The Stacks project

Lemma 58.67.2. 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 58.66.12 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 $\mathcal{O}_{X, \bar\eta } = \kappa (\eta )^{sep}$ 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. $\square$


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