}Q_iG_i$ from the last component, so $T$ is an automorphism. For every section $G$, \begin{align*} \psi(G)=\Psi(TG), \end{align*} hence \begin{align*} \ker \psi=T^{-1}(\ker \Psi). \end{align*} Therefore it suffices to prove local finite generation for $\ker \Psi$. [/step]

[step:Encode the remainder condition as a lower-dimensional kernel]Let $\pi:P=P'\times D\to P'$ be the projection map $\pi(z',z_n)=z'$. Let $\mathcal{R}_{P'}:=\mathcal{O}_{P'}^d$ denote the free $\mathcal{O}_{P'}$-module of remainders of degree $<d$ in $z_n$. We identify a section $B=(B_0,\ldots,B_{d-1})$ of $\mathcal{R}_{P'}$ with the holomorphic function \begin{align*} \iota(B):P&\to\mathbb{C}\\ (z',z_n)&\mapsto \sum_{k=0}^{d-1}B_k(z')z_n^k. \end{align*} For a section $B=(B_1,\ldots,B_{r-1})$ of $\mathcal{R}_{P'}^{\,r-1}$, the product $\sum_{i=1}^{r-1}\iota(B_i)\rho_i$ is a polynomial in $z_n$ with holomorphic coefficients on $P'$. Dividing it by $h$ gives a unique quotient and a unique remainder: \begin{align*} \sum_{i=1}^{r-1}\iota(B_i)\rho_i = A_Bh+\iota(R_B), \end{align*} where $R_B$ is a section of $\mathcal{R}_{P'}$. Define the $\mathcal{O}_{P'}$-module morphism \begin{align*} \Theta:\mathcal{R}_{P'}^{\,r-1}&\to\mathcal{R}_{P'}\\ B&\mapsto R_B. \end{align*} By the finite-target reduction and $\mathsf{Coh}(n-1)$, the sheaf $\ker \Theta$ is locally finitely generated near $0'$. Shrink $P'$ to a neighbourhood $W'$ of $0'$ and choose sections \begin{align*} B^1,\ldots,B^M \in \mathcal{R}_{P'}^{\,r-1}(W') \end{align*} which generate $\ker \Theta|_{W'}$. Put $W:=W'\times D$.[/step]

[guided]The coefficients $\rho_i$ are already polynomials of degree $<d$ in the last variable. The only remaining obstruction to building syzygies is this: if the first $r-1$ components of a syzygy are reduced modulo $h$, then the expression $\sum B_i\rho_i$ must be divisible by $h$. We formalize that obstruction over the base $P'\subset\mathbb{C}^{n-1}$. Let \begin{align*} \pi:P=P'\times D&\to P'\\ (z',z_n)&\mapsto z' \end{align*} be the projection. Let $\mathcal{R}_{P'}=\mathcal{O}_{P'}^d$ be the sheaf whose sections represent polynomials of degree $<d$ in $z_n$. A section $B=(B_0,\ldots,B_{d-1})$ corresponds to \begin{align*} \iota(B):P&\to\mathbb{C}\\ (z',z_n)&\mapsto \sum_{k=0}^{d-1}B_k(z')z_n^k. \end{align*} Now take a tuple $B=(B_1,\ldots,B_{r-1})$ of such remainders. Since each $\rho_i$ is also a polynomial in $z_n$ with coefficients on $P'$, the product $\sum_{i=1}^{r-1}\iota(B_i)\rho_i$ is again a polynomial in $z_n$ with holomorphic coefficients on $P'$. Dividing by the monic polynomial $h$ gives a unique quotient and remainder: \begin{align*} \sum_{i=1}^{r-1}\iota(B_i)\rho_i = A_Bh+\iota(R_B), \end{align*} where $R_B$ has degree $<d$ in $z_n$. The assignment $B\mapsto R_B$ is $\mathcal{O}_{P'}$-linear, so it defines \begin{align*} \Theta:\mathcal{R}_{P'}^{\,r-1}&\to\mathcal{R}_{P'}\\ B&\mapsto R_B. \end{align*} The kernel of $\Theta$ consists exactly of those reduced tuples $B$ for which $\sum B_i\rho_i$ is divisible by $h$. This is now a problem in $n-1$ variables, with finite free source and finite free target over $\mathcal{O}_{P'}$. By the finite-target reduction and the induction hypothesis $\mathsf{Coh}(n-1)$, $\ker\Theta$ is locally finitely generated. Hence, after shrinking $P'$ to a neighbourhood $W'$ of $0'$, there are sections \begin{align*} B^1,\ldots,B^M\in \mathcal{R}_{P'}^{\,r-1}(W') \end{align*} which generate $\ker\Theta|_{W'}$. We set $W=W'\times D$.[/guided]

