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 36.22.5. Observe that since V is étale over A, it is flat over A. \square
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