18.13 Morphisms of ringed topoi and modules
We have now introduced enough notation so that we are able to define the pullback and pushforward of modules along a morphism of ringed topoi.
Definition 18.13.1. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi or ringed sites.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {C}$-modules. We define the pushforward of $\mathcal{F}$ as the sheaf of $\mathcal{O}_\mathcal {D}$-modules which as a sheaf of abelian groups equals $f_*\mathcal{F}$ and with module structure given by the restriction via $f^\sharp : \mathcal{O}_\mathcal {D} \to f_*\mathcal{O}_\mathcal {C}$ of the module structure
\[ f_*\mathcal{O}_\mathcal {C} \times f_*\mathcal{F} \longrightarrow f_*\mathcal{F} \]
from Lemma 18.12.1.
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_\mathcal {D}$-modules. We define the pullback $f^*\mathcal{G}$ to be the sheaf of $\mathcal{O}_\mathcal {C}$-modules defined by the formula
\[ f^*\mathcal{G} = \mathcal{O}_\mathcal {C} \otimes _{f^{-1}\mathcal{O}_\mathcal {D}} f^{-1}\mathcal{G} \]
where the ring map $f^{-1}\mathcal{O}_\mathcal {D} \to \mathcal{O}_\mathcal {C}$ is $f^\sharp $, and where the module structure is given by Lemma 18.12.2.
Thus we have defined functors
\begin{eqnarray*} f_* : \textit{Mod}(\mathcal{O}_\mathcal {C}) & \longrightarrow & \textit{Mod}(\mathcal{O}_\mathcal {D}) \\ f^* : \textit{Mod}(\mathcal{O}_\mathcal {D}) & \longrightarrow & \textit{Mod}(\mathcal{O}_\mathcal {C}) \end{eqnarray*}
The final result on these functors is that they are indeed adjoint as expected.
Lemma 18.13.2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi or ringed sites. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {C}$-modules. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_\mathcal {D}$-modules. There is a canonical bijection
\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}(f^*\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, f_*\mathcal{F}). \]
In other words: the functor $f^*$ is the left adjoint to $f_*$.
Proof.
This follows from the work we did before:
\begin{eqnarray*} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}(f^*\mathcal{G}, \mathcal{F}) & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O}_\mathcal {C})}( \mathcal{O}_\mathcal {C} \otimes _{f^{-1}\mathcal{O}_\mathcal {D}} f^{-1}\mathcal{G}, \mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(f^{-1}\mathcal{O}_\mathcal {D})}( f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}\mathcal{O}_\mathcal {D}}) \\ & = & \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*}
Here we use Lemmas 18.11.3 and 18.12.3.
$\square$
Lemma 18.13.3. $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_1), \mathcal{O}_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2)$ and $(g, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_3), \mathcal{O}_3)$ be morphisms of ringed topoi. There are canonical isomorphisms of functors $(g \circ f)_* \cong g_* \circ f_*$ and $(g \circ f)^* \cong f^* \circ g^*$.
Proof.
This is clear from the definitions.
$\square$
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