## 13.29 Unbounded complexes

A reference for the material in this section is . The following lemma is useful to find “good” left resolutions of unbounded complexes.

Lemma 13.29.1. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be a subset. Assume $\mathcal{P}$ contains $0$, is closed under (finite) direct sums, and 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

1. the vertical arrows are quasi-isomorphisms and termwise surjective,

2. $P_ n^\bullet$ is a bounded above complex with terms in $\mathcal{P}$,

3. 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. We are going to use that the homotopy category $K(\mathcal{A})$ is a triangulated category, see Proposition 13.10.3. By Lemma 13.15.4 we can find a termwise surjective map of complexes $P_1^\bullet \to \tau _{\leq 1}K^\bullet$ which is a quasi-isomorphism such that the terms of $P_1^\bullet$ are in $\mathcal{P}$. 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 $P_{n + 1}^\bullet \to \tau _{\leq n + 1}K^\bullet$.

Choose a distinguished triangle $P_ n^\bullet \to \tau _{\leq n + 1}K^\bullet \to C^\bullet \to P_ n^\bullet $ in $K(\mathcal{A})$. Applying Lemma 13.15.4 we choose a map of complexes $Q^\bullet \to C^\bullet$ which is a quasi-isomorphism such that the terms of $Q^\bullet$ are in $\mathcal{P}$. By the axioms of triangulated categories we may fit the composition $Q^\bullet \to C^\bullet \to P_ n^\bullet $ into a distinguished triangle $P_ n^\bullet \to P_{n + 1}^\bullet \to Q^\bullet \to P_ n^\bullet $ in $K(\mathcal{A})$. By Lemma 13.10.7 we may and do assume $0 \to P_ n^\bullet \to P_{n + 1}^\bullet \to Q^\bullet \to 0$ is a termwise split short exact sequence. This implies that the terms of $P_{n + 1}^\bullet$ are in $\mathcal{P}$ and that $P_ n^\bullet \to P_{n + 1}^\bullet$ is a termwise split injection whose cokernels are in $\mathcal{P}$. By the axioms of triangulated categories we obtain a map of distinguished triangles

$\xymatrix{ P_ n^\bullet \ar[r] \ar[d] & P_{n + 1}^\bullet \ar[r] \ar[d] & Q^\bullet \ar[r] \ar[d] & P_ n^\bullet  \ar[d] \\ P_ n^\bullet \ar[r] & \tau _{\leq n + 1}K^\bullet \ar[r] & C^\bullet \ar[r] & P_ n^\bullet  }$

in the triangulated category $K(\mathcal{A})$. 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 $Q^\bullet \to C^\bullet$ are.

At this point we have all the properties we want, except we don't know that the map $f : P_{n + 1}^\bullet \to \tau _{\leq n + 1}K^\bullet$ is termwise surjective. Since we have the commutative diagram

$\xymatrix{ P_ n^\bullet \ar[d] \ar[r] & P_{n + 1}^\bullet \ar[d] \\ \tau _{\leq n}K^\bullet \ar[r] & \tau _{\leq n + 1}K^\bullet }$

of complexes, by induction hypothesis we see that $f$ is surjective on terms in all degrees except possibly $n$ and $n + 1$. Choose an object $P \in \mathcal{P}$ and a surjection $q : P \to K^ n$. Consider the map

$g : P^\bullet = (\ldots \to 0 \to P \xrightarrow {1} P \to 0 \to \ldots ) \longrightarrow \tau _{\leq n + 1}K^\bullet$

with first copy of $P$ in degree $n$ and maps given by $q$ in degree $n$ and $d_ K \circ q$ in degree $n + 1$. This is a surjection in degree $n$ and the cokernel in degree $n + 1$ is $H^{n + 1}(\tau _{\leq n + 1}K^\bullet )$; to see this recall that $\tau _{\leq n + 1}K^\bullet$ has $\mathop{\mathrm{Ker}}(d_ K^{n + 1})$ in degree $n + 1$. However, since $f$ is a quasi-isomorphism we know that $H^{n + 1}(f)$ is surjective. Hence after replacing $f : P_{n + 1}^\bullet \to \tau _{\leq n + 1}K^\bullet$ by $f \oplus g : P_{n + 1}^\bullet \oplus P^\bullet \to \tau _{\leq n + 1}K^\bullet$ we win. $\square$

In some cases we can use the lemma above to show that a left derived functor is everywhere defined.

Proposition 13.29.2. Let $F : \mathcal{A} \to \mathcal{B}$ be a right exact functor of abelian categories. Let $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be a subset. Assume

1. $\mathcal{P}$ contains $0$, is closed under (finite) direct sums, and every object of $\mathcal{A}$ is a quotient of an element of $\mathcal{P}$,

2. for any bounded above acyclic complex $P^\bullet$ of $\mathcal{A}$ with $P^ n \in \mathcal{P}$ for all $n$ the complex $F(P^\bullet )$ is exact,

3. $\mathcal{A}$ and $\mathcal{B}$ have colimits of systems over $\mathbf{N}$,

4. colimits over $\mathbf{N}$ are exact in both $\mathcal{A}$ and $\mathcal{B}$, and

5. $F$ commutes with colimits over $\mathbf{N}$.

Then $LF$ is defined on all of $D(\mathcal{A})$.

Proof. By (1) and Lemma 13.15.4 for any bounded above complex $K^\bullet$ there exists a quasi-isomorphism $P^\bullet \to K^\bullet$ with $P^\bullet$ bounded above and $P^ n \in \mathcal{P}$ for all $n$. Suppose that $s : P^\bullet \to (P')^\bullet$ is a quasi-isomorphism of bounded above complexes consisting of objects of $\mathcal{P}$. Then $F(P^\bullet ) \to F((P')^\bullet )$ is a quasi-isomorphism because $F(C(s)^\bullet )$ is acyclic by assumption (2). This already shows that $LF$ is defined on $D^{-}(\mathcal{A})$ and that a bounded above complex consisting of objects of $\mathcal{P}$ computes $LF$, see Lemma 13.14.15.

