# The Stacks Project

## Tag 05T7

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

1. 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 $n > a$.
2. 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 $n > a$.

Proof. This lemma is dual to Lemma 13.16.4. $\square$

The code snippet corresponding to this tag is a part of the file derived.tex and is located in lines 5261–5276 (see updates for more information).

\begin{lemma}
\label{lemma-subcategory-left-resolution}
Let $\mathcal{A}$ be an abelian category. Let
$\mathcal{P} \subset \Ob(\mathcal{A})$ be a subset containing $0$
such that every object of $\mathcal{A}$ is a quotient of an element of
$\mathcal{P}$. Let $a \in \mathbf{Z}$.
\begin{enumerate}
\item 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 $n > a$.
\item 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 $n > a$.
\end{enumerate}
\end{lemma}

\begin{proof}
This lemma is dual to
Lemma \ref{lemma-subcategory-right-resolution}.
\end{proof}

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