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
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...