59.52 Colimits and complexes
In this section we discuss taking cohomology of systems of complexes in various settings, continuing the discussion for sheaves started in Section 59.51. We strongly urge the reader not to read this section unless absolutely necessary.
Lemma 59.52.1. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be a limit of a directed system of schemes with affine transition morphisms $f_{i'i} : X_{i'} \to X_ i$. We assume that $X_ i$ is quasi-compact and quasi-separated for all $i \in I$. Let $\mathcal{F}_ i^\bullet $ be a complex of abelian sheaves on $X_{i, {\acute{e}tale}}$. Let $\varphi _{i'i} : f_{i'i}^{-1}\mathcal{F}_ i^\bullet \to \mathcal{F}_{i'}^\bullet $ be a map of complexes on $X_{i, {\acute{e}tale}}$ such that $\varphi _{i''i} = \varphi _{i''i'} \circ f_{i'' i'}^{-1}\varphi _{i'i}$ whenever $i'' \geq i' \geq i$. Assume there is an integer $a$ such that $\mathcal{F}_ i^ n = 0$ for $n < a$ and all $i \in I$. Then we have
\[ H^ p_{\acute{e}tale}(X, \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i^\bullet ) = \mathop{\mathrm{colim}}\nolimits H^ p_{\acute{e}tale}(X_ i, \mathcal{F}^\bullet _ i) \]
where $f_ i : X \to X_ i$ is the projection.
Proof.
This is a consequence of Theorem 59.51.3. Set $\mathcal{F}^\bullet = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i^\bullet $. The theorem tells us that
\[ \mathop{\mathrm{colim}}\nolimits _{i \in I} H_{\acute{e}tale}^ p(X_ i, \mathcal{F}_ i^ n) = H_{\acute{e}tale}^ p(X, \mathcal{F}^ n) \]
for all $n, p \in \mathbf{Z}$. Let us use the spectral sequences
\[ E_{1, i}^{s, t} = H_{\acute{e}tale}^ t(X_ i, \mathcal{F}_ i^ s) \Rightarrow H_{\acute{e}tale}^{s + t}(X_ i, \mathcal{F}_ i^\bullet ) \]
and
\[ E_1^{s, t} = H_{\acute{e}tale}^ t(X, \mathcal{F}^ s) \Rightarrow H_{\acute{e}tale}^{s + t}(X, \mathcal{F}^\bullet ) \]
of Derived Categories, Lemma 13.21.3. Since $\mathcal{F}_ i^ n = 0$ for $n < a$ (with $a$ independent of $i$) we see that only a fixed finite number of terms $E_{1, i}^{s, t}$ (independent of $i$) and $E_1^{s, t}$ contribute to $H^ q_{\acute{e}tale}(X_ i, \mathcal{F}_ i^\bullet )$ and $H^ q_{\acute{e}tale}(X, \mathcal{F}^\bullet )$ and $E_1^{s, t} = \mathop{\mathrm{colim}}\nolimits E_{i, i}^{s, t}$. This implies what we want. Some details omitted. (There is an alternative argument using “stupid” truncations of complexes which avoids using spectral sequences.)
$\square$
Lemma 59.52.2. Let $X$ be a quasi-compact and quasi-sepated scheme. Let $K_ i \in D(X_{\acute{e}tale})$, $i \in I$ be a family of objects. Assume given $a \in \mathbf{Z}$ such that $H^ n(K_ i) = 0$ for $n < a$ and $i \in I$. Then $R\Gamma (X, \bigoplus _ i K_ i) = \bigoplus _ i R\Gamma (X, K_ i)$.
Proof.
We have to show that $H^ p(X, \bigoplus _ i K_ i) = \bigoplus _ i H^ p(X, K_ i)$ for all $p \in \mathbf{Z}$. Choose complexes $\mathcal{F}_ i^\bullet $ representing $K_ i$ such that $\mathcal{F}_ i^ n = 0$ for $n < a$. The direct sum of the complexes $\mathcal{F}_ i^\bullet $ represents the object $\bigoplus K_ i$ by Injectives, Lemma 19.13.4. Since $\bigoplus \mathcal{F}^\bullet $ is the filtered colimit of the finite direct sums, the result follows from Lemma 59.52.1.
