What is a good reference for flat morphisms of complex-analytic spaces? (The book by Grauert and Remmert doesn't treat them).
Topics I'm interested in: openness of flat maps, descent for coherent analytic sheaves.
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5$\begingroup$ The key is that a non-empty complex-analytic space admits points that are Cohen-Macaulay (work locally, using finite maps to a ball of suitable dimension as in the setup of dimension theory in the G-R book). For a flat map and a CM point in a fiber one can make a "locally quasi-finite quasi-section" adapting the slicing argument as in the scheme case. But for a map of complex-analytic spaces, near an isolated point in a fiber it is finite after shrinking on source and target. Thus, locally on the base there exists a finite flat quasi-section. Then openness and coherent descent are clear. $\endgroup$– nfdc23Commented Mar 10, 2018 at 3:53
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1 Answer
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A reference for the fact that flat maps are open in the complex-analytic category is Theorem 2.12, p. 180 in
C. Bănică, O. Stănăşilă: Algebraic methods in the global theory of complex spaces.
See also my answer to MO question 41158.
answered Mar 10, 2018 at 7:26