[step:Lift the lower-dimensional generators to syzygies upstairs] Write each generator $B^\alpha$ as \begin{align*} B^\alpha=(B^\alpha_1,\ldots,B^\alpha_{r-1}), \qquad 1\leq \alpha\leq M, \end{align*} where each $B^\alpha_i$ is regarded, through $\iota$, as a holomorphic function on $W$. Since $B^\alpha\in\ker\Theta(W')$, there is a unique holomorphic function $A^\alpha:W\to\mathbb{C}$, polynomial in $z_n$ with holomorphic coefficients on $W'$, such that \begin{align*} \sum_{i=1}^{r-1}B^\alpha_i\rho_i=A^\alpha h. \end{align*} Define sections \begin{align*} S_\alpha:=\left(B^\alpha_1,\ldots,B^\alpha_{r-1},-A^\alpha\right)\in\mathcal{O}_W^r(W), \qquad 1\leq \alpha\leq M. \end{align*} Then $\Psi(S_\alpha)=0$. For $1\leq i\leq r-1$, define the elementary section \begin{align*} E_i:=h e_i-\rho_i e_r\in\mathcal{O}_W^r(W). \end{align*} Since \begin{align*} \Psi(E_i)=\rho_i h+h(-\rho_i)=0, \end{align*} each $E_i$ is also a section of $\ker\Psi$. [/step]

[step:Generate every syzygy by division and lower-dimensional generation]We prove that the finite family \begin{align*} E_1,\ldots,E_{r-1},S_1,\ldots,S_M \end{align*} generates $\ker\Psi|_W$. Fix a point $q=(q',q_n)\in W$ and a germ \begin{align*} G=(G_1,\ldots,G_r)\in(\ker\Psi)_q. \end{align*} For each $1\leq i\leq r-1$, the Weierstrass Division Theorem applied in the local ring $\mathcal{O}_{W,q}$ gives unique germs $C_i\in\mathcal{O}_{W,q}$ and $D_i\in\mathcal{R}_{W',q'}$ such that \begin{align*} G_i=C_ih+D_i. \end{align*} Subtract the elementary syzygies: \begin{align*} G-\sum_{i=1}^{r-1}C_iE_i = \left(D_1,\ldots,D_{r-1},G_r+\sum_{i=1}^{r-1}C_i\rho_i\right). \end{align*} This germ is still in $\ker\Psi_q$. Therefore \begin{align*} \sum_{i=1}^{r-1}D_i\rho_i + h\left(G_r+\sum_{i=1}^{r-1}C_i\rho_i\right) =0. \end{align*} Taking the Weierstrass remainder modulo $h$, the second term has remainder $0$ and the zero germ has remainder $0$; uniqueness of the remainder gives \begin{align*} \Theta(D_1,\ldots,D_{r-1})=0. \end{align*} Since $B^1,\ldots,B^M$ generate $\ker\Theta$ near $q'$, there are germs $\lambda_1,\ldots,\lambda_M\in\mathcal{O}_{W',q'}$ such that \begin{align*} (D_1,\ldots,D_{r-1}) = \sum_{\alpha=1}^M \lambda_\alpha B^\alpha. \end{align*} Pull each $\lambda_\alpha$ back to $\mathcal{O}_{W,q}$ through $\pi$ and denote the pullback by $\widetilde{\lambda}_\alpha$. Then \begin{align*} G-\sum_{i=1}^{r-1}C_iE_i-\sum_{\alpha=1}^M\widetilde{\lambda}_\alpha S_\alpha = (0,\ldots,0,H) \end{align*} for some germ $H\in\mathcal{O}_{W,q}$. The left-hand side lies in $\ker\Psi_q$, so $hH=0$ in $\mathcal{O}_{W,q}$. The local ring of holomorphic germs $\mathcal{O}_{W,q}$ is an integral domain by the Identity Theorem for Holomorphic Functions, and $h$ is a nonzero germ because it is monic in $z_n$. Hence $H=0$. Thus $G$ is an $\mathcal{O}_{W,q}$-linear combination of the displayed finite family. Since $q\in W$ was arbitrary, $\ker\Psi|_W$ is locally generated by $E_1,\ldots,E_{r-1},S_1,\ldots,S_M$.[/step]

