## Tag `06XX`

Chapter 13: Derived Categories > Section 13.28: Unbounded complexes

Lemma 13.28.1. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{P} \subset \mathop{\rm Ob}\nolimits(\mathcal{A})$ be a subset. Assume that every object of $\mathcal{A}$ is a quotient of an element of $\mathcal{P}$. Let $K^\bullet$ be a complex. There exists a commutative diagram $$ \xymatrix{ P_1^\bullet \ar[d] \ar[r] & P_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau_{\leq 1}K^\bullet \ar[r] & \tau_{\leq 2}K^\bullet \ar[r] & \ldots } $$ in the category of complexes such that

- the vertical arrows are quasi-isomorphisms,
- $P_n^\bullet$ is a bounded above complex with terms in $\mathcal{P}$,
- the arrows $P_n^\bullet \to P_{n + 1}^\bullet$ are termwise split injections and each cokernel $P^i_{n + 1}/P^i_n$ is an element of $\mathcal{P}$.

Proof.By Lemma 13.16.5 any bounded above complex has a resolution by a bounded above complex whose terms are in $\mathcal{P}$. Thus we obtain the first complex $P_1^\bullet$. By induction it suffices, given $P_1^\bullet, \ldots, P_n^\bullet$ to construct $P_{n + 1}^\bullet$ and the maps $P_n^\bullet \to P_{n + 1}^\bullet$ and $P_{n + 1}^\bullet \to \tau_{\leq n + 1}K^\bullet$. Consider the cone $C_1^\bullet$ of the composition $P_n^\bullet \to \tau_{\leq n}K^\bullet \to \tau_{\leq n + 1}K^\bullet$. This fits into the distinguished triangle $$ P_n^\bullet \to \tau_{\leq n + 1}K^\bullet \to C_1^\bullet \to P_n^\bullet[1] $$ Note that $C_1^\bullet$ is bounded above, hence we can choose a quasi-isomorphism $Q^\bullet \to C_1^\bullet$ where $Q^\bullet$ is a bounded above complex whose terms are elements of $\mathcal{P}$. Take the cone $C_2^\bullet$ of the map of complexes $Q^\bullet \to P_n^\bullet[1]$ to get the distinguished triangle $$ Q^\bullet \to P_n^\bullet[1] \to C_2^\bullet \to Q^\bullet[1] $$ By the axioms of triangulated categories we obtain a map of distinguished triangles $$ \xymatrix{ P_n^\bullet \ar[r] \ar[d] & C_2^\bullet[-1] \ar[r] \ar[d] & Q^\bullet \ar[r] \ar[d] & P_n^\bullet[1] \ar[d] \\ P_n^\bullet \ar[r] & \tau_{\leq n + 1}K^\bullet \ar[r] & C_1^\bullet \ar[r] & P_n^\bullet[1] } $$ in the triangulated category $K(\mathcal{A})$. Set $P_{n + 1}^\bullet = C_2^\bullet[-1]$. Note that (3) holds by construction. Choose an actual morphism of complexes $f : P_{n + 1}^\bullet \to \tau_{\leq n + 1}K^\bullet$. The left square of the diagram above commutes up to homotopy, but as $P_n^\bullet \to P_{n + 1}^\bullet$ is a termwise split injection we can lift the homotopy and modify our choice of $f$ to make it commute. Finally, $f$ is a quasi-isomorphism, because both $P_n^\bullet \to P_n^\bullet$ and $Q^\bullet \to C_1^\bullet$ are. $\square$

The code snippet corresponding to this tag is a part of the file `derived.tex` and is located in lines 8429–8451 (see updates for more information).

```
\begin{lemma}
\label{lemma-special-direct-system}
Let $\mathcal{A}$ be an abelian category. Let
$\mathcal{P} \subset \Ob(\mathcal{A})$ be a subset.
Assume that every object of $\mathcal{A}$ is a quotient of an
element of $\mathcal{P}$. Let $K^\bullet$ be a complex.
There exists a commutative diagram
$$
\xymatrix{
P_1^\bullet \ar[d] \ar[r] & P_2^\bullet \ar[d] \ar[r] & \ldots \\
\tau_{\leq 1}K^\bullet \ar[r] & \tau_{\leq 2}K^\bullet \ar[r] & \ldots
}
$$
in the category of complexes such that
\begin{enumerate}
\item the vertical arrows are quasi-isomorphisms,
\item $P_n^\bullet$ is a bounded above complex with terms in
$\mathcal{P}$,
\item the arrows $P_n^\bullet \to P_{n + 1}^\bullet$
are termwise split injections and each cokernel
$P^i_{n + 1}/P^i_n$ is an element of $\mathcal{P}$.
\end{enumerate}
\end{lemma}
\begin{proof}
By
Lemma \ref{lemma-subcategory-left-resolution}
any bounded above complex has a resolution by a bounded above complex
whose terms are in $\mathcal{P}$. Thus we obtain the first complex
$P_1^\bullet$. By induction it suffices, given
$P_1^\bullet, \ldots, P_n^\bullet$ to construct
$P_{n + 1}^\bullet$ and the maps
$P_n^\bullet \to P_{n + 1}^\bullet$ and
$P_{n + 1}^\bullet \to \tau_{\leq n + 1}K^\bullet$.
Consider the cone $C_1^\bullet$ of the composition
$P_n^\bullet \to \tau_{\leq n}K^\bullet \to \tau_{\leq n + 1}K^\bullet$.
This fits into the distinguished triangle
$$
P_n^\bullet \to \tau_{\leq n + 1}K^\bullet \to C_1^\bullet \to P_n^\bullet[1]
$$
Note that $C_1^\bullet$ is bounded above, hence we can choose a
quasi-isomorphism $Q^\bullet \to C_1^\bullet$ where $Q^\bullet$ is a
bounded above complex whose terms are elements of $\mathcal{P}$.
Take the cone $C_2^\bullet$ of the map of complexes
$Q^\bullet \to P_n^\bullet[1]$ to get the
distinguished triangle
$$
Q^\bullet \to P_n^\bullet[1] \to C_2^\bullet \to Q^\bullet[1]
$$
By the axioms of triangulated categories we obtain a map
of distinguished triangles
$$
\xymatrix{
P_n^\bullet \ar[r] \ar[d] &
C_2^\bullet[-1] \ar[r] \ar[d] &
Q^\bullet \ar[r] \ar[d] &
P_n^\bullet[1] \ar[d] \\
P_n^\bullet \ar[r] &
\tau_{\leq n + 1}K^\bullet \ar[r] &
C_1^\bullet \ar[r] &
P_n^\bullet[1]
}
$$
in the triangulated category $K(\mathcal{A})$.
Set $P_{n + 1}^\bullet = C_2^\bullet[-1]$.
Note that (3) holds by construction.
Choose an actual morphism of complexes
$f : P_{n + 1}^\bullet \to \tau_{\leq n + 1}K^\bullet$.
The left square of the diagram above commutes up to homotopy, but as
$P_n^\bullet \to P_{n + 1}^\bullet$ is a termwise split injection
we can lift the homotopy and modify our choice of $f$ to make it commute.
Finally, $f$ is a quasi-isomorphism, because both $P_n^\bullet \to P_n^\bullet$
and $Q^\bullet \to C_1^\bullet$ are.
\end{proof}
```

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