Questions about sheaves of $\mathcal{O}_X$ modules.

Question

Let $f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morhism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$ modules and $\mathcal{G}$ a sheaf of $\mathcal{O}_Y$ modules. Then we have $$ \operatorname{Hom}_{\mathcal{O}_X}(f^*\mathcal{G}, \mathcal{F}) = \operatorname{Hom}_{\mathcal{O}_Y}(\mathcal{G}, f_* \mathcal{F}). \qquad (1) $$ This is different from the case of sheaves of rings. If $\mathcal{F}$, $\mathcal{G}$ are sheaves of rings, then $$ \operatorname{Hom}_{X}(f^{-1}\mathcal{G}, \mathcal{F}) = \operatorname{Hom}_{Y}(\mathcal{G}, f_* \mathcal{F}). \qquad (2) $$ Why in the case of sheaves of modules, (2) is not true? Thank you very mcuh.

Answer 1

In his comment Zhen has already given a correct answer: $f^{-1} G$ is a module over $f^{-1} \mathcal{O}_Y$, but not over $\mathcal{O}_X$. Hence, we tensor it with $f^\# : f^{-1} \mathcal{O}_Y \to \mathcal{O}_X$ to optain a module $f^* G = f^{-1} G \otimes_{f^{-1} \mathcal{O}_Y} \mathcal{O}_X$ over $\mathcal{O}_X$.

Basically you just have to remember in which categories you are working, and which forgetful functors are floating around (most of them are pretended to be identities in the literature...). Let me explain this in detail:

Actually (2) states: If $f : X \to Y$ is a continuous map between topological spaces, then $f^{-1} : \mathsf{Sh}(Y) \to \mathsf{Sh}(X)$ is left adjoint to $f_* : \mathsf{Sh}(X) \to \mathsf{Sh}(Y)$.

And (1) states: If $f : X \to Y$ is a morphism of ringed spaces, then $f^* : \mathsf{Mod}(Y) \to \mathsf{Mod}(X)$ is left adjoint to $f_* : \mathsf{Mod}(X) \to \mathsf{Mod}(Y)$.

Note that the setups are different! In the situation of (1), we only have a functor $|f|^{-1} : \mathsf{Sh}(|Y|) \to \mathsf{Sh}(|X|)$, where $|X|$ denotes the underlying topological space of the ringed space $X$ and $|f|$ denotes the underlying continuous map of the morphism of ringed spaces $f$. Of course we still have the adjunction (1) associated to $|f|$, but this should not be confused with (2). The involved functors are connected as follows:

a) If $|F| \in \mathsf{Sh}(|X|)$ denotes the sheaf of abelian groups which underlies a sheaf of modules $F \in \mathsf{Mod}(X)$, we have $|f_* F| = |f|_* |F|$.

b) We have $f^* G = f^{-1} G \otimes_{f^{-1} \mathcal{O}_Y} \mathcal{O}_X$ for all $G \in \mathsf{Mod}(Y)$.

One can infer (2) from (1): A morphism $f^* G \to F$ in $\mathsf{Mod}(X)$ corresponds to a morphism $f^{-1} |G| \to |F|$ in $\mathsf{Sh}(|X|)$ which is $f^{-1} \mathcal{O}_Y$-linear, i.e. by (1) to a morphism $|G| \to |f|_* |F|$ in $\mathsf{Sh}(|Y|)$ which is $\mathcal{O}_Y$-linear, i.e. to a morphism $G \to f_* F$ in $\mathsf{Mod}(Y)$.

Actually the converse is also true, (1) is a special case of (2). For that we endow the involved topological spaces with the constant sheaf $\underline{\mathbb{Z}}$. Then $f^{-1} \mathcal{O}_Y = \mathcal{O}_X$, $\mathsf{Mod}(X) \simeq \mathsf{Sh}(|X|)$, and under this equivalence we have $|f|^{-1} \simeq f^*$, $|f|_* \simeq f_*$.

My emphasis on the forgetful functors seems to be pedantic here, but in my opinion the ignorance of forgetful functors leads to confusions.