## 13.32 Bounded cohomological dimension

There is another case where the unbounded derived functor exists. Namely, when the functor has bounded cohomological dimension.

Lemma 13.32.1. Let $\mathcal{A}$ be an abelian category. Let $d : \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \to \{ 0, 1, 2, \ldots , \infty \}$ be a function. Assume that

1. every object of $\mathcal{A}$ is a subobject of an object $A$ with $d(A) = 0$,

2. $d(A \oplus B) \leq \max \{ d(A), d(B)\}$ for $A, B \in \mathcal{A}$, and

3. if $0 \to A \to B \to C \to 0$ is short exact, then $d(C) \leq \max \{ d(A) - 1, d(B)\}$.

Let $K^\bullet$ be a complex such that $n + d(K^ n)$ tends to $-\infty$ as $n \to -\infty$. Then there exists a quasi-isomorphism $K^\bullet \to L^\bullet$ with $d(L^ n) = 0$ for all $n \in \mathbf{Z}$.

Proof. By Lemma 13.15.5 we can find a quasi-isomorphism $\sigma _{\geq 0}K^\bullet \to M^\bullet$ with $M^ n = 0$ for $n < 0$ and $d(M^ n) = 0$ for $n \geq 0$. Then $K^\bullet$ is quasi-isomorphic to the complex

$\ldots \to K^{-2} \to K^{-1} \to M^0 \to M^1 \to \ldots$

Hence we may assume that $d(K^ n) = 0$ for $n \gg 0$. Note that the condition $n + d(K^ n) \to -\infty$ as $n \to -\infty$ is not violated by this replacement.

We are going to improve $K^\bullet$ by an (infinite) sequence of elementary replacements. An elementary replacement is the following. Choose an index $n$ such that $d(K^ n) > 0$. Choose an injection $K^ n \to M$ where $d(M) = 0$. Set $M' = \mathop{\mathrm{Coker}}(K^ n \to M \oplus K^{n + 1})$. Consider the map of complexes

$\xymatrix{ K^\bullet : \ar[d] & K^{n - 1} \ar[d] \ar[r] & K^ n \ar[d] \ar[r] & K^{n + 1} \ar[d] \ar[r] & K^{n + 2} \ar[d] \\ (K')^\bullet : & K^{n - 1} \ar[r] & M \ar[r] & M' \ar[r] & K^{n + 2} }$

It is clear that $K^\bullet \to (K')^\bullet$ is a quasi-isomorphism. Moreover, it is clear that $d((K')^ n) = 0$ and

$d((K')^{n + 1}) \leq \max \{ d(K^ n) - 1, d(M \oplus K^{n + 1})\} \leq \max \{ d(K^ n) - 1, d(K^{n + 1})\}$

and the other values are unchanged.

To finish the proof we carefuly choose the order in which to do the elementary replacements so that for every integer $m$ the complex $\sigma _{\geq m}K^\bullet$ is changed only a finite number of times. To do this set

$\xi (K^\bullet ) = \max \{ n + d(K^ n) \mid d(K^ n) > 0\}$

and

$I = \{ n \in \mathbf{Z} \mid \xi (K^\bullet ) = n + d(K^ n) \text{ and } d(K^ n) > 0\}$

Our assumption that $n + d(K^ n)$ tends to $-\infty$ as $n \to -\infty$ and the fact that $d(K^ n) = 0$ for $n >> 0$ implies $\xi (K^\bullet ) < +\infty$ and that $I$ is a finite set. It is clear that $\xi ((K')^\bullet ) \leq \xi (K^\bullet )$ for an elementary transformation as above. An elementary transformation changes the complex in degrees $\leq \xi (K^\bullet ) + 1$. Hence if we can find finite sequence of elementary transformations which decrease $\xi (K^\bullet )$, then we win. However, note that if we do an elementary transformation starting with the smallest element $n \in I$, then we either decrease the size of $I$, or we increase $\min I$. Since every element of $I$ is $\leq \xi (K^\bullet )$ we see that we win after a finite number of steps. $\square$

Lemma 13.32.2. Let $F : \mathcal{A} \to \mathcal{B}$ be a left exact functor of abelian categories. Assume

1. every object of $\mathcal{A}$ is a subobject of an object which is right acyclic for $F$,

2. there exists an integer $n \geq 0$ such that $R^ nF = 0$,

Then

1. $RF : D(\mathcal{A}) \to D(\mathcal{B})$ exists,

2. any complex consisting of right acyclic objects for $F$ computes $RF$,

3. any complex is the source of a quasi-isomorphism into a complex consisting of right acyclic objects for $F$,

4. for $E \in D(\mathcal{A})$

1. $H^ i(RF(\tau _{\leq a}E) \to H^ i(RF(E))$ is an isomorphism for $i \leq a$,

2. $H^ i(RF(E)) \to H^ i(RF(\tau _{\geq b - n + 1}E))$ is an isomorphism for $i \geq b$,

3. if $H^ i(E) = 0$ for $i \not\in [a, b]$ for some $-\infty \leq a \leq b \leq \infty$, then $H^ i(RF(E)) = 0$ for $i \not\in [a, b + n - 1]$.

Proof. Note that the first assumption implies that $RF : D^+(\mathcal{A}) \to D^+(\mathcal{B})$ exists, see Proposition 13.16.8. Let $A$ be an object of $\mathcal{A}$. Choose an injection $A \to A'$ with $A'$ acyclic. Then we see that $R^{n + 1}F(A) = R^ nF(A'/A) = 0$ by the long exact cohomology sequence. Hence we conclude that $R^{n + 1}F = 0$. Continuing like this using induction we find that $R^ mF = 0$ for all $m \geq n$.

