## 13.32 Derived limits

In a triangulated category there is a notion of derived limit.

Definition 13.32.1. Let $\mathcal{D}$ be a triangulated category. Let $(K_ n, f_ n)$ be an inverse system of objects of $\mathcal{D}$. We say an object $K$ is a *derived limit*, or a *homotopy limit* of the system $(K_ n)$ if the product $\prod K_ n$ exists and there is a distinguished triangle

\[ K \to \prod K_ n \to \prod K_ n \to K[1] \]

where the map $\prod K_ n \to \prod K_ n$ is given by $(k_ n) \mapsto (k_ n - f_{n+1}(k_{n + 1}))$. If this is the case, then we sometimes indicate this by the notation $K = R\mathop{\mathrm{lim}}\nolimits K_ n$.

By TR3 a derived limit, if it exists, is unique up to (non-unique) isomorphism. Moreover, by TR1 a derived limit $R\mathop{\mathrm{lim}}\nolimits K_ n$ exists as soon as $\prod K_ n$ exists. The derived category $D(\textit{Ab})$ of the category of abelian groups is an example of a triangulated category where all derived limits exist.

The nonuniqueness makes it hard to pin down the derived limit. In More on Algebra, Lemma 15.77.3 the reader finds an exact sequence

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (L, K_ n[-1]) \to \mathop{\mathrm{Hom}}\nolimits (L, R\mathop{\mathrm{lim}}\nolimits K_ n) \to \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (L, K_ n) \to 0 \]

describing the $\mathop{\mathrm{Hom}}\nolimits $s into a derived limit in terms of the usual $\mathop{\mathrm{Hom}}\nolimits $s.

Lemma 13.32.2. Let $\mathcal{A}$ be an abelian category with exact countable products. Then

$D(\mathcal{A})$ has countable products,

countable products $\prod K_ i$ in $D(\mathcal{A})$ are obtained by taking termwise products of any complexes representing the $K_ i$, and

$H^ p(\prod K_ i) = \prod H^ p(K_ i)$.

**Proof.**
Let $K_ i^\bullet $ be a complex representing $K_ i$ in $D(\mathcal{A})$. Let $L^\bullet $ be a complex. Suppose given maps $\alpha _ i : L^\bullet \to K_ i^\bullet $ in $D(\mathcal{A})$. This means there exist quasi-isomorphisms $s_ i : K_ i^\bullet \to M_ i^\bullet $ of complexes and maps of complexes $f_ i : L^\bullet \to M_ i^\bullet $ such that $\alpha _ i = s_ i^{-1}f_ i$. By assumption the map of complexes

\[ s : \prod K_ i^\bullet \longrightarrow \prod M_ i^\bullet \]

is a quasi-isomorphism. Hence setting $f = \prod f_ i$ we see that $\alpha = s^{-1}f$ is a map in $D(\mathcal{A})$ whose composition with the projection $\prod K_ i^\bullet \to K_ i^\bullet $ is $\alpha _ i$. We omit the verification that $\alpha $ is unique.
$\square$

The duals of Lemmas 13.31.6, 13.31.7, and 13.31.9 should be stated here and proved. However, we do not know any applications of these lemmas for now.

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 $R\mathop{\mathrm{lim}}\nolimits K_ n$ 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 13.29.5.
$\square$

Lemma 13.32.4. Let $\mathcal{A}$ be an abelian category with countable products and enough injectives. Let $K^\bullet $ be a complex. Let $I_ n^\bullet $ be the inverse system of bounded below complexes of injectives produced by Lemma 13.28.3. Then $I^\bullet = \mathop{\mathrm{lim}}\nolimits I_ n^\bullet $ exists, is K-injective, and the following are equivalent

the map $K^\bullet \to I^\bullet $ is a quasi-isomorphism,

the canonical map $K^\bullet \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K^\bullet $ is an isomorphism in $D(\mathcal{A})$.

**Proof.**
The statement of the lemma makes sense as $R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K^\bullet $ exists by Lemma 13.32.3. Each complex $I_ n^\bullet $ is K-injective by Lemma 13.29.4. Choose direct sum decompositions $I_{n + 1}^ p = C_{n + 1}^ p \oplus I_ n^ p$ for all $n \geq 1$. Set $C_1^ p = I_1^ p$. The complex $I^\bullet = \mathop{\mathrm{lim}}\nolimits I_ n^\bullet $ exists because we can take $I^ p = \prod _{n \geq 1} C_ n^ p$. Fix $p \in \mathbf{Z}$. We claim there is a split short exact sequence

\[ 0 \to I^ p \to \prod I_ n^ p \to \prod I_ n^ p \to 0 \]

of objects of $\mathcal{A}$. Here the first map is given by the projection maps $I^ p \to I_ n^ p$ and the second map by $(x_ n) \mapsto (x_ n - f^ p_{n + 1}(x_{n + 1}))$ where $f^ p_ n : I_ n^ p \to I_{n - 1}^ p$ are the transition maps. The splitting comes from the map $\prod I_ n^ p \to \prod C_ n^ p = I^ p$. We obtain a termwise split short exact sequence of complexes

\[ 0 \to I^\bullet \to \prod I_ n^\bullet \to \prod I_ n^\bullet \to 0 \]

Hence a corresponding distinguished triangle in $K(\mathcal{A})$ and $D(\mathcal{A})$. By Lemma 13.29.5 the products are K-injective and represent the corresponding products in $D(\mathcal{A})$. It follows that $I^\bullet $ represents $R\mathop{\mathrm{lim}}\nolimits I_ n^\bullet $ (Definition 13.32.1). Moreover, it follows that $I^\bullet $ is K-injective by Lemma 13.29.3. By the commutative diagram of Lemma 13.28.3 we obtain a corresponding commutative diagram

\[ \xymatrix{ K^\bullet \ar[r] \ar[d] & R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} K^\bullet \ar[d] \\ I^\bullet \ar[r] & R\mathop{\mathrm{lim}}\nolimits I_ n^\bullet } \]

in $D(\mathcal{A})$. Since the right vertical arrow is an isomorphism (as derived limits are defined on the level of the derived category and since $\tau _{\geq -n}K^\bullet \to I_ n^\bullet $ is a quasi-isomorphism), the lemma follows.
$\square$

Lemma 13.32.5. Let $\mathcal{A}$ be an abelian category having enough injectives and exact countable products. Then for every complex there is a quasi-isomorphism to a K-injective complex.

**Proof.**
By Lemma 13.32.4 it suffices to show that $K \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K$ is an isomorphism for all $K$ in $D(\mathcal{A})$. Consider the defining distinguished triangle

\[ R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K \to \prod \tau _{\geq -n}K \to \prod \tau _{\geq -n}K \to (R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K)[1] \]

By Lemma 13.32.2 we have

\[ H^ p(\prod \tau _{\geq -n}K) = \prod \nolimits _{p \geq -n} H^ p(K) \]

It follows in a straightforward manner from the long exact cohomology sequence of the displayed distinguished triangle that $H^ p(R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K) = H^ p(K)$.
$\square$

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