Suppose $G$ is a locally free sheaf on $P^{d}$. $F\_{1}$,$F\_{2}$ are two subsheaves of  

$G$ and they concide on a dense open subscheme of $P^{d}$ .If the quotients of $G$ corresponding to these two subsheaves are torsion free,can we conclude that $F\_{1}=F\_{2}$
everywhere?



Subsheaves of a locally free sheaf on $P^{d}$
Let $F\_3 = F\_1 + F\_2$. Then $F\_3$ coincides with $F\_1$ and $F\_2$ on the same dense open. Therefore $F\_3/F\_1$ and $F\_3/F\_2$ are torsion. But they are also torsion free, as subsheaves of $G/F\_1$ and $G/F\_2$, so they are zero, and $F\_1 = F\_3 = F\_2$.


