The Stacks project

Lemma 53.17.1. Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$.

  1. If $n \geq 1$ is invertible in $k$, then $\mathop{\mathrm{Pic}}\nolimits (X)[n] \cong (\mathbf{Z}/n\mathbf{Z})^{\oplus 2g}$.

  2. If the characteristic of $k$ is $p > 0$, then there exists an integer $0 \leq f \leq g$ such that $\mathop{\mathrm{Pic}}\nolimits (X)[p^ m] \cong (\mathbf{Z}/p^ m\mathbf{Z})^{\oplus f}$ for all $m \geq 1$.

Proof. Let $\mathop{\mathrm{Pic}}\nolimits ^0(X) \subset \mathop{\mathrm{Pic}}\nolimits (X)$ denote the subgroup of invertible sheaves of degree $0$. In other words, there is a short exact sequence

\[ 0 \to \mathop{\mathrm{Pic}}\nolimits ^0(X) \to \mathop{\mathrm{Pic}}\nolimits (X) \xrightarrow {\deg } \mathbf{Z} \to 0. \]

The group $\mathop{\mathrm{Pic}}\nolimits ^0(X)$ is the $k$-points of the group scheme $\underline{\mathrm{Pic}}^0_{X/k}$, see Picard Schemes of Curves, Lemma 44.6.7. The same lemma tells us that $\underline{\mathrm{Pic}}^0_{X/k}$ is a $g$-dimensional abelian variety over $k$ as defined in Groupoids, Definition 39.9.1. Thus we conclude by the results of Groupoids, Proposition 39.9.11. $\square$

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