Lemma 18.36.1. Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$ or of its associated topos.

1. The functor $p_* : \textit{Ab} \to \textit{Ab}(\mathcal{C})$, $A \mapsto p_*A$ is exact.

2. There is a functorial direct sum decomposition

$p^{-1}p_*A = A \oplus I(A)$

for $A \in \mathop{\mathrm{Ob}}\nolimits (\textit{Ab})$.

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. $\square$

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