TITLE: Flat map with reduced fibers.
QUESTION [5 upvotes]: Hi.
Let $f:X\rightarrow S$ be a flat, surjective morphism of complex spaces with reduced fibers over $S$ reduced. 
Q1: Is $X$ reduced too?
Q2: Is the property " reduced fiber" preserved by base change given by the normalization?
Rk: 1) We have some result of this kind in [EGA4], Cor (3.3.5) p.44. 
2) We can apply [Matsumara], Cor(ii) p.189 but with the additional asumptions: $S$ is normal and $X$ locally pure dimensional.
Thanks.

REPLY [4 votes]: This is a consequence of the following result: if $A \to B$ is a flat local homomorphism of local rings, $A$ and $B/\mathfrak{m}_AB$ are reduced, then $B$ is reduced. Keeping in mind that reduced is equivalent to $R_0$ and $S_1$, this follows from Theorem 23.9 in Matsumura's Commutative Ring Theory.
I don't know what the second question means.
[Edit] I just noticed that the hypotheses in the cited theorem are actually stronger, so that it would imply the result for schemes but not for analytic spaces, at least not immediately. I need to think about it some more.