The Stacks project

Lemma 20.20.2. In the situation discussed above. Let $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and let $U_ i \subset X_ i$ be quasi-compact open. Then

\[ \mathop{\mathrm{colim}}\nolimits _{a : j \to i} H^ p(f_ a^{-1}(U_ i), \mathcal{F}_ j) = H^ p(p_ i^{-1}(U_ i), \mathcal{F}) \]

for all $p \geq 0$. In particular we have $H^ p(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(X_ i, \mathcal{F}_ i)$.

Proof. The case $p = 0$ is Sheaves, Lemma 6.29.4.

In this paragraph we show that we can find a map of systems $(\gamma _ i) : (\mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ with $\mathcal{G}_ i$ an injective abelian sheaf and $\gamma _ i$ injective. For each $i$ we pick an injection $\mathcal{F}_ i \to \mathcal{I}_ i$ where $\mathcal{I}_ i$ is an injective abelian sheaf on $X_ i$. Then we can consider the family of maps

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

where the component maps are the maps adjoint to the maps $f_ b^{-1}\mathcal{F}_ i \to \mathcal{F}_ k \to \mathcal{I}_ k$. 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 $f_ b^{-1}f_{a \circ b, *}\mathcal{I}_ k \to f_{b, *}\mathcal{I}_ k$ 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 20.12.11 and Homology, Lemma 12.24.3. This finishes the construction.

Arguing exactly as in the proof of Lemma 20.20.1 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 quasi-compact open of $X$, it suffices to show that the Čech cohomology of $\mathcal{G}$ for any covering $\mathcal{U}$ of a quasi-compact open of $X$ by finitely many quasi-compact opens is zero, see Lemma 20.12.9. Such a covering is the inverse by $p_ i$ of such a covering $\mathcal{U}_ i$ on the space $X_ i$ for some $i$ by Topology, Lemma 5.24.6. We have

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{G}) = \mathop{\mathrm{colim}}\nolimits _{a : j \to i} \check{\mathcal{C}}^\bullet (f_ a^{-1}(\mathcal{U}_ i), \mathcal{G}_ j) \]

by the case $p = 0$. The right hand side is a filtered colimit of complexes each of which is acyclic in positive degrees by Lemma 20.12.1. Thus we conclude by Algebra, Lemma 10.8.8. $\square$


Comments (0)


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 0A37. Beware of the difference between the letter 'O' and the digit '0'.