Pushforward and pullback are very basic operations in algebraic geometry, as is the adjointness between them. I worked out a very careful of adjointness of sheaves (below) when I was working out of Hartshorne – however, I still find this theorem somewhat mysterious.
$\textbf{Question:}$ While I am comfortable with using this fairly abstract yet basic theorem, I feel like I should understand it a little better. How do you understand adjointness of sheaves? Is it clearly true if we make some (weak?) additional conditions? Is there a way to think about it to make it more transparent, more believable or even obvious? Please feel very free to work in the case of complex algebraic geometry, etc.
I tried to give a shorter, heuristic proof of adjointness using the etale space of a sheaf – but I got lost checking details. I would be very grateful if someone more knowledgeable could tell me if such a proof exists.
$\textbf{Thm}$ Let $(X, \mathcal{O}_X) \xrightarrow{f} (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces and $\mathcal{F}, \mathcal{G}$ be sheaves of $\mathcal{O}_X, \mathcal{O}_Y$ modules respectively. Then, we have a canonical bijection of sets $$ \textrm{Hom}_{\mathcal{O}_X} (f^*\mathcal{G}, F) = \textrm{Hom}_{\mathcal{O}_Y} (\mathcal{G}, f_*\mathcal{F})$$
Your comments and answers will be very appreciated!
Best Answer
You can think about this as about a generalization of the adjunction between the extension and the restriction of scalars --- if $A \to B$ is a morphism of rings, $M$ is an $A$-module and $N$ is a $B$-module then $$ Hom_B(M\otimes_A B,N) = Hom_A(M,Res_A N), $$ where $Res$ is the restriction of scalars. This adjunction coincides with the one tou are interested in in case of affine $X$ and $Y$.