Lemma 59.68.3 (03RK)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.68: Higher vanishing for the multiplicative group
Lemma 59.68.3 (cite)

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

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

The right hand side vanishes for $p + q \geq 1$ by Lemma 59.67.12. The left hand side vanishes for $q \geq 1$ by Lemma 59.68.2. Thus we get the desired vanishing. $\square$

Comments (2)

Comment #10717 by Manolis C. Tsakiris on August 16, 2025 at 23:55

We also need that $R^q j_*G_{m,\eta}=0$ for $q \ge 1$ (Lemma 03RJ).

Comment #10792 by Stacks project on September 15, 2025 at 13:41

Thanks very much! Fixed here.

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