Local description of closed ideals and submodules of analytic functions of one variable. I

Let $P$ be a topological module (over the ring of polynomials) of vector-valued functions $f\colon G\to\mathbf C^q$, holomorphic in a domain $G\subset\mathbf C$.
A closed submodule $I\subset P$ is local (that is, uniquely determined by the collection $I_\lambda$, $\lambda\in G$, of its localized submodules) if and only if $I$ is stable and saturated. A submodule is said to be stable if it admits division by binomials: $f\in I$, $\frac f{z-\lambda}\in I_\lambda\Rightarrow\frac f{z-\lambda}\in I$.
Being saturated amounts to possessing sufficiently many elements.
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