Let $A$ be a commutative unital ring and $X$ a separated, quasi-compact scheme over $A$ where the structure map is denoted $f:X\to \text{Spec }A$. Suppose $\mathcal{F}$ is a quasicoherent $\mathcal{O}_X$-module. Are these hypotheses strong enough to ensure that $f_*\mathcal{F}=\widetilde{\mathcal{F}(X)}$?
The notation on the right hand side refers to the sheaf associated to the module $\mathcal{F}(X)$. I know that $f_*\mathcal{F}$ is quasicoherent, but I am not sure if this is enough to conclude. Certainly I hope this is true. If not, what assumptions could one make on the scheme $X$ to ensure it is true? Any and all help is appreciated.
There's an equivalence of categories between quasicoherent sheaves on $\operatorname{Spec} A$ and $A-$Modules. We can send $\mathcal{F}\mapsto \Gamma(\operatorname{Spec}A,\mathcal{F})$ in one direction and in the other we send $M\mapsto \widetilde{M}$. In particular, since you know that $f_*\mathcal{F}$ is quasicoherent (assuming $X$ Noetherian will do), the correspondence for affine schemes between quasicoherent sheaves and modules shows us $$f_*\mathcal{F}\cong \widetilde{\Gamma(\operatorname{spec} A,f_*\mathcal{F})}=\widetilde{\Gamma(f^{-1}(\operatorname{spec} A),\mathcal{F})}=\widetilde{\Gamma(X,\mathcal{F})}=\widetilde{\mathcal{F}(X)}.$$