The Stacks project

[Expose V bis, 4.1.3, SGA4]

Lemma 20.16.3. 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

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

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 20.16.2 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 20.12.2) 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 20.11.1) 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. $\square$


Comments (0)

There are also:

  • 3 comment(s) on Section 20.16: Cohomology on Hausdorff quasi-compact spaces

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09V3. Beware of the difference between the letter 'O' and the digit '0'.