The Stacks project

Lemma 13.15.7. In Situation 13.15.1. Let $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be a subset with the following properties:

  1. every object of $\mathcal{A}$ is a quotient of an element of $\mathcal{P}$,

  2. for any short exact sequence $0 \to P \to Q \to R \to 0$ of $\mathcal{A}$ with $Q, R \in \mathcal{P}$, then $P \in \mathcal{P}$, and $0 \to F(P) \to F(Q) \to F(R) \to 0$ is exact.

Then every object of $\mathcal{P}$ is acyclic for $LF$.

Proof. Dual to the proof of Lemma 13.15.6. $\square$

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