I'm currently reading a text in algebraic geometry and the author has just defined a sheaf. They then remark that: "Note that the restriction-homomorphism $f \mapsto f \vert_V$ need not be the usual restriction of functions, even if the elements of $\mathscr{G}(U)$ and $\mathscr{G}(V)$ are functions on $U$ and $V$, respectively."
I have read this post Under which conditions do the restriction maps of sheaves of modules occur to be injective?
But I'm not comfortable with the commutative algebra to really understand the answer. I was wondering if someone could provide an illuminating concrete example to justify the remark.
Thanks in advance.
I can concoct a specific example if you'd like, but I think this is actually a lot simpler than you think and a general explanation would be more illuminating than a concrete example (which would be fairly messy to write down in detail).
All that is being said here is that just because we call these maps "restriction maps", that does not mean they are given by the usual notion of restriction of functions. There is nothing complicated about this: this is just like how even though we typically denote the operation on a group as multiplication, it need not actually be the familiar multiplication of numbers. So these so-called "restriction maps" can be any collection of maps you want, as long as they satisfy all the conditions in the definition of a sheaf.