21.25 Bounded cohomological dimension
In this section we ask when a functor $Rf_*$ has bounded cohomological dimension. This is a rather subtle question when we consider unbounded complexes.
Situation 21.25.1. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ be a weak Serre subcategory. We assume the following is true: there exists a subset $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that
every object of $\mathcal{C}$ has a covering whose members are in $\mathcal{B}$, and
for every $V \in \mathcal{B}$ there exists an integer $d_ V$ and a cofinal system $\text{Cov}_ V$ of coverings of $V$ such that
\[ H^ p(V_ i, \mathcal{F}) = 0 \text{ for } \{ V_ i \to V\} \in \text{Cov}_ V,\ p > d_ V, \text{ and } \mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \]
reference
Lemma 21.25.2. In Situation 21.25.1 for any $E \in D_\mathcal {A}(\mathcal{O})$ the map $E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} E$ of Derived Categories, Remark 13.34.4 is an isomorphism in $D(\mathcal{O})$.
Proof.
Follows immediately from Lemma 21.23.8.
$\square$
Lemma 21.25.3. In Situation 21.25.1 let $(K_ n)$ be an inverse system in $D_\mathcal {A}^+(\mathcal{O})$. Assume that for every $j$ the inverse system $(H^ j(K_ n))$ in $\mathcal{A}$ is eventually constant with value $\mathcal{H}^ j$. Then $H^ j(R\mathop{\mathrm{lim}}\nolimits K_ n) = \mathcal{H}^ j$ for all $j$.
Proof.
Let $V \in \mathcal{B}$. Let $\{ V_ i \to V\} $ be in the set $\text{Cov}_ V$ of Situation 21.25.1. Because $K_ n$ is bounded below there is a spectral sequence
\[ E_2^{p, q} = H^ p(V_ i, H^ q(K_ n)) \]
converging to $H^{p + q}(V_ i, K_ n)$. See Derived Categories, Lemma 13.21.3. Observe that $E_2^{p, q} = 0$ for $p > d_ V$ by assumption. Pick $n_0$ such that
\[ \begin{matrix} \mathcal{H}^{j + 1}
& =
& H^{j + 1}(K_ n),
\\ \mathcal{H}^ j
& =
& H^ j(K_ n),
\\ \ldots ,
\\ \mathcal{H}^{j - d_ V - 2}
& =
& H^{j - d_ V - 2}(K_ n)
\end{matrix} \]
for all $n \geq n_0$. Comparing the spectral sequences above for $K_ n$ and $K_{n_0}$, we see that for $n \geq n_0$ the cohomology groups $H^{j - 1}(V_ i, K_ n)$ and $H^ j(V_ i, K_ n)$ are independent of $n$. It follows that the map on sections $H^ j(R\mathop{\mathrm{lim}}\nolimits K_ n)(V) \to H^ j(K_ n)(V)$ is injective for $n$ large enough (depending on $V$), see Lemma 21.23.6. Since every object of $\mathcal{C}$ can be covered by elements of $\mathcal{B}$, we conclude that the map $H^ j(R\mathop{\mathrm{lim}}\nolimits K_ n) \to \mathcal{H}^ j$ is injective.
