Lemma 64.3.2. Let $X$ be a scheme and $g : X \to X$ a morphism. Assume that for all $\varphi : U \to X$ étale, there is an isomorphism

\[ \xymatrix{ U \ar[rd]_\varphi \ar[rr]^-\sim & & {U \times _{\varphi , X, g} X} \ar[ld]^{\text{pr}_2} \\ & X } \]

functorial in $U$. Then $g$ induces the identity on cohomology (for any sheaf).

## Comments (2)

Comment #5116 by Laurent Moret-Bailly on

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