The Stacks project

Lemma 53.18.6. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $\ell $ be a prime number invertible in $k$. Then

\[ \dim _{\mathbf{F}_\ell } \mathop{\mathrm{Pic}}\nolimits (X)[\ell ] \leq \dim _ k H^1(X, \mathcal{O}_ X) + g_{geom}(X/k) \]

where $g_{geom}(X/k)$ is as defined above.

Proof. The map $\mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (X_{\overline{k}})$ is injective by Varieties, Lemma 33.30.3. By Cohomology of Schemes, Lemma 30.5.2 $\dim _ k H^1(X, \mathcal{O}_ X)$ equals $\dim _{\overline{k}} H^1(X_{\overline{k}}, \mathcal{O}_{X_{\overline{k}}})$. Hence we may assume $k$ is algebraically closed.

Let $X_{red}$ be the reduction of $X$. Then the surjection $\mathcal{O}_ X \to \mathcal{O}_{X_{red}}$ induces a surjection $H^1(X, \mathcal{O}_ X) \to H^1(X, \mathcal{O}_{X_{red}})$ because cohomology of quasi-coherent sheaves vanishes in degrees $\geq 2$ by Cohomology, Proposition 20.20.7. Since $X_{red} \to X$ induces an isomorphism on irreducible components over $\overline{k}$ and an isomorphism on $\ell $-torsion in Picard groups (Picard Schemes of Curves, Lemma 44.7.2) we may replace $X$ by $X_{red}$. In this way we reduce to Proposition 53.17.3. $\square$

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