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Tag 05T7

Chapter 13: Derived Categories > Section 13.16: Derived functors on derived categories

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 5254–5269 (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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