Abstract
The main result of this paper is the theorem: Proper flat morphisms of complex spaces are Zariski-open. In general a flat epimorphism of complex spaces f:X → S need not be open relativ to the Zariski-topology on X and S (example 5.3). But it is shown in this paper that for flat morphisms f:X → S to every xOeX there is an open neighbourhood U of xo in X such that for any Zariski-open subset Z of X the set f (Z∩U) is Zariski-open in f (U). An important tool for the proof of this proposition is the notion of the “devissage relatif” introduced by M. RAYNAUD and L. GRUSON in [11]. In case that f is also proper this local result about flat morphisms yields the global one stated above.
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