We are going to use Lemma 13.32.1 with the function $d : \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \to \{ 0, 1, 2, \ldots \}$ given by $d(A) = \max \{ 0\} \cup \{ i \mid R^ iF(A) \not= 0\}$. The first assumption of Lemma 13.32.1 is our assumption (1). The second assumption of Lemma 13.32.1 follows from the fact that $RF(A \oplus B) = RF(A) \oplus RF(B)$. The third assumption of Lemma 13.32.1 follows from the long exact cohomology sequence. Hence for every complex $K^\bullet$ there exists a quasi-isomorphism $K^\bullet \to L^\bullet$ into a complex of objects right acyclic for $F$. This proves statement (3).

We claim that if $L^\bullet \to M^\bullet$ is a quasi-isomorphism of complexes of right acyclic objects for $F$, then $F(L^\bullet ) \to F(M^\bullet )$ is a quasi-isomorphism. If we prove this claim then we get statements (1) and (2) of the lemma by Lemma 13.14.15. To prove the claim pick an integer $i \in \mathbf{Z}$. Consider the distinguished triangle

$\sigma _{\geq i - n - 1}L^\bullet \to \sigma _{\geq i - n - 1}M^\bullet \to Q^\bullet ,$

i.e., let $Q^\bullet$ be the cone of the first map. Note that $Q^\bullet$ is bounded below and that $H^ j(Q^\bullet )$ is zero except possibly for $j = i - n - 1$ or $j = i - n - 2$. We may apply $RF$ to $Q^\bullet$. Using the second spectral sequence of Lemma 13.21.3 and the assumed vanishing of cohomology (2) we conclude that $H^ j(RF(Q^\bullet ))$ is zero except possibly for $j \in \{ i - n - 2, \ldots , i - 1\}$. Hence we see that $RF(\sigma _{\geq i - n - 1}L^\bullet ) \to RF(\sigma _{\geq i - n - 1}M^\bullet )$ induces an isomorphism of cohomology objects in degrees $\geq i$. By Proposition 13.16.8 we know that $RF(\sigma _{\geq i - n - 1}L^\bullet ) = \sigma _{\geq i - n - 1}F(L^\bullet )$ and $RF(\sigma _{\geq i - n - 1}M^\bullet ) = \sigma _{\geq i - n - 1}F(M^\bullet )$. We conclude that $F(L^\bullet ) \to F(M^\bullet )$ is an isomorphism in degree $i$ as desired.

Part (4)(a) follows from Lemma 13.16.1.

For part (4)(b) let $E$ be represented by the complex $L^\bullet$ of objects right acyclic for $F$. By part (2) $RF(E)$ is represented by the complex $F(L^\bullet )$ and $RF(\sigma _{\geq c}L^\bullet )$ is represented by $\sigma _{\geq c}F(L^\bullet )$. Consider the distinguished triangle

$H^{b - n}(L^\bullet )[n - b] \to \tau _{\geq b - n}L^\bullet \to \tau _{\geq b - n + 1}L^\bullet$

of Remark 13.12.4. The vanishing established above gives that $H^ i(RF(\tau _{\geq b - n}L^\bullet ))$ agrees with $H^ i(RF(\tau _{\geq b - n + 1}L^\bullet ))$ for $i \geq b$. Consider the short exact sequence of complexes

$0 \to \mathop{\mathrm{Im}}(L^{b - n - 1} \to L^{b - n})[n - b] \to \sigma _{\geq b - n}L^\bullet \to \tau _{\geq b - n}L^\bullet \to 0$

Using the distinguished triangle associated to this (see Section 13.12) and the vanishing as before we conclude that $H^ i(RF(\tau _{\geq b - n}L^\bullet ))$ agrees with $H^ i(RF(\sigma _{\geq b - n}L^\bullet ))$ for $i \geq b$. Since the map $RF(\sigma _{\geq b - n}L^\bullet ) \to RF(L^\bullet )$ is represented by $\sigma _{\geq b - n}F(L^\bullet ) \to F(L^\bullet )$ we conclude that this in turn agrees with $H^ i(RF(L^\bullet ))$ for $i \geq b$ as desired.

Proof of (4)(c). Under the assumption on $E$ we have $\tau _{\leq a - 1}E = 0$ and we get the vanishing of $H^ i(RF(E))$ for $i \leq a - 1$ from part (4)(a). Similarly, we have $\tau _{\geq b + 1}E = 0$ and hence we get the vanishing of $H^ i(RF(E))$ for $i \geq b + n$ from part (4)(b). $\square$

Lemma 13.32.3. Let $F : \mathcal{A} \to \mathcal{B}$ be a right exact functor of abelian categories. If

1. every object of $\mathcal{A}$ is a quotient of an object which is left acyclic for $F$,

2. there exists an integer $n \geq 0$ such that $L^ nF = 0$,

Then

1. $LF : D(\mathcal{A}) \to D(\mathcal{B})$ exists,

2. any complex consisting of left acyclic objects for $F$ computes $LF$,

3. any complex is the target of a quasi-isomorphism from a complex consisting of left acyclic objects for $F$,

4. for $E \in D(\mathcal{A})$

1. $H^ i(LF(\tau _{\leq a + n - 1}E) \to H^ i(LF(E))$ is an isomorphism for $i \leq a$,

2. $H^ i(LF(E)) \to H^ i(LF(\tau _{\geq b}E))$ is an isomorphism for $i \geq b$,

3. if $H^ i(E) = 0$ for $i \not\in [a, b]$ for some $-\infty \leq a \leq b \leq \infty$, then $H^ i(LF(E)) = 0$ for $i \not\in [a - n + 1, b]$.

Proof. This is dual to Lemma 13.32.2. $\square$

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