Hi. I asked in the last post if, for a flat morphism $f:X\rightarrow S$ of complex spaces with reduced fibers and $S$ reduced, $X$ is reduced or not. In the algebraic setting, Liu said that the answer is yes. In the analytic setting, If $S$ is smooth, Georges said that we have a positive answer in Fischer book p.158 and i have found another reference in Math Ann. 153, Satz (2.4),p. 245 of Grauert-Kerner. Angelo pointed out that the problem is of local nature and then we can use Thm 23.9, p.184 of Matsumura Commutative Ring Theory.
Then, we remark that:
1) the general case ($S$ reduced and not smooth) is a direkt consequenz of the smooth base case (after base change on smooth part of $S$),
2) In Fischer proposition, $f$ is only assumed to be open (or universally open). In fact, with the assumptions, $f$ is necessarily flat because an open morphism with reduced fibers on smooth base is flat (cf in the analytic setting, Banica-Stanasila, Thm (2.11) p.215 or [EGA4], cor(15.2.3) p.227 in the algebraic context).
I think that we can drop the flatness hypothesis on $f$ and replace it by openness assumption for the same conclusion. For this, we use the fact that openness is stable by base change and the smooth base case as said in the previous post.
Question: Have i make a mistake or is it realy true that we need only openness ?
Thank you very much.
