ag.algebraic geometry - torsion freeness of tensor product

Hi.

Let $f:Arightarrow B$ be a morphism of local noetherian rings, $M$ (resp. $N$) a $B$ (resp. $A$-)-module of finite type. We assume that $prof_{A}(M)geq 2$ and $N$ is torsion free.
Then it is true that $Notimes _{A}M$ is torsion free ?

Motivation: Let $f:Xrightarrow S$ 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 prof >1) and $G$ torsion free coherent sheaf on $S$.

Question: Is the coherent sheaf $f^{*}Gotimes omega^{n}_{X/S}$ torsion free fiber wise or on all of $X$?

We have a similar result in EGA3, $6. but only if $omega^{n}_{X/S}$ is flat sheaf over $S$...

Thank you.