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. $\square$

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