Let $X=Spec A$ be affine scheme. Then $O_{X,p}\cong A_p$ for $p\in Spec A$ and taking any $f\not\in p$, I have $A_f\otimes_{A_f}A_p\cong A_p$ as $A_f$ module. In particular $A_f=O_X(D(f))$.
My guess is that $O_X(D(f))\otimes_{A_f}A_p\cong A_p$ and $O_X(U)\otimes_{A_f}(A_p)=A_p$ for any open $U\subset D(f)$. $D(f)$ denotes basis of the topology.
Q1: Is the above assertion true? My guess is that $O_X(D(f))\to O_X(U)\to O_{X,p}$ is the module morphism I would like to consider. However, I did not figure out why $O_X(U)\otimes_{A_f}A_p\cong A_p$ should be the case.
Q2. If above is true, how do I prove it? If $O_X$ is flasque, then it surely holds. Under what circumstances, can I conclude that it holds?
Q3. Does it hold in general scheme construction?