Complex analytic space

Remarks

Example: $\C^n$ with its sheaf of holomorphic functions is a complex space; take $D = \C^n$ and $k = 0$. The double point is the local model with $D = \C$ and $f_1 = z^2$: the underlying space is $\set{0}$ and the ring of functions is $\C[z]/(z^2)$, which contains the nilpotent $z$.

Non-example: $\C$ with the sheaf of continuous complex functions satisfies clauses 1 and 2 but not 3. The stalks of a local model are quotients of the ring of convergent power series, hence Noetherian. Let $\mathfrak{m}$ be the maximal ideal of the ring of germs of continuous functions at $0$. Then $\mathfrak{m} = \mathfrak{m}^2$, since $f = \sqrt{\abs{f}} \cdot (f / \sqrt{\abs{f}})$. By Nakayama's lemma $\mathfrak{m}$ is not finitely generated, so this ring is not Noetherian.

A complex space is a locally ringed space. Morphisms of complex spaces, also called holomorphic maps, are morphisms of locally ringed spaces of $\C$-algebras. Some authors drop the Hausdorff condition.