The Stacks project

21.16 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 7.17.7 for cohomology.

Lemma 21.16.1. 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 7.17.7 (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 21.10.9. 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 21.10.2 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 21.10.9. are satisfied and the claim follows. $\square$

Lemma 21.16.2. Let $\mathcal{C}$ be a site. Let $S \subset \mathop{\mathrm{Ob}}\nolimits (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}))$ be a subset. Denote $*$ the final object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Assume

  1. for some $K \in S$ the map $K \to *$ is surjective,

  2. given a surjective map of sheaves $\mathcal{F} \to K$ with $K \in S$ there exists a $K' \in S$ and a map $K' \to \mathcal{F}$ such that the composition $K' \to K$ is surjective,

  3. given $K, K' \in S$ there is a surjection $K'' \to K \times K'$ with $K'' \in S$,

  4. given $a, b : K \to K'$ with $K, K' \in S$ there exists a surjection $K'' \to \text{Equalizer}(a, b)$ with $K'' \in S$, and

  5. every $K \in S$ is quasi-compact (Sites, Definition 7.17.4).

Then for all $p \geq 0$ the map

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

is an isomorphism for every filtered diagram $\Lambda \to \textit{Ab}(\mathcal{C})$, $\lambda \mapsto \mathcal{F}_\lambda $.

Proof. We will prove this by induction on $p$. The base case $p = 0$ follows from Sites, Lemma 7.17.8 part (4). We check the assumptions hold, but we urge the reader to skip this part. Suppose $\mathcal{F} \to *$ is surjective. Choose $K \in S$ and $K \to *$ surjective as in (1). Then $\mathcal{F} \times K \to K$ is surjective. Choose $K' \to \mathcal{F} \times K$ with $K' \in S$ and $K' \to K$ surjective as in (2). Then there is a map $K' \to \mathcal{F}$ and $K' \to *$ is surjective. Hence Sites, Lemma 7.17.8 assumption (4)(a) is satisfied. By Sites, Lemma 7.17.5, assumptions (3) and (5) we see that $K \times K$ is quasi-compact for all $K \in S$. Hence Sites, Lemma 7.17.8 assumption (4)(b) is satisfied. This finishes the proof of the base case.

Induction step. Assume the result holds for $H^ p$ for $p \leq p_0$ and for all topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ such that a set $S \subset \mathop{\mathrm{Ob}}\nolimits (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}))$ can be found satisfying (1) – (5). Arguing exactly as in the proof of Lemma 21.16.1 we see that it suffices to show: given a filtered colimit $\mathcal{I} = \mathop{\mathrm{colim}}\nolimits \mathcal{I}_\lambda $ with $\mathcal{I}_\lambda $ injective abelian sheaves, we have $H^{p_0 + 1}(\mathcal{C}, \mathcal{I}) = 0$. Choose $K \to *$ surjective with $K \in S$ as in (1). Denote $K^ n$ the $n$-fold self product of $K$. Consider the spectral sequence

\[ E_1^{p, q} = H^ q(K^{p + 1}, \mathcal{I}) \Rightarrow H^{p + q}(*, \mathcal{I}) = H^{p + q}(\mathcal{C}, \mathcal{I}) \]

of Lemma 21.13.2. Recall that $H^ q(K^{p + 1}, \mathcal{F}) = H^ q(\mathcal{C}/K^{p + 1}, j^{-1}\mathcal{F})$, for any abelian sheaf on $\mathcal{C}$, see Lemma 21.13.3. We have $j^{-1}\mathcal{I} = \mathop{\mathrm{colim}}\nolimits j^{-1}\mathcal{I}_\lambda $ as $j^{-1}$ commutes with colimits. The restrictions $j^{-1}\mathcal{I}_\lambda $ are injective abelian sheaves on $\mathcal{C}/K^{p + 1}$ by Lemma 21.7.1. Below we will show that the induction hypothesis applies to $\mathcal{C}/K^{p + 1}$ and hence we see that $H^ q(K^{p + 1}, \mathcal{I}) = \mathop{\mathrm{colim}}\nolimits H^ q(K^{p + 1}, \mathcal{I}_\lambda ) = 0$ for $q < p_0 + 1$ (vanishing as $\mathcal{I}_\lambda $ is injective). It follows that

\[ H^{p_0 + 1}(\mathcal{C}, \mathcal{I}) = H^{p_0 + 1}\left(\ldots \to H^0(K^{p_0}, \mathcal{I}) \to H^0(K^{p_0 + 1}, \mathcal{I}) \to H^0(K^{p_0 + 2}, \mathcal{I}) \to \ldots \right) \]

Again using the induction hypothesis, the complex depicted on the right hand side is the colimit over $\Lambda $ of the complexes

\[ \ldots \to H^0(K^{p_0}, \mathcal{I}_\lambda ) \to H^0(K^{p_0 + 1}, \mathcal{I}_\lambda ) \to H^0(K^{p_0 + 2}, \mathcal{I}_\lambda ) \to \ldots \]

These complexes are exact as $\mathcal{I}_\lambda $ is an injective abelian sheaf (follows from the spectral sequence for example). Since filtered colimits are exact in the category of abelian groups we obtain the desired vanishing.

