The Stacks project

Lemma 53.18.3. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $f : Y \to X$ be a finite morphism such that there exists a dense open $U \subset X$ over which $f$ is a closed immersion. Then

\[ \dim _ k H^1(X, \mathcal{O}_ X) \geq \dim _ k H^1(Y, \mathcal{O}_ Y) \]

Proof. Consider the exact sequence

\[ 0 \to \mathcal{G} \to \mathcal{O}_ X \to f_*\mathcal{O}_ Y \to \mathcal{F} \to 0 \]

of coherent sheaves on $X$. By assumption $\mathcal{F}$ is supported in finitely many closed points and hence has vanishing higher cohomology (Varieties, Lemma 33.33.3). On the other hand, we have $H^2(X, \mathcal{G}) = 0$ by Cohomology, Proposition 20.20.7. It follows formally that the induced map $H^1(X, \mathcal{O}_ X) \to H^1(X, f_*\mathcal{O}_ Y)$ is surjective. Since $H^1(X, f_*\mathcal{O}_ Y) = H^1(Y, \mathcal{O}_ Y)$ (Cohomology of Schemes, Lemma 30.2.4) we conclude the lemma holds. $\square$

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