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Tag 0BK7

Chapter 13: Derived Categories > Section 13.32: Derived limits

Lemma 13.32.3. Let $\mathcal{A}$ be an abelian category with countable products and enough injectives. Let $(K_n)$ be an inverse system of $D^+(\mathcal{A})$. Then $R\mathop{\rm lim}\nolimits K_n$ exists.

Proof. It suffices to show that $\prod K_n$ exists in $D(\mathcal{A})$. For every $n$ we can represent $K_n$ by a bounded below complex $I_n^\bullet$ of injectives (Lemma 13.18.3). Then $\prod K_n$ is represented by $\prod I_n^\bullet$, see Lemma 13.29.5. $\square$

    The code snippet corresponding to this tag is a part of the file derived.tex and is located in lines 9544–9549 (see updates for more information).

    \begin{lemma}
    \label{lemma-inverse-limit-bounded-below}
    Let $\mathcal{A}$ be an abelian category with countable products and
    enough injectives. Let $(K_n)$ be an inverse system of $D^+(\mathcal{A})$.
    Then $R\lim K_n$ exists.
    \end{lemma}
    
    \begin{proof}
    It suffices to show that $\prod K_n$ exists in $D(\mathcal{A})$.
    For every $n$ we can represent $K_n$ by a bounded below complex
    $I_n^\bullet$ of injectives (Lemma \ref{lemma-injective-resolutions-exist}).
    Then $\prod K_n$ is represented by $\prod I_n^\bullet$, see
    Lemma \ref{lemma-product-K-injective}.
    \end{proof}

    Comments (2)

    Comment #2464 by anonymous on March 26, 2017 a 2:41 pm UTC

    In the lemma it should be $D^+$ and not $D^-$.

    Comment #2500 by Johan (site) on April 13, 2017 a 11:30 pm UTC

    Thanks, fixed here.

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