In Hasrtshorne the pullback $f^{*}\mathcal{F}$ of a sheaf $\mathcal{F}$ on $Y$ via a map $f:X \rightarrow Y$ is defined as $f^{-1}\mathcal{F}\otimes_{f^{-1}\mathcal{O}_Y}\mathcal{O}_X$.
It is quite tricky to me to visualize it, since the definition of the functor $f^{-1}$ is pretty nasty.
In particular, in proposition 5.2 the book says $f^{*}(\widetilde{M})=\widetilde{M \otimes_A B}$ when $X=\mathrm{Spec} A$ and $Y=\mathrm{Spec} B$, where $f:X \rightarrow Y$ is a morphism of schemes. How to prove this? Also, an algebra question which came to me thinking about it is the following: is there a nice way of thinking to the localizations of the $A$ module $M \otimes_B A$ as tensors of suitable localizations as $B$-modules?