Hi.

Question 1: If $f:A\rightarrow B$ be a morphism of local noetherian rings with $B$ is $A$-flat. Let $M$ (resp. $N$) be a $B$ (resp. $A$-)-module of finite type (fin. generated). We assume that $depth_{A}(M)\geq 2$, $M$ is $B$-torsion free and N is $A$-torsion free. Then it is true that $N\otimes_{A}M$ is torsion free ?

I remind the motivation: Let $f:X\rightarrow S$ be a proper and flat morphism of reduced finite dimensional complex spaces with n-dimensional fibers. Let $\omega^{n}_{X/S}$ be the canonical relative sheaf which is fiber wise of $depth>1$ and torsion free on $X$, $G$ torsion free coherent sheaf on $S$.

Question: Is the coherent sheaf $f^{*}G\otimes \omega^{n}_{X/S}$ (which is generically torsion free) torsion free fiber wise or on all of X?

Remark: We can reduce this question to smooth $f$ and especially to the projection $S\times U\rightarrow S$ and replace $\omega^{n}_{X/S}$ by torsion free coherent sheaf on $S\times U$ which is of $depth>1$ fiber wise...

Thank you.

P.S: Thanks to Boyarski for his remark. The last question on flatness and torsion freeness is not deleted but in another count of kaddar with the same name "kaddar".