Let $p$ be a point of a site $\mathcal{C}$ or a topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. In this section we study the exactness properties of the functor which associates to an abelian group $A$ the skyscraper sheaf $p_*A$. First, recall that $p_* : \textit{Sets} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ has a lot of exactness properties, see Sites, Lemmas 7.32.9 and 7.32.10.
Lemma 18.37.1. Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$ or of its associated topos.
The functor $p_* : \textit{Ab} \to \textit{Ab}(\mathcal{C})$, $A \mapsto p_*A$ is exact.
There is a functorial direct sum decomposition
for $A \in \mathop{\mathrm{Ob}}\nolimits (\textit{Ab})$.
\[ p^{-1}p_*A = A \oplus I(A) \]
Proof. By Sites, Lemma 7.32.9 there are functorial maps $A \to p^{-1}p_*A \to A$ whose composition equals $\text{id}_ A$. Hence a functorial direct sum decomposition as in (2) with $I(A)$ the kernel of the adjunction map $p^{-1}p_*A \to A$. The functor $p_*$ is left exact by Lemma 18.14.3. The functor $p_*$ transforms surjections into surjections by Sites, Lemma 7.32.10. Hence (1) holds. $\square$
To do the same thing for sheaves of modules, suppose given a point $p$ of a ringed topos $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$. Recall that $p^{-1}$ is just the stalk functor. Hence we can think of $p$ as a morphism of ringed topoi
Thus we get a pullback functor $p^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ p)$ which equals the stalk functor, and which we discussed in Lemma 18.36.3. In this section we consider the functor $p_* : \textit{Mod}(\mathcal{O}_ p) \to \textit{Mod}(\mathcal{O})$.
Lemma 18.37.2. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $p$ be a point of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.
The functor $p_* : \textit{Mod}(\mathcal{O}_ p) \to \textit{Mod}(\mathcal{O})$, $M \mapsto p_*M$ is exact.
The canonical surjection $p^{-1}p_*M \to M$ is $\mathcal{O}_ p$-linear.
The functorial direct sum decomposition $p^{-1}p_*M = M \oplus I(M)$ of Lemma 18.37.1 is not $\mathcal{O}_ p$-linear in general.
Proof. Part (1) and surjectivity in (2) follow immediately from the corresponding result for abelian sheaves in Lemma 18.37.1. Since $p^{-1}\mathcal{O} = \mathcal{O}_ p$ we have $p^{-1} = p^*$ and hence $p^{-1}p_*M \to M$ is the same as the counit $p^*p_*M \to M$ of the adjunction for modules, whence linear.
Proof of (3). Suppose that $G$ is a group. Consider the topos $G\textit{-Sets} = \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G)$ and the point $p : \textit{Sets} \to G\textit{-Sets}$. See Sites, Section 7.9 and Example 7.33.7. Here $p^{-1}$ is the functor forgetting about the $G$-action. And $p_*$ is the right adjoint of the forgetful functor, sending $M$ to $\text{Map}(G, M)$. The maps in the direct sum decomposition are the maps
where the first sends $m \in M$ to the constant map with value $m$ and where the second map is evaluation at the identity element $1$ of $G$. Next, suppose that $R$ is a ring endowed with an action of $G$. This determines a sheaf of rings $\mathcal{O}$ on $\mathcal{T}_ G$. The category of $\mathcal{O}$-modules is the category of $R$-modules $M$ endowed with an action of $G$ compatible with the action on $R$. The $R$-module structure on $\text{Map}(G, M)$ is given by
for $r \in R$ and $f \in \text{Map}(G, M)$. This is true because it is the unique $G$-invariant $R$-module structure compatible with evaluation at $1$. The reader observes that in general the image of $M \to \text{Map}(G, M)$ is not an $R$-submodule (for example take $M = R$ and assume the $G$-action is nontrivial), which concludes the proof. $\square$
Example 18.37.3. Let $G$ be a group. Consider the site $\mathcal{T}_ G$ and its point $p$, see Sites, Example 7.33.7. Let $R$ be a ring with a $G$-action which corresponds to a sheaf of rings $\mathcal{O}$ on $\mathcal{T}_ G$. Then $\mathcal{O}_ p = R$ where we forget the $G$-action. In this case $p^{-1}p_*M = \text{Map}(G, M)$ and $I(M) = \{ f : G \to M \mid f(1_ G) = 0\} $ and $M \to \text{Map}(G, M)$ assigns to $m \in M$ the constant function with value $m$.
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