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

\[ p^{-1}p_*A = A \oplus I(A) \]for $A \in \mathop{\mathrm{Ob}}\nolimits (\textit{Ab})$.

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