Section 59.68 (03RH): Higher vanishing for the multiplicative group

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

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

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59.68 Higher vanishing for the multiplicative group

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