The Stacks project

Lemma 64.3.2. Let $X$ be a scheme and $g : X \to X$ a morphism. Assume that for all $\varphi : U \to X$ ├ętale, 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).

Proof. The proof is formal and without difficulty. $\square$

Comments (2)

Comment #5116 by Laurent Moret-Bailly on

The meaning of the statement is not completely formal. To make sense of "the identity on cohomology" we need to show that we can identify with and/or , for any . This is of course the case in subsequent lemmas where is a constant sheaf.

Comment #5323 by on

Hmm... yes. OK, well, I think Exercise 64.3.1 about topological spaces just above the lemma explains why this would be so. To be precise which is functorially equal to by the assumption of the lemma. So indeed for all sheaves and hence also for all sheaves . Then of course this agrees with what you would do on constant sheaves. Sigh!

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  • 12 comment(s) on Section 64.3: Frobenii

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