Can someone me say if this (perhaps obvious!) claim is true:

let $f:X\rightarrow S$ be an open, surjective morphism of complex spaces reduced or without embedded components and with $n$-pure dimensional fibers. Let $F$, $G$ be a coherent sheaves with $S$-depth bigger than $2$ and s.t:

1) $G$ is $S$-flat

2) there is a canonical injective morphism $F\rightarrow G$

3) $F$ and $G$ are locally isomorphics.

Question: Is $F$ $S$-flat too?

Rk: Of course, the answer is yes if the Coker of 2) is $S$-flat.

Thanks.

locally free? Do you need a morphism to or from a free sheaf for that? – Sándor Kovács Nov 23 '10 at 8:54