Lemma 59.95.3. 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 59.83.10. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: