In a category $\mathcal C$, we will say that $(\mathcal T,\mathcal F)$ is a torsion theory if it satisfies:

(1) $Hom(T,F)=0$ for all $T\in \mathcal T$ and $F\in \mathcal F$.

(2) If $Hom(T,F)=0$ for all $F\in \mathcal F$, then $T\in \mathcal T$.

(3) If $Hom(T,F)=0$ for all $T\in \mathcal T$, then $F\in \mathcal F$.

For modules over a commutative integral domain $A$, torsion modules and torsion free modules form such a pair.

For sheaves of $\mathcal O_X$-modules over an integral scheme, we can take locally torsion and locally torsion free $\mathcal O_X$-modules, i.e., torsion and torsion free respectively on each affine.

My question is : does this still hold for quasi-coherent sheaves on an integral scheme? In other words, do locally torsion and locally torsion free quasi-coherent sheaves still give a torsion theory on $QCoh(X)$?