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In the definition of a complex space (in the sense of Grauert), one defines a model space (one to which we require a complex space to be locally isomorphic) to be the support of the quotient sheaf $\mathscr O_X/\mathscr I$ for an analytic set $X\subset\mathbb C^n$ and a finitely generated sheaf of ideals on it. Indeed there seems to be a reccuring motiff of finite generatedness of ideals within the theory of complex spaces; for example, only finitely generated maximal ideals of a Stein algebra correspond to the points of the associated Stein space.

Likewise a complex space is allways (in the literature I am aware of anyway) defined to be Hausdorff. Those two structural requirements bother me, though admittedly more or less only because with schemes, the algebro-geometric analogues of complex spaces, nothing of such is imposed (though in applications one often deals only with separated schemes of finite type).

Could somebody please explain the reasons for these requirements, where they come into play in essential ways, and potentially if they can be dropped and still make for an interesting theory?

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    $\begingroup$ The Hausdorff condition is a matter of taste. The coherence condition on $\mathscr{I}$ is an essential feature to get a reasonable theory, with coherent structure sheaf, etc. The notion of coherence in holomorphic function theory and its robust properties was one of the greater discoveries of Oka, latter clarified and refined via sheaf-theoretic ideas of Cartan, Serre, et al.; one cannot do anything interesting in the style of complex-analytic geometry without it. Motivation for the objects of study is always a good thing to keep in mind. :) $\endgroup$
    – user76758
    Jan 15, 2014 at 19:01

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If X is a scheme and $\mathcal{I} \subset \mathcal{O}_X$ is a sheaf of ideals, then in general $\mathcal{I}$ does not define a closed subscheme. For example, suppose that $X = \mathbf{A}^n_k$ where $k$ is a field and $\mathcal{I} = j_!\mathcal{O}_U$ where $j : U \to \mathbf{A}^n_k$ is the inclusion of the complement of zero and $j_!$ is the extension by zero functor. To get a closed subscheme you have to assume that $\mathcal{I}$ is locally generated by sections, see Lemma Tag 01QQ.

I think that if $\mathcal{I}$ is a sheaf of ideals in the sheaf of holomorphic functions on an open ball in $\mathbf{C}^n$ generated by global sections, then its restriction to a strictly smaller ball is in fact generated by finitely many sections (didn't work out all details; maybe user76758 can help). Thus the situation (wrt ideal sheaves) is similar.

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    $\begingroup$ Yes, use the "noetherian" property of coherent sheaves on a complex-analytic space $X$ (see Ch. 5, section 6, of the book "Coherent analytic sheaves"): any directed system $\{F_j\}$ of coherent subsheaves of a coherent $O_X$-module is locally stationary (i.e., $X$ admits an open cover $\{U_i\}$ such that for each $i$ there is some $j(i)$ with $F_j|_{U_i} = F_{j'}|_{U_i}$ for all $j, j' \ge j(i)$). I think this is due to Serre. Exhausting your $\mathcal{I}$ by the coherent ideals arising from finite subsets of the global set, this "noetherian" property does the job. $\endgroup$
    – user76758
    Jan 15, 2014 at 22:11

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