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The Stacks project

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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