Lemma 59.69.1. Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$ and let $n \geq 1$ be invertible in $k$. Then there are canonical identifications
\[ H_{\acute{e}tale}^ q(X, \mu _ n) = \left\{ \begin{matrix} \mu _ n(k)
& \text{ if }q = 0,
\\ \mathop{\mathrm{Pic}}\nolimits ^0(X)[n]
& \text{ if }q = 1,
\\ \mathbf{Z}/n\mathbf{Z}
& \text{ if }q = 2,
\\ 0
& \text{ if }q \geq 3.
\end{matrix} \right. \]
Since $\mu _ n \cong \underline{\mathbf{Z}/n\mathbf{Z}}$, this gives (noncanonical) identifications
\[ H_{\acute{e}tale}^ q(X, \underline{\mathbf{Z}/n\mathbf{Z}}) \cong \left\{ \begin{matrix} \mathbf{Z}/n\mathbf{Z}
& \text{ if }q = 0,
\\ (\mathbf{Z}/n\mathbf{Z})^{2g}
& \text{ if }q = 1,
\\ \mathbf{Z}/n\mathbf{Z}
& \text{ if }q = 2,
\\ 0
& \text{ if }q \geq 3.
\end{matrix} \right. \]
Proof.
Theorems 59.24.1 and 59.68.5 determine the étale cohomology of $\mathbf{G}_ m$ on $X$ in terms of the Picard group of $X$. The Kummer sequence $0\to \mu _{n, X} \to \mathbf{G}_{m, X} \to \mathbf{G}_{m, X}\to 0$ (Lemma 59.28.1) then gives us the long exact cohomology sequence
\[ \xymatrix{ 0 \ar[r] & \mu _ n(k) \ar[r] & k^* \ar[r]^{(\cdot )^ n} & k^* \ar@(rd, ul)[rdllllr] \\ & H_{\acute{e}tale}^1(X, \mu _ n) \ar[r] & \mathop{\mathrm{Pic}}\nolimits (X) \ar[r]^{(\cdot )^ n} & \mathop{\mathrm{Pic}}\nolimits (X) \ar@(rd, ul)[rdllllr] \\ & H_{\acute{e}tale}^2(X, \mu _ n) \ar[r] & 0 \ar[r] & 0 \ldots } \]
The $n$th power map $k^* \to k^*$ is surjective since $k$ is algebraically closed. So we need to compute the kernel and cokernel of the map $n : \mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (X)$. Consider the commutative diagram with exact rows
\[ \xymatrix{ 0 \ar[r] & \mathop{\mathrm{Pic}}\nolimits ^0(X) \ar[r] \ar@{>>}[d]^{(\cdot )^ n} & \mathop{\mathrm{Pic}}\nolimits (X) \ar[r]_-\deg \ar[d]^{(\cdot )^ n} & \mathbf{Z} \ar[r] \ar@{^{(}->}[d]^ n & 0 \\ 0 \ar[r] & \mathop{\mathrm{Pic}}\nolimits ^0(X) \ar[r] & \mathop{\mathrm{Pic}}\nolimits (X) \ar[r]^-\deg & \mathbf{Z} \ar[r] & 0 } \]
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. Hence the left vertical map is surjective by Groupoids, Proposition 39.9.11. Applying the snake lemma gives canonical identifications as stated in the lemma.
To get the noncanonical identifications of the lemma we need to show the kernel of $n : \mathop{\mathrm{Pic}}\nolimits ^0(X) \to \mathop{\mathrm{Pic}}\nolimits ^0(X)$ is isomorphic to $(\mathbf{Z}/n\mathbf{Z})^{\oplus 2g}$. This is also part of Groupoids, Proposition 39.9.11.
$\square$
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