The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

21.17 Cohomology and colimits

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \to \textit{Mod}(\mathcal{O})$, $i \mapsto \mathcal{F}_ i$ be a diagram over the index category $\mathcal{I}$, see Categories, Section 4.14. For each $i$ there is a canonical map $\mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ which induces a map on cohomology. Hence we get a canonical map

\[ \mathop{\mathrm{colim}}\nolimits _ i H^ p(U, \mathcal{F}_ i) \longrightarrow H^ p(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i) \]

for every $p \geq 0$ and every object $U$ of $\mathcal{C}$. These maps are in general not isomorphisms, even for $p = 0$.

The following lemma is the analogue of Sites, Lemma 7.17.5 for cohomology.

Lemma 21.17.1. Let $\mathcal{C}$ be a site. Let $\text{Cov}_\mathcal {C}$ be the set of coverings of $\mathcal{C}$ (see Sites, Definition 7.6.2). Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $\text{Cov} \subset \text{Cov}_\mathcal {C}$ be subsets. Assume that

  1. For every $\mathcal{U} \in \text{Cov}$ we have $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ with $I$ finite, $U, U_ i \in \mathcal{B}$ and every $U_{i_0} \times _ U \ldots \times _ U U_{i_ p} \in \mathcal{B}$.

  2. For every $U \in \mathcal{B}$ the coverings of $U$ occurring in $\text{Cov}$ is a cofinal system of coverings of $U$.

Then the map

\[ \mathop{\mathrm{colim}}\nolimits _ i H^ p(U, \mathcal{F}_ i) \longrightarrow H^ p(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i) \]

is an isomorphism for every $p \geq 0$, every $U \in \mathcal{B}$, and every filtered diagram $\mathcal{I} \to \textit{Ab}(\mathcal{C})$.

Proof. To prove the lemma we will argue by induction on $p$. Note that we require in (1) the coverings $\mathcal{U} \in \text{Cov}$ to be finite, so that all the elements of $\mathcal{B}$ are quasi-compact. Hence (2) and (1) imply that any $U \in \mathcal{B}$ satisfies the hypothesis of Sites, Lemma 7.17.5 (4). Thus we see that the result holds for $p = 0$. Now we assume the lemma holds for $p$ and prove it for $p + 1$.

Choose a filtered diagram $\mathcal{F} : \mathcal{I} \to \textit{Ab}(\mathcal{C})$, $i \mapsto \mathcal{F}_ i$. Since $\textit{Ab}(\mathcal{C})$ has functorial injective embeddings, see Injectives, Theorem 19.7.4, we can find a morphism of filtered diagrams $\mathcal{F} \to \mathcal{I}$ such that each $\mathcal{F}_ i \to \mathcal{I}_ i$ is an injective map of abelian sheaves into an injective abelian sheaf. Denote $\mathcal{Q}_ i$ the cokernel so that we have short exact sequences

\[ 0 \to \mathcal{F}_ i \to \mathcal{I}_ i \to \mathcal{Q}_ i \to 0. \]

Since colimits of sheaves are the sheafification of colimits on the level of presheaves, since sheafification is exact, and since filtered colimits of abelian groups are exact (see Algebra, Lemma 10.8.8), we see the sequence

\[ 0 \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{Q}_ i \to 0. \]

is also a short exact sequence. We claim that $H^ q(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = 0$ for all $U \in \mathcal{B}$ and all $q \geq 1$. Accepting this claim for the moment consider the diagram

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits _ i H^ p(U, \mathcal{I}_ i) \ar[d] \ar[r] & \mathop{\mathrm{colim}}\nolimits _ i H^ p(U, \mathcal{Q}_ i) \ar[d] \ar[r] & \mathop{\mathrm{colim}}\nolimits _ i H^{p + 1}(U, \mathcal{F}_ i) \ar[d] \ar[r] & 0 \ar[d] \\ H^ p(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) \ar[r] & H^ p(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{Q}_ i) \ar[r] & H^{p + 1}(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i) \ar[r] & 0 } \]

The zero at the lower right corner comes from the claim and the zero at the upper right corner comes from the fact that the sheaves $\mathcal{I}_ i$ are injective. The top row is exact by an application of Algebra, Lemma 10.8.8. Hence by the snake lemma we deduce the result for $p + 1$.

