A point-supported coherent quotient on a polydisc

For each $D\subseteq W$, restriction of ${\mathcal{Q}}_U$ is the cokernel of the finite free map ${\mathcal{O}}_D^{\oplus N}\to {\mathcal{O}}_D$ given by $(a_1,\dots ,a_N)\mapsto {\sum }_j{a}_j{z}_j$. Coherence of ${\mathcal{O}}_D$ and closure under cokernels in [F2] make $\mathcal{Q}$ coherent. Define evaluation ${\mathcal{O}}_D\to i_{\ast }\mathbb{C}$ by $f\mapsto f(0)$ on opens containing $0$ and by the unique zero section on opens disjoint from $0$. Each $z_j$ maps to zero, so the morphism factors through $\mathcal{Q}$.

On the stalk at $0$, evaluation ${\mathcal{O}}_{D,0}\to \mathbb{C}$ is surjective. Its kernel consists exactly of germs with zero constant coefficient. By [F3], write such a germ as $f(z)={\sum }_{\alpha \mathrm{\ne }0}{c}_{\alpha }{z}^{\alpha }$ on a smaller polydisc. Assign each multi-index to its least $j$ with ${\alpha }_j>0$ and set $g_j(z)={\sum }_{\alpha :j=\min \{k:{\alpha }_k>0\}}{c}_{\alpha }{z}^{\alpha -e_j}$. These subseries converge on every still smaller polydisc, and termwise regrouping gives $f={\sum }_{j=1}^N{z}_j{g}_j$. Thus the kernel is precisely $(z_1,\dots ,z_N){\mathcal{O}}_{D,0}$, and the quotient stalk maps isomorphically to $\mathbb{C}$.

If $y\in D$ and $y\mathrm{\ne }0$, choose $j$ with $y_j\mathrm{\ne }0$. The germ of $z_j$ is a unit at $y$, so the quotient stalk ${\mathcal{Q}}_y$ is zero; [F4] says the target stalk is also zero. At $0$, step 1.2 gives an isomorphism on stalks. Hence [F5] makes the induced morphism $\mathcal{Q}\to i_{\ast }\mathbb{C}$ an isomorphism.

By [F4], $\mathrm{\Gamma }(D,i_{\ast }\mathbb{C})=\mathbb{C}$, so step 2.1 gives $\mathrm{\Gamma }(D,\mathcal{Q})\cong \mathbb{C}$. By step 1.1, $\mathcal{Q}={\mathcal{Q}}_U{\mathrm{\mid }}_D$; apply [F6] with the $W$ chosen for ${\mathcal{Q}}_U$ to obtain $H^q(D,\mathcal{Q})=0$ for every $q>0$.