The Stacks project

59.68 Higher vanishing for the multiplicative group

In this section, we fix an algebraically closed field $k$ and a smooth curve $X$ over $k$. We denote $i_ x : x \hookrightarrow X$ the inclusion of a closed point of $X$ and $j : \eta \hookrightarrow X$ the inclusion of the generic point. We also denote $X_0$ the set of closed points of $X$.

Theorem 59.68.1 (The Fundamental Exact Sequence). There is a short exact sequence of étale sheaves on $X$

\[ 0 \longrightarrow \mathbf{G}_{m, X} \longrightarrow j_* \mathbf{G}_{m, \eta } \longrightarrow \bigoplus \nolimits _{x \in X_0} {i_ x}_* \underline{\mathbf{Z}} \longrightarrow 0. \]

Proof. Let $\varphi : U \to X$ be an étale morphism. Then by properties of étale morphisms (Proposition 59.26.2), $U = \coprod _ i U_ i$ where each $U_ i$ is a smooth curve mapping to $X$. The above sequence for $U$ is a product of the corresponding sequences for each $U_ i$, so it suffices to treat the case where $U$ is connected, hence irreducible. In this case, there is a well known exact sequence

\[ 1 \longrightarrow \Gamma (U, \mathcal{O}_ U^*) \longrightarrow k(U)^* \longrightarrow \bigoplus \nolimits _{y \in U_0} \mathbf{Z}_ y. \]

This amounts to a sequence

\[ 0 \longrightarrow \Gamma (U, \mathcal{O}_ U^*) \longrightarrow \Gamma (\eta \times _ X U, \mathcal{O}_{\eta \times _ X U}^*) \longrightarrow \bigoplus \nolimits _{x \in X_0} \Gamma (x \times _ X U, \underline{\mathbf{Z}}) \]

which, unfolding definitions, is nothing but a sequence

\[ 0 \longrightarrow \mathbf{G}_ m(U) \longrightarrow j_* \mathbf{G}_{m, \eta }(U) \longrightarrow \left(\bigoplus \nolimits _{x \in X_0} {i_ x}_* \underline{\mathbf{Z}}\right)(U). \]

This defines the maps in the Fundamental Exact Sequence and shows it is exact except possibly at the last step. To see surjectivity, let us recall that if $U$ is a nonsingular curve and $D$ is a divisor on $U$, then there exists a Zariski open covering $\{ U_ j \to U\} $ of $U$ such that $D|_{U_ j} = \text{div}(f_ j)$ for some $f_ j \in k(U)^*$. $\square$

Lemma 59.68.2. For any $q \geq 1$, $R^ q j_*\mathbf{G}_{m, \eta } = 0$.

Proof. We need to show that $(R^ q j_*\mathbf{G}_{m, \eta })_{\bar x} = 0$ for every geometric point $\bar x$ of $X$.

Assume that $\bar x$ lies over a closed point $x$ of $X$. Let $\mathop{\mathrm{Spec}}(A)$ be an affine open neighbourhood of $x$ in $X$, and $K$ the fraction field of $A$. Then

\[ \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \bar x}) \times _ X \eta = \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \bar x} \otimes _ A K). \]

The ring $\mathcal{O}^{sh}_{X, \bar x} \otimes _ A K$ is a localization of the discrete valuation ring $\mathcal{O}^{sh}_{X, \bar x}$, so it is either $\mathcal{O}^{sh}_{X, \bar x}$ again, or its fraction field $K^{sh}_{\bar x}$. But since some local uniformizer gets inverted, it must be the latter. Hence

\[ (R^ q j_*\mathbf{G}_{m, \eta })_{(X, \bar x)} = H_{\acute{e}tale}^ q(\mathop{\mathrm{Spec}}K^{sh}_{\bar x}, \mathbf{G}_ m). \]

Now recall that $\mathcal{O}^{sh}_{X, \bar x} = \mathop{\mathrm{colim}}\nolimits _{(U, \bar u) \to \bar x} \mathcal{O}(U) = \mathop{\mathrm{colim}}\nolimits _{A \subset B} B$ where $A \to B$ is étale, hence $K^{sh}_{\bar x}$ is an algebraic extension of $K = k(X)$, and we may apply Lemma 59.67.12 to get the vanishing.