It remains to show that the claim is true. We will use Lemma 21.11.9. By the result for $p = 0$ we see that for $\mathcal{U} \in \text{Cov}$ we have

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = \mathop{\mathrm{colim}}\nolimits _ i \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}_ i) \]

because all the $U_{j_0} \times _ U \ldots \times _ U U_{j_ p}$ are in $\mathcal{B}$. By Lemma 21.11.2 each of the complexes in the colimit of Čech complexes is acyclic in degree $\geq 1$. Hence by Algebra, Lemma 10.8.8 we see that also the Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i)$ is acyclic in degrees $\geq 1$. In other words we see that $\check{H}^ p(\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = 0$ for all $p \geq 1$. Thus the assumptions of Lemma 21.11.9. are satisfied and the claim follows. $\square$

Lemma 21.17.2. Let $\mathcal{I}$ be a cofiltered index category and let $\{ \mathcal{C}_ i, f_ a\} $ be an inverse system of sites over $\mathcal{I}$ as in Sites, Situation 7.18.1. Set $\mathcal{C} = \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i$ as in Sites, Lemmas 7.18.2 and 7.18.3. Moreover, assume given

  1. an abelian sheaf $\mathcal{F}_ i$ on $\mathcal{C}_ i$ for all $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$,

  2. for $a : j \to i$ a map $\varphi _ a : f_ a^{-1}\mathcal{F}_ i \to \mathcal{F}_ j$ of abelian sheaves on $\mathcal{C}_ j$

such that $\varphi _ c = \varphi _ b \circ f_ b^{-1}\varphi _ a$ whenever $c = a \circ b$. Then there exists a map of systems $(\mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ such that $\mathcal{F}_ i \to \mathcal{G}_ i$ is injective and $\mathcal{G}_ i$ is an injective abelian sheaf.

Proof. For each $i$ we pick an injection $\mathcal{F}_ i \to \mathcal{A}_ i$ where $\mathcal{A}_ i$ is an injective abelian sheaf on $\mathcal{C}_ i$. Then we can consider the family of maps

\[ \gamma _ i : \mathcal{F}_ i \longrightarrow \prod \nolimits _{b : k \to i} f_{b, *}\mathcal{A}_ k = \mathcal{G}_ i \]

where the component maps are the maps adjoint to the maps $f_ b^{-1}\mathcal{F}_ i \to \mathcal{F}_ k \to \mathcal{A}_ k$. For $a : j \to i$ in $\mathcal{I}$ there is a canonical map

\[ \psi _ a : f_ a^{-1}\mathcal{G}_ i \to \mathcal{G}_ j \]

whose components are the canonical maps $f_ b^{-1}f_{a \circ b, *}\mathcal{A}_ k \to f_{b, *}\mathcal{A}_ k$ for $b : k \to j$. Thus we find an injection $\{ \gamma _ i\} : \{ \mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ of systems of abelian sheaves. Note that $\mathcal{G}_ i$ is an injective sheaf of abelian groups on $\mathcal{C}_ i$, see Lemma 21.15.2 and Homology, Lemma 12.24.3. This finishes the construction. $\square$

Lemma 21.17.3. In the situation of Lemma 21.17.2 set $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i$. Let $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$, $X_ i \in \text{Ob}(\mathcal{C}_ i)$. Then

\[ \mathop{\mathrm{colim}}\nolimits _{a : j \to i} H^ p(u_ a(X_ i), \mathcal{F}_ j) = H^ p(u_ i(X_ i), \mathcal{F}) \]

for all $p \geq 0$.

Proof. The case $p = 0$ is Sites, Lemma 7.18.4.

Choose $(\mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ as in Lemma 21.17.2. Arguing exactly as in the proof of Lemma 21.17.1 we see that it suffices to prove that $H^ p(X, \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{G}_ i) = 0$ for $p > 0$.

Set $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{G}_ i$. To show vanishing of cohomology of $\mathcal{G}$ on every object of $\mathcal{C}$ we show that the Čech cohomology of $\mathcal{G}$ for any covering $\mathcal{U}$ of $\mathcal{C}$ is zero (Lemma 21.11.9). The covering $\mathcal{U}$ comes from a covering $\mathcal{U}_ i$ of $\mathcal{C}_ i$ for some $i$. We have

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{G}) = \mathop{\mathrm{colim}}\nolimits _{a : j \to i} \check{\mathcal{C}}^\bullet (u_ a(\mathcal{U}_ i), \mathcal{G}_ j) \]

by the case $p = 0$. The right hand side is acyclic in positive degrees as a filtered colimit of acyclic complexes by Lemma 21.11.2. See Algebra, Lemma 10.8.8. $\square$


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