A coherent point quotient and its first syzygies

If $y\in D$ is not the origin, choose an index $k$ with $z_k(y)\mathrm{\ne }0$. Then $z_k$ is a unit in the germ ring ${\mathcal{O}}_{D,y}$, so the ideal generated by the coordinates is the whole stalk and ${\mathcal{F}}_y=0$.

At the origin, evaluation ${\mathrm{ev}}_0:{\mathcal{O}}_{D,0}\to \mathbb{C}$ is surjective. Its kernel is generated by $z_1,\dots ,z_N$: if $f(0)=0$, its convergent power series has no constant term, and grouping each nonconstant monomial according to the first coordinate with positive exponent writes $f={\sum }_i{z}_i{g}_i$ with convergent germs $g_i$; each of the finitely many grouped subseries remains convergent on a smaller polydisc. Conversely every such sum vanishes at $0$. Hence ${\mathcal{F}}_0\cong \mathbb{C}$.

At a point $y\mathrm{\ne }0$, choose $k$ with $z_k(y)\mathrm{\ne }0$. If $a=(a_1,\dots ,a_N)$ is a relation germ in the stalk from [F6], then $a_k=-{\sum }_{i\mathrm{\ne }k}{a}_i{z}_i\mathrm{/}{z}_k$, so

Each vector $z_k{e}_i-z_i{e}_k$ is one of the stated pairwise sections or its negative. Thus those sections generate the relation stalk at every nonzero point, including when $N=1$, where the sum is empty.

Let $a=(a_1,\dots ,a_N)$ be a relation germ at the origin, so ${\sum }_i{a}_i{z}_i=0$. We prove by induction on $N$ that it is generated by the stated pairwise sections. For $N=1$, $a_1{z}_1=0$ and the germ ring is a domain by [F5], so $a_1=0$. For $N>1$, restrict to $z_N=0$; the first $N-1$ coefficients form a relation among $z_1,\dots ,z_{N-1}$. By induction subtract a combination of the pairwise sections with both indices below $N$, extending their coefficient germs independently of $z_N$. The resulting first $N-1$ coefficients vanish at $z_N=0$.

By [F5], each remaining coefficient has a convergent series divisible by $z_N$, so write $a_i=z_N{c}_i$ for $i<N$. The relation becomes $z_N({\sum }_{i<N}{c}_i{z}_i+a_N)=0$. Since the germ ring is a domain, $a_N=-{\sum }_{i<N}{c}_i{z}_i$, and the remaining vector is ${\sum }_{i<N}{c}_i(z_N{e}_i-z_i{e}_N)$, a combination of the stated pairwise sections.

At every point, steps 1.3–2.1 show that the pairwise sections generate the kernel stalk. Therefore [F4] identifies their generated subsheaf with $\ker \mu$. The quotient sequence is exact by the definition of $\mathcal{F}$ as a cokernel.

Under the assumed Axiom of Choice [A1], [F2] makes ${\mathcal{O}}_D$ coherent. Repeatedly applying the finite-extension case of [F3] to the split sequences $0\to {\mathcal{O}}_D^{\oplus (j-1)}\to {\mathcal{O}}_D^{\oplus j}\to {\mathcal{O}}_D\to 0$ shows that ${\mathcal{O}}_D^{\oplus N}$ is coherent. The cokernel case of [F3] applied to $\mu$ then makes $\mathcal{F}$ coherent.