[proofplan] We prove exactness stalkwise. After trivialising the line bundle near a point, the section $s$ becomes multiplication by a [holomorphic function](/page/Holomorphic%20Function) $g$ whose zero set is the smooth hypersurface $D$. Smoothness and reducedness of the effective Cartier divisor $D$ imply that locally $g$ is a unit times one coordinate, so multiplication by $g$ is injective and the quotient by its image is precisely restriction to the hypersurface. These local descriptions are compatible with changes of trivialisation, hence they glue to give the asserted exact sequence of sheaves. [/proofplan]

[step:Define the two sheaf morphisms and reduce exactness to stalks] Let $\mathcal O_D$ denote the structure sheaf of the closed complex submanifold $D$, and for every [open set](/page/Open%20Set) $U\subset X$ write $\mathcal O_{D\cap U}:=\mathcal O_D|_{D\cap U}$ for the restricted structure sheaf. Let \begin{align*} \mu_s:\mathcal O_X \to L \end{align*} be the sheaf morphism defined on every open set $U\subset X$ by \begin{align*} \mu_s(U):\mathcal O_X(U) &\to L(U) \end{align*} and \begin{align*} f &\mapsto fs|_U. \end{align*} Let \begin{align*} \rho_D:L \to i_*(L|_D) \end{align*} be the sheaf morphism defined on every open set $U\subset X$ by restriction, \begin{align*} \rho_D(U):L(U) &\to L(U\cap D) \end{align*} and \begin{align*} \tau &\mapsto \tau|_{U\cap D}. \end{align*} Here $L(U)$ denotes the $\mathcal O_X(U)$-module of holomorphic sections of $L$ over $U$, and $L(U\cap D)=(L|_D)(U\cap D)$ denotes the $\mathcal O_D(U\cap D)$-module of holomorphic sections of the restricted line bundle $L|_D$ over $U\cap D$. A sequence of sheaves is exact if and only if the induced sequence on every stalk is exact. Thus it is enough to prove that, for every point $x\in X$, the stalk sequence \begin{align*} 0 \longrightarrow \mathcal O_{X,x} \xrightarrow{(\mu_s)_x} L_x \xrightarrow{(\rho_D)_x} (i_*L|_D)_x \longrightarrow 0 \end{align*} is exact. [/step]

[step:Compute the stalk sequence in a local trivialisation]Fix $x\in X$. Choose an open neighbourhood $U\subset X$ of $x$ and a holomorphic frame \begin{align*} e\in L(U) \end{align*} for the line bundle $L|_U$. There is a unique holomorphic function \begin{align*} g:U\to \mathbb C \end{align*} such that \begin{align*} s|_U=ge. \end{align*} Under the frame $e$, the sheaf $L|_U$ is identified with $\mathcal O_U$, and the morphism $\mu_s$ becomes \begin{align*} \mathcal O_U &\to \mathcal O_U \end{align*} with \begin{align*} f &\mapsto gf. \end{align*} The restriction morphism $\rho_D$ becomes the ordinary restriction map \begin{align*} \mathcal O_U &\to i_*\mathcal O_{D\cap U} \end{align*} with \begin{align*} h &\mapsto h|_{D\cap U}. \end{align*} Since $D=(s)_0$ and $e$ is nowhere vanishing on $U$, the set $D\cap U$ is exactly the zero set of $g$.[/step]

[guided]The purpose of the frame $e$ is to replace sections of a line bundle by ordinary holomorphic functions. Because $e$ is a nowhere-vanishing holomorphic section of $L|_U$, every section $\tau\in L(U)$ can be written uniquely as $\tau=he$ for a holomorphic function $h:U\to\mathbb C$. In particular, the given section $s|_U$ has the unique form \begin{align*} s|_U=ge \end{align*} for a holomorphic function $g:U\to\mathbb C$. Under this identification, multiplying a holomorphic function $f\in \mathcal O_X(U)$ by $s$ gives \begin{align*} fs|_U=fge. \end{align*} Therefore the map $\mu_s$ is just multiplication by $g$ on holomorphic functions. Similarly, restricting a section $\tau=he$ to $D\cap U$ gives \begin{align*} \tau|_{D\cap U}=(h|_{D\cap U})(e|_{D\cap U}), \end{align*} so the map $\rho_D$ is just the restriction map $h\mapsto h|_{D\cap U}$ after using the restricted frame $e|_{D\cap U}$ of $L|_D$. Finally, because $e$ is nowhere zero, the section $s|_U=ge$ vanishes exactly where $g$ vanishes. Hence $D\cap U=\{y\in U:g(y)=0\}$. This reduces the proof to the local analytic statement that, for a local defining equation $g$ of a smooth hypersurface, the quotient of holomorphic functions by the ideal generated by $g$ is the sheaf of holomorphic functions on the hypersurface.[/guided]

