Lemma 13.34.3 (0BK7)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.34: Derived limits
Lemma 13.34.3 (cite)

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Lemma 13.34.3. 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 $R\mathop{\mathrm{lim}}\nolimits K_ n$ 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 13.31.5. $\square$

Comments (2)

Comment #2464 by anonymous on March 26, 2017 at 14:41

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

Comment #2500 by Johan on April 13, 2017 at 23:30

Thanks, fixed here.

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