Coherent-section quotient topologies are Fréchet and presentation-independent

In chart coordinates, choose a compact exhaustion $K_1\subseteq K_2\subseteq \cdots$ of $V$, for example $K_n=\{z:\mathrm{\mid }z\mathrm{\mid }\le n,\mathrm{dist}(z,{\mathbb{C}}^m\setminus V)\ge 1\mathrm{/}n\}$, with distance to the empty set taken as $+\mathrm{\infty }$. The seminorms $p_n(f)={\sup }_{K_n}\mathrm{\parallel }f\mathrm{\parallel }$ generate compact-convergence. The resulting topology on $\mathcal{O}(V{)}^{p_0}$ is Hausdorff and locally convex. A Cauchy sequence converges uniformly on each $K_n$ to a compatible continuous limit; it converges locally uniformly on $V$, so [F3] makes the limit holomorphic. Thus $\mathcal{O}(V{)}^{p_0}$ is Fréchet.

Let $R={\mathcal{O}}_{M,x}$ and let $L_x\subseteq R^{p_0}$ be the stalk of the relation sheaf $\ker ({\mathcal{O}}_V^{p_0}\to \mathcal{F}{\mathrm{\mid }}_V)$. By [F1], $L_x$ is finitely generated. Artin–Rees in [F2] identifies the topology induced on $L_x$ from $R^{p_0}$ with its intrinsic Krull topology, while Krull intersection makes $R^{p_0}\mathrm{/}{L}_x$ separated; thus $L_x$ is closed in the analytic Krull topology. If relation sections converge uniformly on compact subsets, [F3] makes every finite Taylor jet converge. For each $k$, the image of $L_x$ in the finite-dimensional jet space $R^{p_0}\mathrm{/}{\mathfrak{m}}_x^k{R}^{p_0}$ is a linear subspace and hence closed. The limiting jet therefore lies in that image, so the limit germ belongs to $L_x+{\mathfrak{m}}_x^k{R}^{p_0}$ for every $k$. Krull intersection applied to the finite local module $R^{p_0}\mathrm{/}{L}_x$ gives ${\bigcap }_k(L_x+{\mathfrak{m}}_x^k{R}^{p_0})=L_x$, proving the claimed membership explicitly. This holds at every $x$, and the sheaf-kernel condition puts the limit in the relation section space. Therefore the kernel of the global section map is closed in $\mathcal{O}(V{)}^{p_0}$.

A quotient of a Fréchet space by a closed linear subspace is Hausdorff and metrizable by the quotient seminorms. For completeness, choose a decreasing absolutely convex neighborhood basis $U_n$ in the original space such that $p_j(u)\le 2^{-n}$ for $u\in U_n$ and $j\le n$, for a defining countable seminorm family $(p_j)$. From a Cauchy sequence of cosets choose a subsequence whose $n$th successive difference lies in the open quotient image of $U_n$, and lift that difference to some $u_n\in U_n$. For each fixed $j$, ${\sum }_n{p}_j(u_n)$ converges, so $(u_n)$ is summable in the Fréchet topology. Its sum gives a limit of the subsequence's cosets, and the original Cauchy sequence converges to the same coset. Local convexity descends through the open quotient map. Applying this argument to the closed kernel in step 1.2 proves the quotient topology is Fréchet.

Let two finite presentations be given near a point $x$, with coherent relation kernels $K,K^{\mathrm{\prime }}$ by [A2]. Apply [A3] to both kernels and intersect the resulting polydisc neighborhoods with the presentation neighborhoods. On a sufficiently small coordinate polydisc $D$ in this intersection, $H^1(D,K)=H^1(D,K^{\mathrm{\prime }})=0$. The long exact sequence [F5] for each presentation then proves that both section maps on $D$ are surjective. More generally, on any common open set $W$ where both presentations are defined and their section maps are surjective, lift the images of the finitely many basis sections of the first free module through the second section surjection. These lifts form a holomorphic matrix $A$ on $W$ with ${\pi }^{\mathrm{\prime }}A=\pi$. Multiplication by $A$ is continuous for compact-convergence because its entries are bounded on compact subsets; it induces the identity continuously between the quotient topologies. Reversing the presentations gives the reverse continuous identity. Thus the quotient topologies agree on each common presentation neighborhood. A holomorphic change of chart preserves compact-convergence because it and its inverse send compact subsets to compact subsets.

First consider a presentation neighborhood $V$ with its surjective section map and quotient topology. If a small chart polydisc $D\subseteq V$ has a target presentation whose section map is surjective, the restricted images of the finitely many source free generators lift to target free sections on $D$. As in step 3.1, these lifts give a holomorphic matrix on $D$. Restricting source free sections and multiplying by that matrix is continuous for compact-convergence, so passage through the two quotient maps proves continuity of $\mathrm{\Gamma }(V,\mathcal{F})\to \mathrm{\Gamma }(D,\mathcal{F})$. For any countable cover $(V_i)$ by such presentation neighborhoods, choose countably many sufficiently small polydiscs $D_{ij\nu }$ covering every overlap $V_i\cap V_j$, with surjective local section maps as in step 3.1. The restrictions from each $V_i,V_j$ into each such polydisc are continuous by the matrix argument. Therefore the compatible families are the intersection of the kernels of the continuous differences into $\mathrm{\Gamma }(D_{ij\nu },\mathcal{F})$. Those targets are Hausdorff Fréchet spaces by step 2.1, so this intersection is closed in the countable product of the local section spaces. A countable product of Fréchet spaces is Fréchet by enumerating their seminorms and taking componentwise limits; a closed subspace is complete for its restricted metric. Thus the compatible-family space is an actual Fréchet space. Compatibility checked on these overlap covers is full compatibility by the sheaf axiom.

To compare the quotient topology on an arbitrary presentation neighborhood $V$ with its topology from local sections, cover $V$ by countably many sufficiently small presentation polydiscs $(D_{\nu })$ as in step 3.1. The restriction map from the already Fréchet quotient $\mathrm{\Gamma }(V,\mathcal{F})$ into the compatible-family space for this cover is continuous by step 4.1. It is a linear bijection by unique sheaf gluing. Both its source and target are Fréchet by step 2.1 and step 4.1, so [F6] makes it a topological isomorphism. In particular, inverse gluing is continuous, and the original arbitrary-$V$ presentation quotient agrees with the compatible-family topology; no unproved identification of these topologies is made.

Any two countable presentation covers have a common countable refinement by sufficiently small presentation polydiscs; choose them inside the local vanishing neighborhoods of the finitely many relation kernels involved at each source-target comparison, as in step 3.1. Restriction from either compatible-family space into the refined compatible-family space is continuous by the local matrix argument of step 4.1 and is a bijection by sheaf gluing. All these spaces are Fréchet by step 4.1, so [F6] makes both maps topological isomorphisms. The identity on ordinary sections therefore identifies the two cover topologies. Together with the presentation and chart comparison in step 3.1 and the quotient-to-family comparison in step 5.1, this proves every asserted independence.