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Lemma 47.24.6. Let $R \to R'$ be a homomorphism of Noetherian rings. Let $\varphi : R \to A$ be flat of finite type. Let $\varphi ' : R' \to A' = A \otimes _ R R'$ be the map induced by $\varphi $. Then we have a functorial isomorphism

\[ \varphi ^!(K) \otimes _ A^\mathbf {L} A' = (\varphi ')^!(K \otimes _ R^\mathbf {L} R') \]

for $K$ in $D(R)$.

Proof. Special case of Lemma 47.24.5 by More on Algebra, Lemma 15.82.4. $\square$

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