## 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 9513–9518 (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}
```

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