Flat map with reduced fibers. - MathOverflow

Flat map with reduced fibers.

2011-01-03T10:46:59Z

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.

Answer by Georges Elencwajg for Flat map with reduced fibers.

2011-01-03T14:56:40Z

Dear kaddar, here is a partial answer.

According to a theorem of Douady, a flat map $f:X\to S$ between complex analytic spaces is always open . So if you assume that the fibers of $f$ are reduced and that your reduced space $S$ is actually smooth (i.e. is a manifold), then $X$ is indeed reduced : this follows from the Proposition on page 158 of Gerd Fischer's Complex Analytic Geometry (Springer, LNM 538, 1976).

Edit $\;$ On the evoked relation between flat and open, let me add the following. It is not true that an open morphism $f:X\to S$ of complex spaces is flat: the simplest counter-example is the immersion of a simple point into a double point i.e. the morphism of schemes $Spec \;\mathbb C \to Spec \; \mathbb C[\epsilon ] \quad(\epsilon^2=0)$ seen analytically. However if $X$ and $S$ are complex manifolds then it is true that $f$ open implies $f$ flat (Fischer, same page 158) and so for morphisms between manifolds you have the easy to remember equivalence flat=open, which helps understand the notoriously unintuitive notion of flatness.

Answer by Angelo for Flat map with reduced fibers.

2011-01-03T19:44:35Z

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.