Lemma 47.25.1. Let $R \to R'$ be a homomorphism of Noetherian rings. Let $R \to A$ be of finite type. Set $A' = A \otimes _ R R'$. If
$R \to R'$ is flat, or
$R \to A$ is flat, or
$R \to A$ is perfect and $R'$ and $A$ are tor independent over $R$,
then there is an isomorphism $\omega _{A/R}^\bullet \otimes _ A^\mathbf {L} A' \to \omega ^\bullet _{A'/R'}$ in $D(A')$.
Proof. Follows from Lemmas 47.24.4, 47.24.6, and 47.24.5 and the definitions. $\square$
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