# Descent for complex-analytic spaces

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.

• what's a quasi-coherent sheaf on a complex space? – Yosemite Sam Feb 8 '17 at 22:22
• @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 $J$. – Qfwfq Feb 9 '17 at 0:30
• Edit: in my previous comment, I meant locally on $X$ – Qfwfq Feb 9 '17 at 0:38
• 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 $i$. – nfdc23 Feb 9 '17 at 5:14
• 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 $R$. – nfdc23 Feb 9 '17 at 5:31