Lemma 13.16.5. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be a subset containing $0$ such that every object of $\mathcal{A}$ is a quotient of an element of $\mathcal{P}$. Let $a \in \mathbf{Z}$.

Given $K^\bullet $ with $K^ n = 0$ for $n > a$ there exists a quasi-isomorphism $P^\bullet \to K^\bullet $ with $P^ n \in \mathcal{P}$ and $P^ n \to K^ n$ surjective for all $n$ and $P^ n = 0$ for $n > a$.

Given $K^\bullet $ with $H^ n(K^\bullet ) = 0$ for $n > a$ there exists a quasi-isomorphism $P^\bullet \to K^\bullet $ with $P^ n \in \mathcal{P}$ for all $n$ and $P^ n = 0$ for $n > a$.

## Comments (0)

There are also: