Lemma 51.19.2. 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. Let $I \subset A$ be a finitely generated ideal. Set $Z = V(I)$. There exists an isomorphism $R\Gamma _ Z(A) \otimes _{A, F}^\mathbf {L} A \cong R\Gamma _ Z(A)$.

Proof. Follows from Dualizing Complexes, Lemma 47.9.3 and the fact that $Z = V(f_1^ p, \ldots , f_ r^ p)$ if $I = (f_1, \ldots , f_ r)$. $\square$

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