## Tag `09V3`

Chapter 20: Cohomology of Sheaves > Section 20.17: Cohomology on Hausdorff quasi-compact spaces

Lemma 20.17.3. Let $X$ be a topological space. Let $Z \subset X$ be a quasi-compact subset such that any two points of $Z$ have disjoint open neighbourhoods in $X$. For every abelian sheaf $\mathcal{F}$ on $X$ the canonical map $$ \mathop{\rm colim}\nolimits H^p(U, \mathcal{F}) \longrightarrow H^p(Z, \mathcal{F}|_Z) $$ where the colimit is over open neighbourhoods $U$ of $Z$ in $X$ is an isomorphism.

Proof.We first prove this for $p = 0$. Injectivity follows from the definition of $\mathcal{F}|_Z$ and holds in general (for any subset of any topological space $X$). Next, suppose that $s \in H^0(Z, \mathcal{F}|_Z)$. Then we can find opens $U_i \subset X$ such that $Z \subset \bigcup U_i$ and such that $s|_{Z \cap U_i}$ comes from $s_i \in \mathcal{F}(U_i)$. It follows that there exist opens $W_{ij} \subset U_i \cap U_j$ with $W_{ij} \cap Z = U_i \cap U_j \cap Z$ such that $s_i|_{W_{ij}} = s_j|_{W_{ij}}$. Applying Topology, Lemma 5.13.7 we find opens $V_i$ of $X$ such that $V_i \subset U_i$ and such that $V_i \cap V_j \subset W_{ij}$. Hence we see that $s_i|_{V_i}$ glue to a section of $\mathcal{F}$ over the open neighbourhood $\bigcup V_i$ of $Z$.To finish the proof, it suffices to show that if $\mathcal{I}$ is an injective abelian sheaf on $X$, then $H^p(Z, \mathcal{I}|_Z) = 0$ for $p > 0$. This follows using short exact sequences and dimension shifting; details omitted. Thus, suppose $\overline{\xi}$ is an element of $H^p(Z, \mathcal{I}|_Z)$ for some $p > 0$. By Lemma 20.17.2 the element $\overline{\xi}$ comes from $\check{H}^p(\mathcal{V}, \mathcal{I}|_Z)$ for some open covering $\mathcal{V} : Z = \bigcup V_i$ of $Z$. Say $\overline{\xi}$ is the image of the class of a cocycle $\xi = (\xi_{i_0 \ldots i_p})$ in $\check{\mathcal{C}}^p(\mathcal{V}, \mathcal{I}|_Z)$.

Let $\mathcal{I}' \subset \mathcal{I}|_Z$ be the subpresheaf defined by the rule $$ \mathcal{I}'(V) = \{s \in \mathcal{I}|_Z(V) \mid \exists (U, t),~U \subset X\text{ open}, ~t \in \mathcal{I}(U),~V = Z \cap U,~s = t|_{Z \cap U} \} $$ Then $\mathcal{I}|_Z$ is the sheafification of $\mathcal{I}'$. Thus for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can find an open covering $V_{i_0 \ldots i_p} = \bigcup W_{i_0 \ldots i_p, k}$ such that $\xi_{i_0 \ldots i_p}|_{W_{i_0 \ldots i_p, k}}$ is a section of $\mathcal{I}'$. Applying Topology, Lemma 5.13.5 we may after refining $\mathcal{V}$ assume that each $\xi_{i_0 \ldots i_p}$ is a section of the presheaf $\mathcal{I}'$.

