TITLE: What is a simple definition of the pullback of a section?
QUESTION [16 upvotes]: I am simply asking for a definition for something everyone uses but nobody defines. Really, this is used in class and in Hartshorne, and I have tried to look for a definition in Hartshorne, Qing Liu, Wikipedia, nothing comes up, so I am wondering whether somebody on this planet knows a definition of this.
Let $X,Y$ be topological spaces and $F, G$ be sheaves of modules over $X,Y$ respectively.
The pull-back of a sheaf is very well-documented and defined everywhere with high precision:
If $f:X\rightarrow Y$ is a continuous map, then
$f^*G=f^{-1}G\otimes_{f^{-1}O_Y} O_X$
So I know what $f^*G$ and what $(f^*G)(U)$ are (with $U \subset X$).
But what is $f^*s$ if $s\in G(Y)$, or more generally $s \in G(U)$ where $U \subset Y$ is some open subset of $Y$?
I know there is already a discussion in this thread and apparently the definition is given in a comment for affine schemes (it is just the image by the induced ring map), but I don't find it particularly enlightening. Could somebody please provide a straightforward definition for the pull-back of a section of a sheaf of modules on a general scheme? Can it be defined in a simple way (with e.g. a formula) without using high-powered, unintelligible stuff? In particular I don't know what adjunction correspondance is...

REPLY [2 votes]: We treat pullbacks of global sections, nothing changes if we restrict to some other open set. We will be using the fact that tensor product of sheaves commutes with restriction to open sets. 
Call $f: X \rightarrow Y$ our morphism of schemes. Let $\mathscr{G}$ be an $\mathcal{O}_Y$-module. There is a canonical morphism $\mathscr{G} \rightarrow f_* f^* \mathscr{G}$ of $\mathcal{O}_Y$-modules. This induces a morphism of global sections $\psi:\Gamma(Y,\mathscr{G}) \rightarrow \Gamma(X,f^*\mathscr{G})$. The question is how to describe concretely the image of $s \in \Gamma(Y,\mathscr{G})$ under this map.  
It is instructive to begin with the special case $\mathscr{G}=\mathcal{O}_Y$. Consider the following rings $A=\Gamma(Y,\mathcal{O}_Y), \, B = \Gamma(X,f^{-1} \mathcal{O}_Y), \, C = \Gamma(X,\mathcal{O}_X)$. The data of $f$ include a morphism $\mathcal{O}_Y \rightarrow f_* \mathcal{O}_X$. This induces a ring homomorphism $\phi_1: A \rightarrow C$ on global sections. Now the canonical morphism $\mathcal{O}_Y \rightarrow f_* f^{*} \mathcal{O}_Y$ induces a ring homomorphism $\phi_1':A \rightarrow \Gamma(X,f^* \mathscr{G})$. The key is that $\phi_1$ and $\phi_1'$ have the same image. To see this note that we have a ring homomorphism $\phi_2: A \rightarrow B$ induced by $\mathcal{O}_Y \rightarrow f_* f^{-1} \mathcal{O}_Y$. We also have a $\phi_3 : B \rightarrow C$ induced by $f^{-1} \mathcal{O}_Y \rightarrow f^{-1} f_* \mathcal{O}_X \rightarrow \mathcal{O}_X$. Moreover $\phi_1 = \phi_3 \circ \phi_2$. Now $\Gamma(X,f^* \mathcal{O}_Y) = B \otimes_B C$. Finally by checking the definition of $\phi_1'$ we can see that $\phi_1'$ sends $s$ to $\phi_2(s) \otimes 1 \in B \otimes_B C$ which we can identify with $\phi_3(\phi_2(s))=\phi_1(s)$. 
For general $\mathscr{G}$ the picture is analogous with the note that $\Gamma(X,f^*\mathscr{G}) = \Gamma(X,f^{-1}\mathscr{G}) \otimes_B C$. That is $\psi$ sends global sections of $\mathscr{G}$ to their canonical image in the $B$-module $\Gamma(X,f^{-1} \mathscr{G})$ and then to their image in the $C$-module $\Gamma(X,f^{-1} \mathscr{G}) \otimes_B C$, the latter being an extended module by scalars according to $\phi_3: B \rightarrow C$.