Surjectivity is shown in a similar manner. Namely, pick $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $\gamma \in \mathcal{H}^ j(U)$. We want to lift $\gamma $ to a section of $H^ j(R\mathop{\mathrm{lim}}\nolimits K_ n)$ after replacing $U$ by the members of a covering. Hence we may assume $U = V \in \mathcal{B}$ by property (1) of Situation 21.25.1. Pick $n_0$ such that
\[ \begin{matrix} \mathcal{H}^{j + 1}
& =
& H^{j + 1}(K_ n),
\\ \mathcal{H}^ j
& =
& H^ j(K_ n),
\\ \ldots ,
\\ \mathcal{H}^{j - d_ V - 2}
& =
& H^{j - d_ V - 2}(K_ n)
\end{matrix} \]
for all $n \geq n_0$. Choose an element $\{ V_ i \to V\} $ of $\text{Cov}_ V$ such that $\gamma |_{V_ i} \in \mathcal{H}^ j(V_ i) = H^ j(K_{n_0})(V_ i)$ lifts to an element $\gamma _{n_0, i} \in H^ j(V_ i, K_{n_0})$. This is possible because $H^ j(K_{n_0})$ is the sheafification of $U \mapsto H^ j(U, K_{n_0})$ by Lemma 21.20.3. By the discussion in the first paragraph of the proof we have that $H^{j - 1}(V_ i, K_ n)$ and $H^ j(V_ i, K_ n)$ are independent of $n \geq n_0$. Hence $\gamma _{n_0, i}$ lifts to an element $\gamma _ i \in H^ j(V_ i, R\mathop{\mathrm{lim}}\nolimits K_ n)$ by Lemma 21.23.2. This finishes the proof.
$\square$
reference
Lemma 21.25.4. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$ be weak Serre subcategories. Assume there is an integer $N$ such that
$\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation 21.25.1,
$\mathcal{C}', \mathcal{O}', \mathcal{A}'$ satisfy the assumption of Situation 21.25.1,
$R^ pf_*\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$ for $p \geq 0$ and $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$,
$R^ pf_*\mathcal{F} = 0$ for $p > N$ and $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$,
Then for $K$ in $D_\mathcal {A}(\mathcal{O})$ we have
$Rf_*K$ is in $D_{\mathcal{A}'}(\mathcal{O}')$,
the map $H^ j(Rf_*K) \to H^ j(Rf_*(\tau _{\geq -n}K))$ is an isomorphism for $j \geq N - n$.
Proof.
By Lemma 21.25.2 we have $K = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K$. By Lemma 21.23.3 we have $Rf_*K = R\mathop{\mathrm{lim}}\nolimits Rf_*\tau _{\geq -n}K$. The complexes $Rf_*\tau _{\geq -n}K$ are bounded below. The spectral sequence
\[ E_2^{p, q} = R^ pf_*H^ q(\tau _{\geq -n}K) \]
converging to $H^{p + q}(Rf_*\tau _{\geq -n}K)$ (Derived Categories, Lemma 13.21.3) and assumption (3) show that $Rf_*\tau _{\geq -n}K$ lies in $D^+_{\mathcal{A}'}(\mathcal{O}')$, see Homology, Lemma 12.24.11. Observe that for $m \geq n$ the map
\[ Rf_*(\tau _{\geq -m}K) \longrightarrow Rf_*(\tau _{\geq -n}K) \]
induces an isomorphism on cohomology sheaves in degrees $j \geq -n + N$ by the spectral sequences above. Hence we may apply Lemma 21.25.3 to conclude.
$\square$
It turns out that we sometimes need a variant of the lemma above where the assumptions are slightly different.
Situation 21.25.5. Let $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ be a morphism of ringed sites. Let $u : \mathcal{C}' \to \mathcal{C}$ be the corresponding continuous functor of sites. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ be a weak Serre subcategory. We assume the following is true: there exists a subset $\mathcal{B}' \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ such that
every object of $\mathcal{C}'$ has a covering whose members are in $\mathcal{B}'$, and
for every $V' \in \mathcal{B}'$ there exists an integer $d_{V'}$ and a cofinal system $\text{Cov}_{V'}$ of coverings of $V'$ such that
\[ H^ p(u(V'_ i), \mathcal{F}) = 0 \text{ for } \{ V'_ i \to V'\} \in \text{Cov}_{V'},\ p > d_{V'}, \text{ and } \mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \]
reference
Lemma 21.25.6. Let $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ be a morphism of ringed sites. assume moreover there is an integer $N$ such that
$\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation 21.25.1,
$f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ and $\mathcal{A}$ satisfy the assumption of Situation 21.25.5,
$R^ pf_*\mathcal{F} = 0$ for $p > N$ and $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$,
Then for $K$ in $D_\mathcal {A}(\mathcal{O})$ the map $H^ j(Rf_*K) \to H^ j(Rf_*(\tau _{\geq -n}K))$ is an isomorphism for $j \geq N - n$.
