Let $f:X\rightarrow S$ be a surjective proper, open morphism of reduced or without embedded component complex spaces (or, in alg.geom, surjective proper, universally open morphism of excellent, locally noetherian reduced or without embedded component schemes). Let $g:T\rightarrow S$ be a flat morphism with $T$ reduced or without embedded component.

Question: It is true that the fiber product $X_{T}:=X\times_{S} T$ is reduced or without embedded component?

I think that is true essentially because that the projection $X_{T}\rightarrow X$ is flat and then contract an irreducible component of $X_{T}$ on an irreducible component of $X$.

But universally open map have this propertie and reduced or without embedded component are not preserved by universally open base change....

Thank you.

  • $\begingroup$ For reducedness the answer is negative in the algebraic setting. But in the analytic setting, reducedness implies the smooth locus is "schematically dense", so reducedness should be preserved (loosely, if $T'$ is smooth locus in $T$ and $X'$ is $S$-smooth locus in $X$ then $X'_T$ should be "schematically dense" in $X_T$ due to $S$-flatness of $T$ and "schematic density" of $X'$ in $X$ due to reducedness of $S$ and $X$, and then $X'_{T'}$ should be "schematically dense" in $X'_T$ because $X'$ is $S$-flat). Works algebraically if have enough char-0 or separability stuff flying around. $\endgroup$ – BCnrd Sep 8 '10 at 21:52
  • $\begingroup$ Oh, by the way, for the suggested strategy for reducedness I was implicitly using that $g$ is automatically open, and so $S$ is forced to be reduced along the open image of $g$ (and hence one may replace $S$ with that to arrange $S$ is reduced). For embedded component stuff, look in section 3 of EGA IV$_2$ for some inspiration from the algebraic case; they discuss how associated points behave with respect to flat base change (and those arguments should adapt to analytic setting). $\endgroup$ – BCnrd Sep 8 '10 at 21:56
  • $\begingroup$ Thank you Brian. If i understand, you say that, in the analytic setting (and perhaps in the alg.setting for char 0 with the above assumption) reducedness is preserved by flat base change. In your arguments, it seems to me that we only use the universal openess of the flatness. But i know that reducedness is not preserved by universally open base change. I dont understand (for the moment) how the full concept of flatness play an important role in this problem? $\endgroup$ – kaddar Sep 9 '10 at 7:15
  • $\begingroup$ For different reasons, I see, now, the necessity of flatness for the reduced case (see, for example, Banica Lect.Notes 743, p.389). $\endgroup$ – kaddar Sep 9 '10 at 9:06
  • $\begingroup$ Dear Kaddar: I was using more than universal openness. I was using that flatness preserves injectivity, loosely speaking, in the sense of preserving "schematic density" (so reducedness can be checked over certain open subspaces where smoothness methods may be applied). $\endgroup$ – BCnrd Sep 10 '10 at 1:24

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