[step:Use smoothness of the divisor to identify the local ideal] Assume first that $x\in D$. Since $D$ is a smooth divisor, after shrinking $U$ there is a holomorphic coordinate chart \begin{align*} \varphi:U\to \varphi(U)\subset \mathbb C^n \end{align*} with coordinate functions $z_1,\dots,z_n$ such that \begin{align*} D\cap U=\{y\in U:z_1(y)=0\}. \end{align*} Since $g$ and $z_1$ are local defining equations for the same smooth reduced Cartier divisor, the equality of effective Cartier divisors means that they generate the same invertible ideal sheaf of $D\cap U$ in $\mathcal O_U$. Two generators of the same principal ideal in the local ring differ by a unit. Equivalently, after shrinking $U$ again if necessary, there is a nowhere-vanishing holomorphic function \begin{align*} u:U\to \mathbb C \end{align*} such that \begin{align*} g=uz_1. \end{align*} Thus the ideal sheaf generated by $g$ equals the ideal sheaf generated by $z_1$. We now show that the kernel of restriction $\mathcal O_U\to i_*\mathcal O_{D\cap U}$ is the principal ideal $(z_1)$. If $h\in \mathcal O_U(V)$ for an open set $V\subset U$ and $h|_{V\cap D}=0$, then in the coordinate chart, choose each sufficiently small coordinate polydisc $P\subset V$ so that $(tz_1,z_2,\dots,z_n)\in P$ whenever $(z_1,\dots,z_n)\in P$ and $0\le t\le 1$. The condition $h|_{V\cap D}=0$ gives $h(0,z_2,\dots,z_n)=0$ on $P\cap D$. Let $\mathcal L^1$ denote one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $[0,1]$, and define the function \begin{align*} q:P\to \mathbb C \end{align*} by \begin{align*} q(z_1,\dots,z_n)=\int_0^1 \frac{\partial h}{\partial z_1}(tz_1,z_2,\dots,z_n)\,d\mathcal L^1(t). \end{align*} The integrand is holomorphic in $(z_1,\dots,z_n)$ and continuous in $t$ on the compact interval $[0,1]$. On each compact subset of $P$, all coordinate derivatives of the integrand are uniformly bounded for $t\in[0,1]$, so differentiating under the parameter integral is justified by [uniform convergence](/page/Uniform%20Convergence) on compact subsets. Hence $q$ is holomorphic. The one-variable [fundamental theorem of calculus](/theorems/632) along the segment $t\mapsto (tz_1,z_2,\dots,z_n)$ gives \begin{align*} h=z_1q \end{align*} on $P$. Hence the germ of $h$ at every point of $D\cap U$ lies in the ideal generated by $z_1$, and therefore in the ideal generated by $g$. Away from $D$, the function $g$ is a unit, so the ideal generated by $g$ is all of $\mathcal O_U$. Consequently, \begin{align*} \ker(\mathcal O_U\to i_*\mathcal O_{D\cap U})=g\mathcal O_U. \end{align*} [/step]

[step:Verify injectivity and surjectivity on stalks] At any point $x\in X$, the local ring $\mathcal O_{X,x}$ of holomorphic function germs is an [integral domain](/page/Integral%20Domain). If $x\notin D$, then $g_x$ is a unit in $\mathcal O_{X,x}$, so multiplication by $g_x$ is injective and the stalk $(i_*L|_D)_x$ is zero after shrinking to a neighbourhood disjoint from $D$. If $x\in D$, then $g_x$ is a nonzero germ in the integral domain $\mathcal O_{X,x}$, so multiplication by $g_x$ is injective. The previous step identifies the kernel of restriction with $g_x\mathcal O_{X,x}$. The restriction map \begin{align*} \mathcal O_{X,x}\to \mathcal O_{D,x} \end{align*} where $\mathcal O_{D,x}$ is the stalk of the structure sheaf $\mathcal O_D$ at $x$, is surjective because, in the coordinates with $D=\{z_1=0\}$, a holomorphic germ \begin{align*} a\in \mathcal O_{D,x} \end{align*} is represented by a holomorphic function in the variables $z_2,\dots,z_n$, and it is the restriction of the holomorphic germ on $X$ represented by the same expression independent of $z_1$. Transporting this statement through the frame $e$, we obtain exactness of \begin{align*} 0 \longrightarrow \mathcal O_{X,x} \xrightarrow{(\mu_s)_x} L_x \xrightarrow{(\rho_D)_x} (i_*L|_D)_x \longrightarrow 0 \end{align*} for every $x\in X$. [/step]

[step:Conclude that the local exact sequences glue globally] The argument above was made after choosing a local holomorphic frame $e$ for $L$. On an overlap of two trivialising neighbourhoods, the two frames differ by multiplication by a nowhere-vanishing holomorphic function. Multiplication by such a unit does not change the ideal generated by the local defining equation of $D$ and carries the local restriction map for one frame to the local restriction map for the other. Hence the stalkwise exact sequences are independent of the chosen trivialisation. Since the sequence is exact on every stalk, the sheaf sequence \begin{align*} 0 \longrightarrow \mathcal O_X \xrightarrow{\cdot s} L \xrightarrow{\operatorname{res}_D} i_*(L|_D) \longrightarrow 0 \end{align*} is exact. This proves the divisor restriction exact sequence. [/step]