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

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

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. $\square$

Your email address will not be published.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.

## Comments (0)