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A common technique for constructing objects (sheaves) and morphisms in algebraic geometry is faithfully flat descent. Roughly speaking this consists on constructing an object or a morphism "locally" in a fpqc covering of a scheme, verifying that the local constructions form a descent datum, and using the theorem that says that fpqc descent data are effective to show that there is a global object or morphism defined over the original scheme.

Usually, a lot of constructions from algebraic geometry can be easily done in the complex geometric setting. Moreover, usual "analytic" opens of a complex variety are "smaller" than Zariski opens, so most constructions work better in the analytic topology. For example, to develop a good theory of principal bundles on the algebraic setting, one needs to work with étale-locally trivial objects, rather than with Zariski-locally trivial ones.

My question is if there is a "canonical" way to translate an argument or construction in the algebraic setting using fpqc descent to an argument in complex geometry.

Roughly speaking my question would be if, given a construction over a complex algebraic variety using fpqc descent, one could do the same construction over the analytification by, for example, gluing local data over a covering by usual opens.

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    $\begingroup$ What example do you have in your mind? Note that analytic localizations are probably not flat on quasicoherent sheaves in a reasonable framework (e.g. condensed mathematics). The following is given by Clausen–Scholze: consider a disc $D$ and two disjoint discs $D_1,D_2$ inside $D$. Then the base change along $\mathcal O(D)\to\mathcal O(D_1)$ is not flat: the map $\mathcal O(D)\to\mathcal O(D_2)$ is injective by unicity, but after base change, it becomes $\mathcal O(D_1)\to\mathcal O(D_1\cap D_2=\varnothing)=0$. $\endgroup$
    – Z. M
    Commented Jan 26, 2023 at 16:55

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There will not be a way to directly translate faithfully flat descent into gluing over analytic opens; the former is genuinely stronger than the latter in the analytic category. However basically any faithfully flat descent statement that holds for schemes and makes sense for complex analytic spaces (including those involving coherent sheaves) does hold. See Kiehl's paper Äquivalenzrelationen in analytischen Räumen, for example.

The greater flexibility of analytic opens does manifest itself in improvements to effective descent results (or, more generally, effectivity of quotients). The aforementioned paper more or less shows that every separated flat equivalence relation is effective, which is certainly not true for schemes (you'd get an algebraic space in general).

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