Inverse image sheaf of an $\mathcal{O}_Y$ Module

Question

I am trying to understand the inverse image sheaf since I am not comfortable with sheafification and taking direct limits. So I tried to prove the following to help my understanding:

Let $f:X \rightarrow Y$ be a map of (locally) ringed topological spaces. Let $\mathcal{G}$ be an $\mathcal{O}_Y$ module. Show that $f^{-1}\mathcal{G}$ is an $f^{-1}\mathcal{O}_Y$ module.

My attempt:

Given any $U \in X$ open, we know that the inverse image presheaf $f^{-1,pre}\mathcal{G}(U)$ is given by lim$_{V \supseteq f(U)}\mathcal{G}(V)$. So any element in the sheafification of this will be given by a function $s:U\rightarrow \bigcup_{p \in U}f^{-1,pre}\mathcal{G}_p$.

Now given that stalks on the inverse image presheaf $f^{-1,pre}\mathcal G_p$ are isomorphic to stalks on $\mathcal G_{f(p)}$, we get that any element of the sheafification is given by a function $s: U \rightarrow \bigcup _{p \in U}\mathcal{G}_{f(p)}$. Similarly, any element in $f^{-1}\mathcal{O}_Y(U)$ is given by a function $r:U \rightarrow \bigcup_{p \in U} \mathcal{O}_{Y,f(p)}$.

So we can define the module structure as $(r \cdot s)(p) = r(p)\cdot s(p)$ where the multiplication on the right is given by the module structure of $\mathcal{G}_{f(p)}$ as an $\mathcal{O}_{Y,f(p)}$ module induced by the $\mathcal{O}_Y$-module structure on $\mathcal G$.

Since we are using the induced module structure on stalks, this does indeed define a module and we only need to check that the module structure defined respects restrictions. But $(rs)|_{V}(p) = r(p)s(p)|_V = r|_V(p) = s|_V(p)$ for any $V \subseteq U$ open. And we are done.

My questions:

1. Is the module structure and reasoning for why it defines a module structure correct?
2. I'm a bit unsure about how to explicitly write out the restriction maps and so would appreciate any explanation on how the restriction map explicitly works (viewed as restrictions of the functions r,s - if my understanding till there was fine)
3. Is there anything else that needs to be checked aside from the module structure and restriction maps that I am missing.

Thanks

Answer 1

If $f:X \rightarrow Y$ is a map of topological spaces and if $A$ is a sheaf of rings on $Y$ you define the presheaf on $X$:

$$F_A(U):=lim_{f(U) \subseteq V}A(V).$$

Since $A(V)$ is a ring for all $V$ and since $F_A(U)$ is a direct limit of rings it follows $F_A(U)$ is a presheaf of rings on $X$. It follows $F_A^+:=f^{-1}(A)$ is a sheaf of rings on $X$. Hence the inverse image of a sheaf of rings is a sheaf of rings. If $f: X \rightarrow Y$ is a map of schemes there is a canonical map of sheaves of rings

$$ f^{\#}: f^{-1}(\mathcal{O}_Y) \rightarrow \mathcal{O}_X.$$

If $E$ is a sheaf of $\mathcal{O}_Y$-modules it follows $f^{-1}(E)$ is a sheaf of $f^{-1}(\mathcal{O}_Y)$-modules:

Since $E(V)$ is an $\mathcal{O}_Y(V)$-module for all $V$ it follows $lim_{f(U) \subseteq V} E(V)$ is a $lim_{f(U)\subseteq V}\mathcal{O}_Y(V)$-module for all $U$, hence $f^{-1}(E)(U)$ is an $f^{-1}(\mathcal{O}_Y)(U)$-module for all $U$. It follows that $f^{-1}(E)$ is an $f^{-1}(\mathcal{O}_Y)$-module. You define the following presheaf

$$ F_{f^*E}(U):= \mathcal{O}_X(U)\otimes_{f^{-1}(\mathcal{O}_Y(U)}f^{-1}(E)(U)$$

and define

$$ f^*E:=F^+_{f^*E}:=\mathcal{O}_X \otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(E)$$

which is canonically a sheaf of $\mathcal{O}_X$-modules.

Question: "1. Is the module structure and reasoning for why it defines a module structure correct? 2. I'm a bit unsure about how to explicitly write out the restriction maps and so would appreciate any explanation on how the restriction map explicitly works (viewed as restrictions of the functions r,s - if my understanding till there was fine) 3. Is there anything else that needs to be checked aside from the module structure and restriction maps that I am missing."

Answer: Instead of working with stalks it is better to work with direct limits and tensor products. Then the module structure is "canonical".