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The Stacks project

Lemma 18.37.1. 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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