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Tag 0739

Chapter 21: Cohomology on Sites > Section 21.16: Cohomology and colimits

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 $\mathcal{B} \subset \mathop{\rm Ob}\nolimits(\mathcal{C})$, 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 $$ \mathop{\rm colim}\nolimits_i H^p(U, \mathcal{F}_i) \longrightarrow H^p(U, \mathop{\rm colim}\nolimits_i \mathcal{F}_i) $$ 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 7.11.2 (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 10.8.9), we see the sequence $$ 0 \to \mathop{\rm colim}\nolimits_i \mathcal{F}_i \to \mathop{\rm colim}\nolimits_i \mathcal{I}_i \to \mathop{\rm colim}\nolimits_i \mathcal{Q}_i \to 0. $$ is also a short exact sequence. We claim that $H^q(U, \mathop{\rm colim}\nolimits_i \mathcal{I}_i) = 0$ for all $U \in \mathcal{B}$ and all $q \geq 1$. Accepting this claim for the moment consider the diagram $$ \xymatrix{ \mathop{\rm colim}\nolimits_i H^p(U, \mathcal{I}_i) \ar[d] \ar[r] & \mathop{\rm colim}\nolimits_i H^p(U, \mathcal{Q}_i) \ar[d] \ar[r] & \mathop{\rm colim}\nolimits_i H^{p + 1}(U, \mathcal{F}_i) \ar[d] \ar[r] & 0 \ar[d] \\ H^p(U, \mathop{\rm colim}\nolimits_i \mathcal{I}_i) \ar[r] & H^p(U, \mathop{\rm colim}\nolimits_i \mathcal{Q}_i) \ar[r] & H^{p + 1}(U, \mathop{\rm colim}\nolimits_i \mathcal{F}_i) \ar[r] & 0 } $$ 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 10.8.9. 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 $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathop{\rm colim}\nolimits_i \mathcal{I}_i) = \mathop{\rm colim}\nolimits_i \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}_i) $$ 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 Cech complexes is acyclic in degree $\geq 1$. Hence by Algebra, Lemma 10.8.9 we see that also the Cech complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathop{\rm colim}\nolimits_i \mathcal{I}_i)$ is acyclic in degrees $\geq 1$. In other words we see that $\check{H}^p(\mathcal{U}, \mathop{\rm colim}\nolimits_i \mathcal{I}_i) = 0$ for all $p \geq 1$. Thus the assumptions of Lemma 21.11.9. are satisfied and the claim follows. $\square$

The code snippet corresponding to this tag is a part of the file sites-cohomology.tex and is located in lines 2413–2437 (see updates for more information).

\begin{lemma}
\label{lemma-colim-works-over-collection}
Let $\mathcal{C}$ be a site. Let $\text{Cov}_\mathcal{C}$ be the set
of coverings of $\mathcal{C}$ (see
Sites, Definition \ref{sites-definition-site}). Let
$\mathcal{B} \subset \Ob(\mathcal{C})$, and
$\text{Cov} \subset \text{Cov}_\mathcal{C}$
be subsets. Assume that
\begin{enumerate}
\item 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
$U_{i_0} \times_U \ldots \times_U U_{i_p} \in \mathcal{B}$.
\item For every $U \in \mathcal{B}$ the coverings of $U$
occurring in $\text{Cov}$ is a cofinal system of coverings of $U$.
\end{enumerate}
Then the map
$$
\colim_i H^p(U, \mathcal{F}_i)
\longrightarrow
H^p(U, \colim_i \mathcal{F}_i)
$$
is an isomorphism for every $p \geq 0$, every $U \in \mathcal{B}$, and
every filtered diagram $\mathcal{I} \to \textit{Ab}(\mathcal{C})$.
\end{lemma}

\begin{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 \ref{sites-lemma-directed-colimits-sections} (4).
Thus we see that the result holds for $p = 0$.
Now we assume the lemma holds for $p$ and prove it for $p + 1$.

\medskip\noindent
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 \ref{injectives-theorem-sheaves-injectives},
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 \ref{algebra-lemma-directed-colimit-exact}),
we see the sequence
$$
0 \to
\colim_i \mathcal{F}_i \to
\colim_i \mathcal{I}_i \to
\colim_i \mathcal{Q}_i \to 0.
$$
is also a short exact sequence. We claim that
$H^q(U, \colim_i \mathcal{I}_i) = 0$ for all $U \in \mathcal{B}$
and all $q \geq 1$. Accepting this claim
for the moment consider the diagram
$$
\xymatrix{
\colim_i H^p(U, \mathcal{I}_i) \ar[d] \ar[r] &
\colim_i H^p(U, \mathcal{Q}_i) \ar[d] \ar[r] &
\colim_i H^{p + 1}(U, \mathcal{F}_i) \ar[d] \ar[r] &
0 \ar[d] \\
H^p(U, \colim_i \mathcal{I}_i) \ar[r] &
H^p(U, \colim_i \mathcal{Q}_i) \ar[r] &
H^{p + 1}(U, \colim_i \mathcal{F}_i) \ar[r] &
0
}
$$
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 \ref{algebra-lemma-directed-colimit-exact}.
Hence by the snake lemma we deduce the
result for $p + 1$.

\medskip\noindent
It remains to show that the claim is true. We will use
Lemma \ref{lemma-cech-vanish-collection}.
By the result for $p = 0$ we see that for $\mathcal{U} \in \text{Cov}$
we have
$$
\check{\mathcal{C}}^\bullet(\mathcal{U}, \colim_i \mathcal{I}_i)
=
\colim_i \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}_i)
$$
because all the $U_{j_0} \times_U \ldots \times_U U_{j_p}$
are in $\mathcal{B}$. By
Lemma \ref{lemma-injective-trivial-cech}
each of the complexes in the colimit of Cech complexes is
acyclic in degree $\geq 1$. Hence by
Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact}
we see that also the Cech complex
$\check{\mathcal{C}}^\bullet(\mathcal{U}, \colim_i \mathcal{I}_i)$
is acyclic in degrees $\geq 1$. In other words we see that
$\check{H}^p(\mathcal{U}, \colim_i \mathcal{I}_i) = 0$
for all $p \geq 1$. Thus the assumptions of
Lemma \ref{lemma-cech-vanish-collection}.
are satisfied and the claim follows.
\end{proof}

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