Write $V_i = Z \cap U_i$ for some opens $U_i \subset X$. Since $\mathcal{I}$ is flasque (Lemma 20.13.2) and since $\xi_{i_0 \ldots i_p}$ is a section of $\mathcal{I}'$ for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can choose a section $s_{i_0 \ldots i_p} \in \mathcal{I}(U_{i_0 \ldots i_p})$ which restricts to $\xi_{i_0 \ldots i_p}$ on $V_{i_0 \ldots i_p} = Z \cap U_{i_0 \ldots i_p}$. (This appeal to injectives being flasque can be avoided by an additional application of Topology, Lemma 5.13.7.) Let $s = (s_{i_0 \ldots i_p})$ be the corresponding cochain for the open covering $U = \bigcup U_i$. Since $\text{d}(\xi) = 0$ we see that the sections $\text{d}(s)_{i_0 \ldots i_{p + 1}}$ restrict to zero on $Z \cap U_{i_0 \ldots i_{p + 1}}$. Hence, by the initial remarks of the proof, there exists open subsets $W_{i_0 \ldots i_{p + 1}} \subset U_{i_0 \ldots i_{p + 1}}$ with $Z \cap W_{i_0 \ldots i_{p + 1}} = Z \cap U_{i_0 \ldots i_{p + 1}}$ such that $\text{d}(s)_{i_0 \ldots i_{p + 1}}|_{W_{i_0 \ldots i_{p + 1}}} = 0$. By Topology, Lemma 5.13.7 we can find $U'_i \subset U_i$ such that $Z \subset \bigcup U'_i$ and such that $U'_{i_0 \ldots i_{p + 1}} \subset W_{i_0 \ldots i_{p + 1}}$. Then $s' = (s'_{i_0 \ldots i_p})$ with $s'_{i_0 \ldots i_p} = s_{i_0 \ldots i_p}|_{U'_{i_0 \ldots i_p}}$ is a cocycle for $\mathcal{I}$ for the open covering $U' = \bigcup U'_i$ of an open neighbourhood of $Z$. Since $\mathcal{I}$ has trivial higher Čech cohomology groups (Lemma 20.12.1) we conclude that $s'$ is a coboundary. It follows that the image of $\xi$ in the Čech complex for the open covering $Z = \bigcup Z \cap U'_i$ is a coboundary and we are done. $\square$