[guided]We now show that the constructed syzygies generate all local syzygies. Fix a point $q=(q',q_n)\in W$ and a germ \begin{align*} G=(G_1,\ldots,G_r)\in(\ker\Psi)_q. \end{align*} The equation $G\in\ker\Psi_q$ means \begin{align*} \sum_{i=1}^{r-1}\rho_iG_i+hG_r=0 \end{align*} in the local ring $\mathcal{O}_{W,q}$. The purpose of the elementary syzygies $E_i=h e_i-\rho_i e_r$ is to remove all multiples of $h$ from the first $r-1$ components. For each $i$, Weierstrass division by $h$ gives unique germs $C_i\in\mathcal{O}_{W,q}$ and $D_i\in\mathcal{R}_{W',q'}$ such that \begin{align*} G_i=C_ih+D_i. \end{align*} Subtracting $\sum_{i=1}^{r-1}C_iE_i$ replaces $G_i$ by its remainder $D_i$: \begin{align*} G-\sum_{i=1}^{r-1}C_iE_i = \left(D_1,\ldots,D_{r-1},G_r+\sum_{i=1}^{r-1}C_i\rho_i\right). \end{align*} Because each $E_i$ is itself a syzygy, this new germ is still in $\ker\Psi_q$. Therefore \begin{align*} \sum_{i=1}^{r-1}D_i\rho_i + h\left(G_r+\sum_{i=1}^{r-1}C_i\rho_i\right) =0. \end{align*} Now take the remainder after division by $h$. The term multiplied by $h$ has zero remainder, and the zero germ also has zero remainder. By uniqueness in the Weierstrass Division Theorem, the remainder of $\sum_{i=1}^{r-1}D_i\rho_i$ is zero. In the notation of the previous step, this says \begin{align*} \Theta(D_1,\ldots,D_{r-1})=0. \end{align*} Thus the reduced tuple $(D_1,\ldots,D_{r-1})$ belongs to the lower-dimensional kernel $\ker\Theta$ at $q'$. Since the sections $B^1,\ldots,B^M$ generate $\ker\Theta$ near $q'$, there exist germs $\lambda_1,\ldots,\lambda_M\in\mathcal{O}_{W',q'}$ such that \begin{align*} (D_1,\ldots,D_{r-1}) = \sum_{\alpha=1}^M \lambda_\alpha B^\alpha. \end{align*} Pull these coefficients back along $\pi$ to germs $\widetilde{\lambda}_\alpha\in\mathcal{O}_{W,q}$. Subtracting the lifted syzygies gives \begin{align*} G-\sum_{i=1}^{r-1}C_iE_i-\sum_{\alpha=1}^M\widetilde{\lambda}_\alpha S_\alpha = (0,\ldots,0,H) \end{align*} for some germ $H\in\mathcal{O}_{W,q}$. The left-hand side is a syzygy because it is obtained from $G$ by subtracting syzygies. Applying $\Psi$ to $(0,\ldots,0,H)$ gives $hH$, so \begin{align*} hH=0 \end{align*} in $\mathcal{O}_{W,q}$. The ring $\mathcal{O}_{W,q}$ is an integral domain by the Identity Theorem for Holomorphic Functions, and $h$ is not the zero germ because it is monic as a polynomial in $z_n$. Hence $H=0$. Therefore \begin{align*} G= \sum_{i=1}^{r-1}C_iE_i+\sum_{\alpha=1}^M\widetilde{\lambda}_\alpha S_\alpha. \end{align*} This proves that the finite family $E_1,\ldots,E_{r-1},S_1,\ldots,S_M$ generates $\ker\Psi$ at every point of $W$.[/guided]

[step:Return to the original morphism and complete the induction] We have proved local finite generation for $\ker\Psi$ on $W$. Since \begin{align*} \ker\psi=T^{-1}(\ker\Psi), \end{align*} and $T$ is an automorphism of $\mathcal{O}_W^r$, the sheaf $\ker\psi|_W$ is locally finitely generated. Multiplication of the target by the unit $u^{-1}$ did not change the kernel, and the preliminary reindexing and biholomorphic coordinate change preserve finite generation under the corresponding sheaf isomorphisms. Therefore $\ker\phi$ is locally finitely generated in a neighbourhood of the original point $p$. The point $p\in U$ was arbitrary, so $\mathsf{Coh}(n)$ follows from $\mathsf{Coh}(n-1)$. Since $\mathsf{Coh}(0)$ holds, induction gives $\mathsf{Coh}(n)$ for every $n\geq 0$. In particular, for every open set $U\subset\mathbb{C}^n$, every $r\geq 1$, and every $\mathcal{O}_U$-module morphism $\phi:\mathcal{O}_U^r\to\mathcal{O}_U$, the sheaf $\ker\phi$ is locally finitely generated. Thus $\mathcal{O}_{\mathbb{C}^n}$ is coherent as an $\mathcal{O}_{\mathbb{C}^n}$-module. [/step]