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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