TITLE: Inverse image of the sheaf associated to a module
QUESTION [18 upvotes]: In Hartshorne, Algebraic geometry it's written, that for every scheme morphism $f: Spec B \to Spec A$ and $A$-module $M$ $f^*(\tilde M) = \tilde {(M \otimes_A B)}$. And that it immediately follows from the definition. But I don't know how to prove it in simple way. Could you help me?

REPLY [6 votes]: As Martin indicates in his answer, it is easiest to prove that $f_*$ corresponds to the forgetful functor from $B$-modules to $A$-modules, and then use adjointness to understand $f^*$.
Again as Martin notes, once we know that $f_*$ takes the quasi-coherent sheaf $\widetilde{N}$ to a quasi-coherent sheaf, we can determine which sheaf we get by taking global sections.
However, it is not quite automatic that $f_*$ takes a quasi-coherent sheaf to a quasi-coherent sheaf (e.g. this is not true in complete generality for arbitrary maps between arbitrary schemes).  
In the affine case, one can get around this issue as follows:

To determine $f_*\widetilde{N}$, it is enough to understand its behaviour on the basis of open sets of the form Spec $A_a$ ($a \in A$).  
The preimage of Spec $A_a$ is Spec $B_a$.
So, by definition of $f_*$, the sections of $f_*\widetilde N$ on Spec $A_a$ are $N_a$, with the evident restriction maps if $a \mid a$'.  (Note that $N_a$ has the same meaning whether we regard $a$ as an element of $A$ and regard $N$ as an $A$-module, or regard $a$ as an element of $B$, via the morphism $A \to B$, and regard $N$ as a $B$-module.)
When we regard $N$ as an $A$-module, then the associated sheaf again attaches $N_a$ to Spec $A_a$, with the evident restriction maps.  
Thus indeed $f_* \widetilde N$ is naturally isomorphic to the sheaf attached to $N$ regarded as an $A$-module.  

One reason for spelling this argument out is to point out (a) that $f_*$ can be computed directly in terms of its sections (there is no sheafification process, unlike in the construction of $f^*$), and so is sometimes more accessible; and
(b) that computing $f_*$ requires understanding preimages of open subsets, and one has the important fact in the affine setting that the preimage of Spec $A_a$ is simply Spec $B_a$.  This fact has an obvious geometric meaning (say in the case of varieties) which is worth making clear for yourself.