Hi.
Question 1: let $f:X\rightarrow S$ be a proper and surjective morphism of complex reduced spaces with $X$ pure dimensional. Let $F$ be a $S$-flat coherent sheaf on $X$. Is it true that the two condition a) and b) are equivalent:
a) There $s$ in $S$ for which $F_{s}$ is ${\cal O}_{X_{s}$-torsion free.
b) $F$ is ${\cal O}_{X}$-torsion free.
Remark: Of course, we can suppose that $Supp(F)=X$ and then $f$ is universally open.
Question 2: Suppose that $f$ is open of fiber dimension constant $n$ and let $U$ be an open polydisk in $C^{n}$, $g:X\rightarrow S\times U$ a relative local parametrization of $f$ which is finite, surjective and universally open morphism. Consider the relative canonical sheaf $\omega^{n}_{X/S}$ and the isomorphism $g_{*}\omega^{n}_{X/S}\simeq {\cal H}om(g_{*}({\cal O}_{X}), {\cal O}_{S\times U})$
then it is true that (i) $S$ reduced imply $\omega^{n}_{X/S}$ is torsion free on $X$.
(ii) $S$ normal imply $\omega^{n}_{X/S}$ is of depth > 1 on $X$.
Remark: (i) and (ii) are true for $S$-torsion freeness and $S$-depth.
Thank you very much.