The code snippet corresponding to this tag is a part of the file `cohomology.tex` and is located in lines 2790–2806 (see updates for more information).

```
\begin{lemma}
\label{lemma-cohomology-of-closed}
\begin{reference}
\cite[Expose V bis, 4.1.3]{SGA4}
\end{reference}
Let $X$ be a topological space. Let $Z \subset X$ be a quasi-compact subset
such that any two points of $Z$ have disjoint open neighbourhoods in $X$.
For every abelian sheaf $\mathcal{F}$ on $X$ the canonical
map
$$
\colim H^p(U, \mathcal{F})
\longrightarrow
H^p(Z, \mathcal{F}|_Z)
$$
where the colimit is over open neighbourhoods $U$ of $Z$ in $X$
is an isomorphism.
\end{lemma}
\begin{proof}
We first prove this for $p = 0$. Injectivity follows from
the definition of $\mathcal{F}|_Z$ and holds in general
(for any subset of any topological space $X$). Next, suppose that
$s \in H^0(Z, \mathcal{F}|_Z)$. Then we can find opens $U_i \subset X$
such that $Z \subset \bigcup U_i$ and such that $s|_{Z \cap U_i}$
comes from $s_i \in \mathcal{F}(U_i)$. It follows that
there exist opens $W_{ij} \subset U_i \cap U_j$ with
$W_{ij} \cap Z = U_i \cap U_j \cap Z$ such that
$s_i|_{W_{ij}} = s_j|_{W_{ij}}$. Applying
Topology, Lemma
\ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset}
we find opens $V_i$ of $X$ such that $V_i \subset U_i$ and
such that $V_i \cap V_j \subset W_{ij}$. Hence we see that
$s_i|_{V_i}$ glue to a section of $\mathcal{F}$ over the
open neighbourhood $\bigcup V_i$ of $Z$.
\medskip\noindent
To finish the proof, it suffices to show that if $\mathcal{I}$ is an
injective abelian sheaf on $X$, then $H^p(Z, \mathcal{I}|_Z) = 0$
for $p > 0$. This follows using short exact sequences and dimension
shifting; details omitted. Thus, suppose $\overline{\xi}$ is an element
of $H^p(Z, \mathcal{I}|_Z)$ for some $p > 0$.
By Lemma \ref{lemma-cech-Hausdorff-quasi-compact}
the element $\overline{\xi}$ comes from
$\check{H}^p(\mathcal{V}, \mathcal{I}|_Z)$
for some open covering $\mathcal{V} : Z = \bigcup V_i$ of $Z$.
Say $\overline{\xi}$ is the image of the class of a cocycle
$\xi = (\xi_{i_0 \ldots i_p})$ in
$\check{\mathcal{C}}^p(\mathcal{V}, \mathcal{I}|_Z)$.
\medskip\noindent
Let $\mathcal{I}' \subset \mathcal{I}|_Z$ be the subpresheaf
defined by the rule
$$
\mathcal{I}'(V) =
\{s \in \mathcal{I}|_Z(V) \mid
\exists (U, t),\ U \subset X\text{ open},
\ t \in \mathcal{I}(U),\ V = Z \cap U,\ s = t|_{Z \cap U} \}
$$
Then $\mathcal{I}|_Z$ is the sheafification of $\mathcal{I}'$.
Thus for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can find an
open covering $V_{i_0 \ldots i_p} = \bigcup W_{i_0 \ldots i_p, k}$
such that $\xi_{i_0 \ldots i_p}|_{W_{i_0 \ldots i_p, k}}$ is
a section of $\mathcal{I}'$. Applying
Topology, Lemma \ref{topology-lemma-refine-covering}
we may after refining $\mathcal{V}$ assume that each
$\xi_{i_0 \ldots i_p}$ is a section of the presheaf $\mathcal{I}'$.
\medskip\noindent
Write $V_i = Z \cap U_i$ for some opens $U_i \subset X$.
Since $\mathcal{I}$ is flasque (Lemma \ref{lemma-injective-flasque})
and since $\xi_{i_0 \ldots i_p}$ is a section of $\mathcal{I}'$
for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can choose
a section $s_{i_0 \ldots i_p} \in \mathcal{I}(U_{i_0 \ldots i_p})$
which restricts to $\xi_{i_0 \ldots i_p}$ on
$V_{i_0 \ldots i_p} = Z \cap U_{i_0 \ldots i_p}$.
(This appeal to injectives being flasque can be avoided by an
additional application of
Topology, Lemma
\ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset}.)
Let $s = (s_{i_0 \ldots i_p})$ be the corresponding cochain
for the open covering $U = \bigcup U_i$.
Since $\text{d}(\xi) = 0$ we see that the sections
$\text{d}(s)_{i_0 \ldots i_{p + 1}}$ restrict to zero
on $Z \cap U_{i_0 \ldots i_{p + 1}}$. Hence, by the initial
remarks of the proof, there exists open subsets
$W_{i_0 \ldots i_{p + 1}} \subset U_{i_0 \ldots i_{p + 1}}$
with $Z \cap W_{i_0 \ldots i_{p + 1}} = Z \cap U_{i_0 \ldots i_{p + 1}}$
such that $\text{d}(s)_{i_0 \ldots i_{p + 1}}|_{W_{i_0 \ldots i_{p + 1}}} = 0$.
By Topology, Lemma
\ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset}
we can find $U'_i \subset U_i$ such that $Z \subset \bigcup U'_i$
and such that $U'_{i_0 \ldots i_{p + 1}} \subset W_{i_0 \ldots i_{p + 1}}$.
Then $s' = (s'_{i_0 \ldots i_p})$ with
$s'_{i_0 \ldots i_p} = s_{i_0 \ldots i_p}|_{U'_{i_0 \ldots i_p}}$
is a cocycle for $\mathcal{I}$ for the open covering
$U' = \bigcup U'_i$ of an open neighbourhood of $Z$.
Since $\mathcal{I}$ has trivial higher {\v C}ech cohomology groups
(Lemma \ref{lemma-injective-trivial-cech})
we conclude that $s'$ is a coboundary. It follows that the image of
$\xi$ in the {\v C}ech complex for the open covering
$Z = \bigcup Z \cap U'_i$ is a coboundary and we are done.
\end{proof}
```

## References

[SGA4, Expose V bis, 4.1.3]

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