Torsion freeness and injectivity of morphism of some coherent sheaves

Hi, everybody.

My problem is to give a relative version of the well known fact which say that a generical isomorphism between torsion free coh.sheaves on complex space (or in more general setting with some reasonnable assumptions) is necessarily injectiv.

The claim: Let $X$ and $S$ be a complex spaces of finite dimension (perhaps reduced) (or locally noetherian excellent schemes), $\pi:X\rightarrow S$ be an open and surjective map with constant fiber dimension. Let $A$ and $B$ two coherent sheaves on $X$ which satisfies the properties:

1) There is an open dense subset $U$ of $X$ on which the restriction of $A$ and $B$ are canonically isomorphics

2) For every open set $V$ of $X$ s.t $V\cap \pi^{-1}(s)$ is dense in $\pi^{-1}(s)$, the natural restriction morphism $\Gamma(X, A)\rightarrow \Gamma(V, A)$ is injectiv (the same is true for $B$).

Then any morphism $f:A\rightarrow B$ which is an isomorphism on $U$, is injectiv.

P.S: Of course, the problem is local and we can translate this claim in terms of analytic algebra or modules..

Thank you.

-