$\square$
Lemma 59.52.3. Let $S$ be a scheme. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be a limit of a directed system of schemes over $S$ with affine transition morphisms $f_{i'i} : X_{i'} \to X_ i$. We assume that $X_ i$ is quasi-compact and quasi-separated for all $i \in I$. Let $K \in D^+(S_{\acute{e}tale})$. Then
\[ \mathop{\mathrm{colim}}\nolimits _{i \in I} H_{\acute{e}tale}^ p(X_ i, K|_{X_ i}) = H_{\acute{e}tale}^ p(X, K|_ X). \]
for all $p \in \mathbf{Z}$ where $K|_{X_ i}$ and $K|_ X$ are the pullbacks of $K$ to $X_ i$ and $X$.
Proof.
We may represent $K$ by a bounded below complex $\mathcal{G}^\bullet $ of abelian sheaves on $S_{\acute{e}tale}$. Say $\mathcal{G}^ n = 0$ for $n < a$. Denote $\mathcal{F}^\bullet _ i$ and $\mathcal{F}^\bullet $ the pullbacks of this complex of $X_ i$ and $X$. These complexes represent the objects $K|_{X_ i}$ and $K|_ X$ and we have $\mathcal{F}^\bullet = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i^\bullet $ termwise. Hence the lemma follows from Lemma 59.52.1.
$\square$
Lemma 59.52.4. Let $I$, $g_ i : X_ i \to S_ i$, $g : X \to S$, $f_ i$, $g_ i$, $h_ i$ be as in Lemma 59.51.8. Let $0 \in I$ and $K_0 \in D^+(X_{0, {\acute{e}tale}})$. For $i \geq 0$ denote $K_ i$ the pullback of $K_0$ to $X_ i$. Denote $K$ the pullback of $K$ to $X$. Then
\[ R^ pg_*K = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} h_ i^{-1}R^ pg_{i, *}K_ i \]
for all $p \in \mathbf{Z}$.
Proof.
Fix an integer $p_0 \in \mathbf{Z}$. Let $a$ be an integer such that $H^ j(K_0) = 0$ for $j < a$. We will prove the formula holds for all $p \leq p_0$ by descending induction on $a$. If $a > p_0$, then we see that the left and right hand side of the formula are zero for $p \leq p_0$ by trivial vanishing, see Derived Categories, Lemma 13.16.1. Assume $a \leq p_0$. Consider the distinguished triangle
\[ H^ a(K_0)[-a] \to K_0 \to \tau _{\geq a + 1}K_0 \]
Pulling back this distinguished triangle to $X_ i$ and $X$ gives compatible distinguished triangles for $K_ i$ and $K$. For $p \leq p_0$ we consider the commutative diagram
\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} h_ i^{-1}R^{p - 1}g_{i, *}(\tau _{\geq a + 1}K_ i) \ar[r]_-\alpha \ar[d] & R^{p - 1}g_*(\tau _{\geq a + 1}K) \ar[d] \\ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} h_ i^{-1}R^ pg_{i, *}(H^ a(K_ i)[-a]) \ar[r]_-\beta \ar[d] & R^ pg_*(H^ a(K)[-a]) \ar[d] \\ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} h_ i^{-1}R^ pg_{i, *}K_ i \ar[r]_-\gamma \ar[d] & R^ pg_*K \ar[d] \\ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} R^ pg_{i, *}\tau _{\geq a + 1}K_ i \ar[r]_-\delta \ar[d] & R^ pg_*\tau _{\geq a + 1}K \ar[d] \\ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} R^{p + 1}g_{i, *}(H^ a(K_ i)[-a]) \ar[r]^-\epsilon & R^{p + 1}g_*(H^ a(K)[-a]) } \]
with exact columns. The arrows $\beta $ and $\epsilon $ are isomorphisms by Lemma 59.51.8. The arrows $\alpha $ and $\delta $ are isomorphisms by induction hypothesis. Hence $\gamma $ is an isomorphism as desired.
