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. $\square$

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