I'm basically interested in knowing the difference between complex spaces and schemes when studying stacks. I'd like to use stacks to study moduli problems in complex analytic geometry.

Citing a proposition from the stacks project:

Let $S$ be a scheme. Let $\{X_i \rightarrow X\}$ be an fpqc covering of algebraic spaces over $S$. Any descent datum on quasi-coherent sheaves for $\{X_i \rightarrow X\}$ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_X$-modules to the category of descent data with respect to $\{X_i\rightarrow X\}$ is fully faithful.

Are there similar/different results for complex spaces? I don't seem to grasp the underlying difficulties. Any references or explanations would be appreciated.

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    $\begingroup$ what's a quasi-coherent sheaf on a complex space? $\endgroup$ – Yosemite Sam Feb 8 '17 at 22:22
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    $\begingroup$ @Yosemite Sam:I think it's the same as for ringed spaces: a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ such that $\mathcal{O}_X^{\oplus I}\to \mathcal{O}_X^{\oplus J}\to \mathcal{F}\to 0$ is exact for possibly infinite index sets $I$ and $J$. $\endgroup$ – Qfwfq Feb 9 '17 at 0:30
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    $\begingroup$ Edit: in my previous comment, I meant locally on $X$ $\endgroup$ – Qfwfq Feb 9 '17 at 0:38
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    $\begingroup$ I assume you know material as in the book Coherent Analytic Sheaves; if not, read it. The existence of finite flat sections locally on the base for a flat analytic surjection is proved via slicing through Cohen-Macaulay points in fibers (argue from there as in EGA IV$_4$,17.6.2); the existence of such points is proved via the analytic version of Noether normalization. All descent questions for coherent sheaves are thereby settled well. For a flat equivalence relations $d:R\to X\times X$ to have effective quotient, $d$ must factor as $i\circ j$ for open immersion $j$ and closed immersion $i$. $\endgroup$ – nfdc23 Feb 9 '17 at 5:14
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    $\begingroup$ The reason for the necessity of that condition on $d$ is that diagonals of analytic spaces satisfy that property (since analytic spaces are locally Hausdorff). Under that assumption, I have a hazy memory of once working out for myself that the descent is effective; in the setting of etale equivalence relations this is modeled on the proof of Prop. 5.18 in section 5 of Ch. 1 of D. Knutson's book Algebraic Spaces, and one adapts that to the flat setting by cutting out "finite flat charts" for $X/R$ via slicing of $R \rightarrow X$ through CM-points on $X$ and then CM-points over them in $R$. $\endgroup$ – nfdc23 Feb 9 '17 at 5:31

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