The Stacks project

Lemma 57.89.3. Let $K$ be a field. Let $X$ be a $1$-dimensional affine scheme of finite type over $K$. Then $\text{cd}(X) \leq 1 + \text{cd}(K)$.

Proof. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$. Consider the Leray spectral sequence for the morphism $f : X \to \mathop{\mathrm{Spec}}(K)$. We obtain

\[ E_2^{p, q} = H^ p(\mathop{\mathrm{Spec}}(K), R^ qf_*\mathcal{F}) \]

converging to $H^{p + q}_{\acute{e}tale}(X, \mathcal{F})$. The stalk of $R^ qf_*\mathcal{F}$ at a geometric point $\mathop{\mathrm{Spec}}(\overline{K}) \to \mathop{\mathrm{Spec}}(K)$ is the cohomology of the pullback of $\mathcal{F}$ to $X_{\overline{K}}$. Hence it vanishes in degrees $\geq 2$ by Theorem 57.80.9. $\square$


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Your email address will not be published.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 0F0S. Beware of the difference between the letter 'O' and the digit '0'.