We still have to show that the induction hypothesis applies to the site $\mathcal{C}/K^ n$ for all $n \geq 1$. Recall that $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K^ n) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/K^ n$, see Sites, Lemma 7.30.3. Thus we may work in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/K^ n$. Denote $S_ n \subset \mathop{\mathrm{Ob}}\nolimits (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K^ n)$ the set of objects of the form $K' \to K^ n$. We check each property in turn:

  1. By (3) and induction there exists a surjection $K' \to K^ n$ with $K' \in S$. Then $(K' \to K^ n) \to (K^ n \to K^ n)$ is a surjection in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/K^ n$ and $K^ n \to K^ n$ is the final object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/K^ n$. Hence (1) holds for $S_ n$,

  2. Property (2) for $S_ n$ is an immediate consquence of (2) for $S$.

  3. Let $a : K_1 \to K^ n$ and $b : K_2 \to K^ n$ be in $S_ n$. Then $(K_1 \to K^ n) \times (K_2 \to K^ n)$ is the object $K_1 \times _{K^ n} K_2 \to K^ n$ of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/K^ n$. The subsheaf $K_1 \times _{K^ n} K_2 \subset K_1 \times K_2$ is the equalizer of $a \circ \text{pr}_1$ and $b \circ \text{pr}_2$. Write $a = (a_1, \ldots , a_ n)$ and $b = (b_1, \ldots , b_ n)$. Pick $K_3 \to K_1 \times K_2$ surjective with $K_3 \in S$; this is possibly by assumption (3) for $\mathcal{C}$. Pick

    \[ K_4 \longrightarrow \text{Equalizer}(K_3 \to K_1 \times K_2 \xrightarrow {a_1, b_1} K) \]

    surjective with $K_4 \in S$. This is possible by assumption (4) for $\mathcal{C}$. Pick

    \[ K_5 \longrightarrow \text{Equalizer}(K_4 \to K_1 \times K_2 \xrightarrow {a_2, b_2} K) \]

    surjective with $K_5 \in S$. Again this is possible. Continue in this fashion until we get

    \[ K_{3 + n} \longrightarrow \text{Equalizer}(K_{2 + n} \to K_1 \times K_2 \xrightarrow {a_ n, b_ n} K) \]

    surjective with $K_{3 + n} \in S$. By construction $K_{3 + n} \to K_1 \times _{K^ n} K_2$ is surjective. Hence $(K_{3 + n} \to K^ n)$ is in $S_ n$ and surjects onto the product $(K_1 \to K^ n) \times (K_2 \to K^ n)$. Thus (3) holds for $S_ n$.

  4. Property (4) for $S_ n$ is an immediate consequence of property (4) for $S$.

  5. Property (5) for $S_ n$ is a consequence of property (5) for $S$. Namely, an object $\mathcal{F} \to K^ n$ of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/K^ n$ corresponds to a quasi-compact object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K^ n)$ if and only if $\mathcal{F}$ is a quasi-compact object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

This finishes the proof of the lemma. $\square$

Remark 21.16.3. Let $\mathcal{C}$ be a site. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Let $S \subset \mathop{\mathrm{Ob}}\nolimits (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}))$ be the set of sheaves $K$ which have the form

\[ K = \coprod \nolimits _{i = 1, \ldots , n} h_{U_ i}^\# \]

with $U_1, \ldots , U_ n \in \mathcal{B}$. Then we can ask: when does this set satisfy the assumptions of Lemma 21.16.2? One answer is that it suffices if

  1. for some $n \geq 0$, $U_1, \ldots , U_ n \in \mathcal{B}$ the map $\coprod _{i = 1, \ldots , n} h_{U_ i}^\# \to *$ is surjective,

  2. every covering of $U \in \mathcal{B}$ can be refined by a covering of the form $\{ U_ i \to U\} _{i = 1, \ldots , n}$ with $U_ i \in \mathcal{B}$,

  3. given $U, U' \in \mathcal{B}$ there exist $n \geq 0$, $U_1, \ldots , U_ n \in \mathcal{B}$, maps $U_ i \to U$ and $U_ i \to U'$ such that $\coprod _{i = 1, \ldots , n} h_{U_ i}^\# \to h_ U^\# \times h_{U'}^\# $ is surjective,

  4. given morphisms $a, b : U \to U'$ in $\mathcal{C}$ with $U, U' \in \mathcal{B}$, there exist $U_1, \ldots , U_ n \in \mathcal{B}$, maps $U_ i \to U$ equalizing $a, b$ such that $\coprod _{i = 1, \ldots , n} h_{U_ i}^\# \to \text{Equalizer}(h_ a^\# , h_ b^\# : h_ U^\# \to h_{U'}^\# )$ is surjective.

We omit the detailed verification, except to mention that part (2) above insures that every element of $\mathcal{B}$ is quasi-compact and hence every $K \in S$ is quasi-compact as well by Sites, Lemma 7.17.6.

Lemma 21.16.4. 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 21.14.2 and Homology, Lemma 12.27.3. This finishes the construction. $\square$

Lemma 21.16.5. In the situation of Lemma 21.16.4 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 21.16.4. Arguing exactly as in the proof of Lemma 21.16.1 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 21.10.9). 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 21.10.2. See Algebra, Lemma 10.8.8. $\square$

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