Oka coherence and local coherent free resolutions

Work under [A1]. In a holomorphic chart, prove the relation assertion by induction on dimension $n$, and for a fixed $n$ by induction on tuple length $q$. On ${\mathbb{C}}^0=\{0\}$ each relation space is a subspace of the finite-dimensional vector space ${\mathbb{C}}^q$, so a finite basis gives local generators. For any dimension, the empty tuple has zero relation sheaf.

If some tuple entry has zero germ at the chosen point, [F13] gives a neighborhood on which that entry vanishes identically; its coordinate in a relation is then arbitrary, and the relation sheaf is the direct sum of one copy of $\mathcal{O}$ with the shorter tuple's relation sheaf. The induction on $q$ gives finite local generators, and repeating this finite reduction removes every zero germ. If some remaining entry, say $f_1$, is nonzero at the point, shrink until $f_1$ is a unit; the sections $e_j-(f_j\mathrm{/}{f}_1)e_1$, $2\le j\le q$, generate the relation sheaf because the relation equation uniquely determines its first coordinate.

It remains to treat a tuple of nonzero germs all vanishing at the point. By [F11], the product $G={\prod }_j{f}_j$ is a nonzero germ. Apply [F8] to $G$; after the resulting linear coordinate change the restriction of $G$ to the last coordinate line has finite order. Since this restriction is the product of the corresponding one-variable restrictions of the $f_j$, each factor is nonzero and hence has a finite order; each $f_j$ is therefore regular in the last variable.

Apply [F9] to each of the finitely many $f_j$ and intersect the resulting neighborhoods. On one common product polydisc each $f_j=u_j{P}_j$, where $u_j$ is a unit and $P_j$ is a monic Weierstrass polynomial. Multiplication of the relation coefficients by the units $u_j$ is an isomorphism from the relation sheaf of $(f_j)$ to that of $(P_j)$. Set $d={\max }_j{\mathrm{deg}}_w{P}_j$.

By [F4], whose quotient/remainder step uses the uniqueness in [F10], every relation germ of $(P_1,\dots ,P_q)$ is generated by polynomial relation germs whose components have $w$-degree at most $d$. Such vectors are identified with coefficient vectors in ${\mathcal{O}}_{{\mathrm{\Delta }}^{\mathrm{\prime }}}^{\oplus q(d+1)}$, where ${\mathrm{\Delta }}^{\mathrm{\prime }}$ is the base polydisc. Multiplication by the $P_j$ defines a finite matrix map $\mathrm{\Psi }:{\mathcal{O}}_{{\mathrm{\Delta }}^{\mathrm{\prime }}}^{\oplus q(d+1)}\to {\mathcal{O}}_{{\mathrm{\Delta }}^{\mathrm{\prime }}}^{\oplus (2d+1)}$ by taking coefficients of ${\sum }_j{Q}_j{P}_j$. Its kernel is exactly the bounded-degree polynomial relation sheaf.

By dimension induction, ${\mathcal{O}}_{{\mathrm{\Delta }}^{\mathrm{\prime }}}$ is coherent; [F3] therefore makes $\mathcal{K}=\ker \mathrm{\Psi }$ locally of finite type. At the base point, its stalk is a submodule of $A^{\oplus q(d+1)}$, where $A={\mathcal{O}}_{n-1,0}$ after translation. If $n>1$, [F6] makes $A$ Noetherian; if $n=1$, $A=\mathbb{C}$ is a field and hence Noetherian. The finite free module is finitely generated, so [F7] makes it Noetherian and its submodule ${\mathcal{K}}_0$ finitely generated.

The sheaf $\mathcal{K}$ is coherent: it is locally of finite type by step 6.1, and for any finite free map into $\mathcal{K}$, composition with its inclusion into the finite free domain of $\mathrm{\Psi }$ has the same kernel, locally of finite type by [F3]. A finite-rank free module over the coherent structure sheaf is coherent, since its standard basis gives finite type and [F3] gives the kernel condition. Choose finitely many germs generating ${\mathcal{K}}_0$ and represent them by sections on one smaller base neighborhood. Their cokernel in $\mathcal{K}$ is coherent by [F5]; its stalk at the base point is zero by the chosen generators and the stalk definition [F13]. Since that cokernel is locally of finite type, a finite generating family has zero germs there and vanishes after a common shrinking. Thus the selected sections generate every kernel stalk on a smaller base neighborhood.

Regard these finite coefficient sections as polynomial relation sections on the product polydisc. Together with the finitely many elementary polynomial relations $P_q{e}_j-P_j{e}_q$, they generate every bounded-degree relation stalk over the smaller base neighborhood; [F4] then makes them generate every relation stalk of $(P_1,\dots ,P_q)$ throughout a smaller product neighborhood. The unit isomorphisms from step 4.1 transfer these generators back to $(f_1,\dots ,f_q)$. Equality of the generated subsheaf with the relation sheaf follows from their equal stalks by [F12]. Thus the relation sheaf is locally of finite type at the chosen point.

Since the point and tuple were arbitrary, every finite tuple of holomorphic functions on every chart has a locally finite-type relation sheaf. A holomorphic coordinate change induces an isomorphism of structure sheaves and transports the relation map and its kernel, so the chart-local property is independent of coordinates. Hence [F2] makes ${\mathcal{O}}_M$ coherent on all of $M$.

Apply [F5] to the coherent structure sheaf ${\mathcal{O}}_M$. It gives closure of coherent ${\mathcal{O}}_M$-modules under kernels, cokernels, and finite extensions, as stated.

Fix coherent $\mathcal{F}$, $x\in M$, and finite $\mathrm{\ell }\ge 0$. Local finite type gives a finite-rank epimorphism ${\mathcal{O}}^{\oplus p_0}\to \mathcal{F}$ near $x$. The finite free source is coherent by [F3] and the coherence of ${\mathcal{O}}_M$ from step 9.1; hence its kernel is coherent by [F5]. Choose a finite-rank epimorphism onto that kernel, compose with its inclusion into ${\mathcal{O}}^{\oplus p_0}$, and repeat this operation until $\mathrm{\ell }+1$ free modules have been chosen. At each intermediate stage the kernel is coherent by [F5], so the next finite-rank epimorphism exists locally.

Only finitely many neighborhoods are used for the finite length $\mathrm{\ell }+1$ construction; their intersection is a chart neighborhood $W$ of $x$. Each new free module maps onto the preceding kernel, so the resulting sequence is exact at $\mathcal{F}{\mathrm{\mid }}_W$ and every free term before the leftmost one. No further kernel is chosen at the leftmost term, and no global choices of generators or global resolution are asserted.