Proof.
Let $K$ be in $D_\mathcal {A}(\mathcal{O})$. By Lemma 21.25.2 we have $K = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K$. By Lemma 21.23.3 we have $Rf_*K = R\mathop{\mathrm{lim}}\nolimits Rf_*(\tau _{\geq -n}K)$. Let $V' \in \mathcal{B}'$ and let $\{ V'_ i \to V'\} $ be an element of $\text{Cov}_{V'}$. Then we consider
\[ H^ j(V'_ i, Rf_*K) = H^ j(u(V'_ i), K) \quad \text{and}\quad H^ j(V'_ i, Rf_*(\tau _{\geq -n}K)) = H^ j(u(V'_ i), \tau _{\geq -n}K) \]
The assumption in Situation 21.25.5 implies that the last group is independent of $n$ for $n$ large enough depending on $j$ and $d_{V'}$. Some details omitted. We apply this for $j$ and $j - 1$ and via Lemma 21.23.2 this gives that
\[ H^ j(V'_ i, Rf_*K) = \mathop{\mathrm{lim}}\nolimits H^ j(V'_ i, Rf_*(\tau _{\geq -n} K)) \]
and the system on the right is constant for $n$ larger than a constant depending only on $d_{V'}$ and $j$. Thus Lemma 21.23.6 implies that
\[ H^ j(Rf_*K)(V') \longrightarrow \left(\mathop{\mathrm{lim}}\nolimits H^ j(Rf_*(\tau _{\geq -n}K))\right)(V') \]
is injective. Since the elements $V' \in \mathcal{B}'$ cover every object of $\mathcal{C}'$ we conclude that the map $H^ j(Rf_*K) \to \mathop{\mathrm{lim}}\nolimits H^ j(Rf_*(\tau _{\geq -n}K))$ is injective. The spectral sequence
\[ E_2^{p, q} = R^ pf_*H^ q(\tau _{\geq -n}K) \]
converging to $H^{p + q}(Rf_*(\tau _{\geq -n}K))$ (Derived Categories, Lemma 13.21.3) and assumption (3) show that $H^ j(Rf_*(\tau _{\geq -n}K))$ is constant for $n \geq N - j$. Hence $H^ j(Rf_*K) \to H^ j(Rf_*(\tau _{\geq -n}K))$ is injective for $j \geq N - n$.
Thus we proved the lemma with “isomorphism” in the last line of the lemma replaced by “injective”. However, now choose $j$ and $n$ with $j \geq N - n$. Then consider the distinguished triangle
\[ \tau _{\leq -n - 1}K \to K \to \tau _{\geq -n}K \to (\tau _{\leq -n - 1}K)[1] \]
See Derived Categories, Remark 13.12.4. Since $\tau _{\geq -n}\tau _{\leq -n -1}K = 0$, the injectivity already proven for $\tau _{-n - 1}K$ implies
\[ 0 = H^ j(Rf_*(\tau _{\leq -n - 1}K)) = H^{j + 1}(Rf_*(\tau _{\leq -n - 1}K)) = H^{j + 2}(Rf_*(\tau _{\leq -n - 1}K)) = \ldots \]
By the long exact cohomology sequence associated to the distinguished triangle
\[ Rf_*(\tau _{\leq -n - 1}K) \to Rf_*K \to Rf_*(\tau _{\geq -n}K) \to Rf_*(\tau _{\leq -n - 1}K)[1] \]
this implies that $H^ j(Rf_*K) \to H^ j(Rf_*(\tau _{\geq -n}K))$ is an isomorphism.
$\square$
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