$\square$
Lemma 59.52.5. Let $I$, $g_ i : X_ i \to S_ i$, $g : X \to S$, $f_{ii'}$, $f_ i$, $g_ i$, $h_ i$ be as in Lemma 59.51.8. Let $\mathcal{F}_ i^\bullet $ be a complex of abelian sheaves on $X_{i, {\acute{e}tale}}$. Let $\varphi _{i'i} : f_{i'i}^{-1}\mathcal{F}_ i^\bullet \to \mathcal{F}_{i'}^\bullet $ be a map of complexes on $X_{i, {\acute{e}tale}}$ such that $\varphi _{i''i} = \varphi _{i''i'} \circ f_{i'' i'}^{-1}\varphi _{i'i}$ whenever $i'' \geq i' \geq i$. Assume there is an integer $a$ such that $\mathcal{F}_ i^ n = 0$ for $n < a$ and all $i \in I$. Then
\[ R^ pg_*(\mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i^\bullet ) = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} h_ i^{-1}R^ pg_{i, *}\mathcal{F}_ i^\bullet \]
for all $p \in \mathbf{Z}$.
Proof.
This is a consequence of Lemma 59.51.8. Set $\mathcal{F}^\bullet = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i^\bullet $. The lemma tells us that
\[ \mathop{\mathrm{colim}}\nolimits _{i \in I} h_ i^{-1}R^ pg_{i, *}\mathcal{F}_ i^ n = R^ pg_*\mathcal{F}^ n \]
for all $n, p \in \mathbf{Z}$. Let us use the spectral sequences
\[ E_{1, i}^{s, t} = R^ tg_{i, *}\mathcal{F}_ i^ s \Rightarrow R^{s + t}g_{i, *}\mathcal{F}_ i^\bullet \]
and
\[ E_1^{s, t} = R^ tg_*\mathcal{F}^ s \Rightarrow R^{s + t}g_*\mathcal{F}^\bullet \]
of Derived Categories, Lemma 13.21.3. Since $\mathcal{F}_ i^ n = 0$ for $n < a$ (with $a$ independent of $i$) we see that only a fixed finite number of terms $E_{1, i}^{s, t}$ (independent of $i$) and $E_1^{s, t}$ contribute and $E_1^{s, t} = \mathop{\mathrm{colim}}\nolimits E_{i, i}^{s, t}$. This implies what we want. Some details omitted. (There is an alternative argument using “stupid” truncations of complexes which avoids using spectral sequences.)
$\square$
Lemma 59.52.6. Let $f : X \to Y$ be a quasi-compact and quasi-sepated morphism of schemes. Let $K_ i \in D(X_{\acute{e}tale})$, $i \in I$ be a family of objects. Assume given $a \in \mathbf{Z}$ such that $H^ n(K_ i) = 0$ for $n < a$ and $i \in I$. Then $Rf_*(\bigoplus _ i K_ i) = \bigoplus _ i Rf_*K_ i$.
Proof.
We have to show that $R^ pf_*(\bigoplus _ i K_ i) = \bigoplus _ i R^ pf_*K_ i$ for all $p \in \mathbf{Z}$. Choose complexes $\mathcal{F}_ i^\bullet $ representing $K_ i$ such that $\mathcal{F}_ i^ n = 0$ for $n < a$. The direct sum of the complexes $\mathcal{F}_ i^\bullet $ represents the object $\bigoplus K_ i$ by Injectives, Lemma 19.13.4. Since $\bigoplus \mathcal{F}^\bullet $ is the filtered colimit of the finite direct sums, the result follows from Lemma 59.52.5.
$\square$
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