The Stacks project

Lemma 51.19.5. Let $p$ be a prime number. Let $A$ be a ring with $p = 0$. Denote $F : A \to A$, $a \mapsto a^ p$ the Frobenius endomorphism. If $V \to \mathop{\mathrm{Spec}}(A)$ is quasi-compact, quasi-separated, and étale, then there exists an isomorphism $R\Gamma (V, \mathcal{O}_ V) \otimes _{A, F}^\mathbf {L} A \cong R\Gamma (V, \mathcal{O}_ V)$.

Proof. Observe that the relative Frobenius morphism

\[ V \longrightarrow V \times _{\mathop{\mathrm{Spec}}(A), \mathop{\mathrm{Spec}}(F)} \mathop{\mathrm{Spec}}(A) \]

of $V$ over $A$ is an isomorphism, see Étale Morphisms, Lemma 41.14.3. Thus the lemma follows from cohomology and base change, see Derived Categories of Schemes, Lemma 36.21.5. Observe that since $V$ is étale over $A$, it is flat over $A$. $\square$


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