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.