Next, let $K^\bullet$ be an arbitrary complex of $\mathcal{A}$. Choose a diagram

$\xymatrix{ P_1^\bullet \ar[d] \ar[r] & P_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau _{\leq 1}K^\bullet \ar[r] & \tau _{\leq 2}K^\bullet \ar[r] & \ldots }$

as in Lemma 13.29.1. Note that the map $\mathop{\mathrm{colim}}\nolimits P_ n^\bullet \to K^\bullet$ is a quasi-isomorphism because colimits over $\mathbf{N}$ in $\mathcal{A}$ are exact and $H^ i(P_ n^\bullet ) = H^ i(K^\bullet )$ for $n > i$. We claim that

$F(\mathop{\mathrm{colim}}\nolimits P_ n^\bullet ) = \mathop{\mathrm{colim}}\nolimits F(P_ n^\bullet )$

(termwise colimits) is $LF(K^\bullet )$, i.e., that $\mathop{\mathrm{colim}}\nolimits P_ n^\bullet$ computes $LF$. To see this, by Lemma 13.14.15, it suffices to prove the following claim. Suppose that

$\mathop{\mathrm{colim}}\nolimits Q_ n^\bullet = Q^\bullet \xrightarrow {\ \alpha \ } P^\bullet = \mathop{\mathrm{colim}}\nolimits P_ n^\bullet$

is a quasi-isomorphism of complexes, such that each $P_ n^\bullet$, $Q_ n^\bullet$ is a bounded above complex whose terms are in $\mathcal{P}$ and the maps $P_ n^\bullet \to \tau _{\leq n}P^\bullet$ and $Q_ n^\bullet \to \tau _{\leq n}Q^\bullet$ are quasi-isomorphisms. Claim: $F(\alpha )$ is a quasi-isomorphism.

The problem is that we do not assume that $\alpha$ is given as a colimit of maps between the complexes $P_ n^\bullet$ and $Q_ n^\bullet$. However, for each $n$ we know that the solid arrows in the diagram

$\xymatrix{ & R^\bullet \ar@{..>}[d] \\ P_ n^\bullet \ar[d] & L^\bullet \ar@{..>}[l] \ar@{..>}[r] & Q_ n^\bullet \ar[d] \\ \tau _{\leq n}P^\bullet \ar[rr]^{\tau _{\leq n}\alpha } & & \tau _{\leq n}Q^\bullet }$

are quasi-isomorphisms. Because quasi-isomorphisms form a multiplicative system in $K(\mathcal{A})$ (see Lemma 13.11.2) we can find a quasi-isomorphism $L^\bullet \to P_ n^\bullet$ and map of complexes $L^\bullet \to Q_ n^\bullet$ such that the diagram above commutes up to homotopy. Then $\tau _{\leq n}L^\bullet \to L^\bullet$ is a quasi-isomorphism. Hence (by the first part of the proof) we can find a bounded above complex $R^\bullet$ whose terms are in $\mathcal{P}$ and a quasi-isomorphism $R^\bullet \to L^\bullet$ (as indicated in the diagram). Using the result of the first paragraph of the proof we see that $F(R^\bullet ) \to F(P_ n^\bullet )$ and $F(R^\bullet ) \to F(Q_ n^\bullet )$ are quasi-isomorphisms. Thus we obtain a isomorphisms $H^ i(F(P_ n^\bullet )) \to H^ i(F(Q_ n^\bullet ))$ fitting into the commutative diagram

$\xymatrix{ H^ i(F(P_ n^\bullet )) \ar[r] \ar[d] & H^ i(F(Q_ n^\bullet )) \ar[d] \\ H^ i(F(P^\bullet )) \ar[r] & H^ i(F(Q^\bullet )) }$

The exact same argument shows that these maps are also compatible as $n$ varies. Since by (4) and (5) we have

$H^ i(F(P^\bullet )) = H^ i(F(\mathop{\mathrm{colim}}\nolimits P_ n^\bullet )) = H^ i(\mathop{\mathrm{colim}}\nolimits F(P_ n^\bullet )) = \mathop{\mathrm{colim}}\nolimits H^ i(F(P_ n^\bullet ))$

and similarly for $Q^\bullet$ we conclude that $H^ i(\alpha ) : H^ i(F(P^\bullet ) \to H^ i(F(Q^\bullet )$ is an isomorphism and the claim follows. $\square$

Lemma 13.29.3. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{I} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be a subset. Assume $\mathcal{I}$ contains $0$, is closed under (finite) products, and every object of $\mathcal{A}$ is a subobject of an element of $\mathcal{I}$. Let $K^\bullet$ be a complex. There exists a commutative diagram

$\xymatrix{ \ldots \ar[r] & \tau _{\geq -2}K^\bullet \ar[r] \ar[d] & \tau _{\geq -1}K^\bullet \ar[d] \\ \ldots \ar[r] & I_2^\bullet \ar[r] & I_1^\bullet }$

in the category of complexes such that

1. the vertical arrows are quasi-isomorphisms and termwise injective,

2. $I_ n^\bullet$ is a bounded below complex with terms in $\mathcal{I}$,

3. the arrows $I_{n + 1}^\bullet \to I_ n^\bullet$ are termwise split surjections and $\mathop{\mathrm{Ker}}(I^ i_{n + 1} \to I^ i_ n)$ is an element of $\mathcal{I}$.

Proof. This lemma is dual to Lemma 13.29.1. $\square$

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