The Stacks project

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18.12 Morphisms of topoi and sheaves of modules

All of this material is completely straightforward. We formulate everything in the case of morphisms of topoi, but of course the results also hold in the case of morphisms of sites.

Lemma 18.12.1. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. There is a natural map of sheaves of sets

\[ f_*\mathcal{O} \times f_*\mathcal{F} \longrightarrow f_*\mathcal{F} \]

which turns $f_*\mathcal{F}$ into a sheaf of $f_*\mathcal{O}$-modules. This construction is functorial in $\mathcal{F}$.

Proof. Denote $\mu : \mathcal{O} \times \mathcal{F} \to \mathcal{F}$ the multiplication map. Recall that $f_*$ (on sheaves of sets) is left exact and hence commutes with products. Hence $f_*\mu $ is a map as indicated. This proves the lemma. $\square$

Lemma 18.12.2. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{D}$. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules. There is a natural map of sheaves of sets

\[ f^{-1}\mathcal{O} \times f^{-1}\mathcal{G} \longrightarrow f^{-1}\mathcal{G} \]

which turns $f^{-1}\mathcal{G}$ into a sheaf of $f^{-1}\mathcal{O}$-modules. This construction is functorial in $\mathcal{G}$.

Proof. Denote $\mu : \mathcal{O} \times \mathcal{G} \to \mathcal{G}$ the multiplication map. Recall that $f^{-1}$ (on sheaves of sets) is exact and hence commutes with products. Hence $f^{-1}\mu $ is a map as indicated. This proves the lemma. $\square$

Lemma 18.12.3. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{D}$. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules. Let $\mathcal{F}$ be a sheaf of $f^{-1}\mathcal{O}$-modules. Then

\[ \mathop{Mor}\nolimits _{\textit{Mod}(f^{-1}\mathcal{O})}(f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{Mod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). \]

Here we use Lemmas 18.12.2 and 18.12.1, and we think of $f_*\mathcal{F}$ as an $\mathcal{O}$-module by restriction via $\mathcal{O} \to f_*f^{-1}\mathcal{O}$.

Proof. First we note that we have

\[ \mathop{Mor}\nolimits _{\textit{Ab}(\mathcal{C})}(f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{Ab}(\mathcal{D})}(\mathcal{G}, f_*\mathcal{F}). \]

by Sites, Proposition 7.44.3. Suppose that $\alpha : f^{-1}\mathcal{G} \to \mathcal{F}$ and $\beta : \mathcal{G} \to f_*\mathcal{F}$ are morphisms of abelian sheaves which correspond via the formula above. We have to show that $\alpha $ is $f^{-1}\mathcal{O}$-linear if and only if $\beta $ is $\mathcal{O}$-linear. For example, suppose $\alpha $ is $f^{-1}\mathcal{O}$-linear, then clearly $f_*\alpha $ is $f_*f^{-1}\mathcal{O}$-linear, and hence (as restriction is a functor) is $\mathcal{O}$-linear. Hence it suffices to prove that the adjunction map $\mathcal{G} \to f_*f^{-1}\mathcal{G}$ is $\mathcal{O}$-linear. Using that both $f_*$ and $f^{-1}$ commute with products (on sheaves of sets) this comes down to showing that

\[ \xymatrix{ \mathcal{O} \times \mathcal{G} \ar[r] \ar[d] & f_*f^{-1}(\mathcal{O} \times \mathcal{G}) \ar[d] \\ \mathcal{G} \ar[r] & f_*f^{-1}\mathcal{G} } \]

is commutative. This holds because the adjunction mapping $\text{id}_{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})} \to f_*f^{-1}$ is a transformation of functors. We omit the proof of the implication $\beta $ linear $\Rightarrow $ $\alpha $ linear. $\square$

Lemma 18.12.4. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Let $\mathcal{G}$ be a sheaf of $f_*\mathcal{O}$-modules. Then

\[ \mathop{Mor}\nolimits _{\textit{Mod}(\mathcal{O})}( \mathcal{O} \otimes _{f^{-1}f_*\mathcal{O}} f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{Mod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). \]

Here we use Lemmas 18.12.2 and 18.12.1, and we use the canonical map $f^{-1}f_*\mathcal{O} \to \mathcal{O}$ in the definition of the tensor product.

Proof. Note that we have

\[ \mathop{Mor}\nolimits _{\textit{Mod}(\mathcal{O})}( \mathcal{O} \otimes _{f^{-1}f_*\mathcal{O}} f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{Mod}(f^{-1}f_*\mathcal{O})}( f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}f_*\mathcal{O}}) \]

by Lemma 18.11.3. Hence the result follows from Lemma 18.12.3. $\square$


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