The Stacks project

Lemma 59.51.5. Let $A$ be a ring, $(I, \leq )$ a directed set and $(B_ i, \varphi _{ij})$ a system of $A$-algebras. Set $B = \mathop{\mathrm{colim}}\nolimits _{i\in I} B_ i$. Let $X \to \mathop{\mathrm{Spec}}(A)$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}$ an abelian sheaf on $X_{\acute{e}tale}$. Denote $Y_ i = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(B_ i)$, $Y = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(B)$, $\mathcal{G}_ i = (Y_ i \to X)^{-1}\mathcal{F}$ and $\mathcal{G} = (Y \to X)^{-1}\mathcal{F}$. Then

\[ H_{\acute{e}tale}^ p(Y, \mathcal{G}) = \mathop{\mathrm{colim}}\nolimits _{i\in I} H_{\acute{e}tale}^ p (Y_ i, \mathcal{G}_ i). \]

Proof. This is a special case of Theorem 59.51.3. We also outline a direct proof as follows.

  1. Given $V \to Y$ étale with $V$ quasi-compact and quasi-separated, there exist $i\in I$ and $V_ i \to Y_ i$ such that $V = V_ i \times _{Y_ i} Y$. If all the schemes considered were affine, this would correspond to the following algebra statement: if $B = \mathop{\mathrm{colim}}\nolimits B_ i$ and $B \to C$ is étale, then there exist $i \in I$ and $B_ i \to C_ i$ étale such that $C \cong B \otimes _{B_ i} C_ i$. This is proved in Algebra, Lemma 10.143.3.

  2. In the situation of (1) show that $\mathcal{G}(V) = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \mathcal{G}_{i'}(V_{i'})$ where $V_{i'}$ is the base change of $V_ i$ to $Y_{i'}$.

  3. By (1), we see that for every étale covering $\mathcal{V} = \{ V_ j \to Y\} _{j\in J}$ with $J$ finite and the $V_ j$s quasi-compact and quasi-separated, there exists $i \in I$ and an étale covering $\mathcal{V}_ i = \{ V_{ij} \to Y_ i\} _{j \in J}$ such that $\mathcal{V} \cong \mathcal{V}_ i \times _{Y_ i} Y$.

  4. Show that (2) and (3) imply

    \[ \check H^*(\mathcal{V}, \mathcal{G})= \mathop{\mathrm{colim}}\nolimits _{i\in I} \check H^*(\mathcal{V}_ i, \mathcal{G}_ i). \]
  5. Cleverly use the Čech-to-cohomology spectral sequence (Theorem 59.19.2).


Comments (2)

Comment #2355 by Yu-Liang Huang on

In the last sentence of the lemma, the right side should be , rather than .

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Your email address will not be published.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 03Q6. Beware of the difference between the letter 'O' and the digit '0'.