The Stacks project

Theorem 64.3.3 (The Baffling Theorem). Let $X$ be a scheme in characteristic $p > 0$. Then the absolute frobenius induces (by pullback) the trivial map on cohomology, i.e., for all integers $j\geq 0$,

\[ F_ X^* : H^ j (X, \underline{\mathbf{Z}/n\mathbf{Z}}) \longrightarrow H^ j (X, \underline{\mathbf{Z}/n\mathbf{Z}}) \]

is the identity.

Proof of Theorem 64.3.3. We need to verify the existence of a functorial isomorphism as above. For an étale morphism $\varphi : U \to X$, consider the diagram

\[ \xymatrix{ U \ar@{-->}[rd] \ar@/^1pc/[rrd]^{F_ U} \ar@/_1pc/[rdd]_\varphi \\ & {U \times _{\varphi , X, F_ X} X} \ar[r]_-{\text{pr}_1} \ar[d]^{\text{pr}_2} & U \ar[d]^\varphi \\ & X \ar[r]^{F_ X} & X. } \]

The dotted arrow is an étale morphism and a universal homeomorphism, so it is an isomorphism. See Étale Morphisms, Lemma 41.14.3. $\square$

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