Assume that $\bar x = \bar\eta $ lies over the generic point $\eta $ of $X$ (in fact, this case is superfluous). Then $\mathcal{O}^{sh}_{X, \bar\eta } = \kappa (\eta )^{sep}$ and thus

\begin{eqnarray*} (R^ q j_*\mathbf{G}_{m, \eta })_{\bar\eta } & = & H_{\acute{e}tale}^ q(\mathop{\mathrm{Spec}}(\kappa (\eta )^{sep}) \times _ X \eta , \mathbf{G}_ m) \\ & = & H_{\acute{e}tale}^ q (\mathop{\mathrm{Spec}}(\kappa (\eta )^{sep}), \mathbf{G}_ m) \\ & = & 0 \ \ \text{ for } q \geq 1 \end{eqnarray*}

since the corresponding Galois group is trivial. $\square$

Lemma 59.68.3. For all $p \geq 1$, $H_{\acute{e}tale}^ p(X, j_*\mathbf{G}_{m, \eta }) = 0$.

Proof. The Leray spectral sequence reads

\[ E_2^{p, q} = H_{\acute{e}tale}^ p(X, R^ qj_*\mathbf{G}_{m, \eta }) \Rightarrow H_{\acute{e}tale}^{p+q}(\eta , \mathbf{G}_{m, \eta }), \]

which vanishes for $p+q \geq 1$ by Lemma 59.67.12. Taking $q = 0$, we get the desired vanishing. $\square$

Lemma 59.68.4. For all $q \geq 1$, $H_{\acute{e}tale}^ q(X, \bigoplus _{x \in X_0} {i_ x}_* \underline{\mathbf{Z}}) = 0$.

Proof. For $X$ quasi-compact and quasi-separated, cohomology commutes with colimits, so it suffices to show the vanishing of $H_{\acute{e}tale}^ q(X, {i_ x}_* \underline{\mathbf{Z}})$. But then the inclusion $i_ x$ of a closed point is finite so $R^ p {i_ x}_* \underline{\mathbf{Z}} = 0$ for all $p \geq 1$ by Proposition 59.55.2. Applying the Leray spectral sequence, we see that $H_{\acute{e}tale}^ q(X, {i_ x}_* \underline{\mathbf{Z}}) = H_{\acute{e}tale}^ q(x, \underline{\mathbf{Z}})$. Finally, since $x$ is the spectrum of an algebraically closed field, all higher cohomology on $x$ vanishes. $\square$

Concluding this series of lemmata, we get the following result.

Theorem 59.68.5. Let $X$ be a smooth curve over an algebraically closed field. Then

\[ H_{\acute{e}tale}^ q(X, \mathbf{G}_ m) = 0 \ \ \text{ for all } q \geq 2. \]

Proof. See discussion above. $\square$

We also get the cohomology long exact sequence

\[ 0 \to H_{\acute{e}tale}^0(X, \mathbf{G}_ m) \to H_{\acute{e}tale}^0(X, j_*\mathbf{G}_{m\eta }) \to H_{\acute{e}tale}^0(X, \bigoplus {i_ x}_*\underline{\mathbf{Z}}) \to H_{\acute{e}tale}^1(X, \mathbf{G}_ m) \to 0 \]

although this is the familiar

\[ 0 \to H_{Zar}^0(X, \mathcal{O}_ X^*) \to k(X)^* \to \text{Div}(X) \to \mathop{\mathrm{Pic}}\nolimits (X) \to 0. \]

Comments (0)


Post a comment

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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03RH. Beware of the difference between the letter 'O' and the digit '0'.