The Stacks project

Lemma 47.24.10. Let $\varphi : R \to A$ be a perfect homomorphism of Noetherian rings (for example $\varphi $ is flat of finite type). Then $\varphi ^!(K) = K \otimes _ R^\mathbf {L} \varphi ^!(R)$ for $K \in D(R)$.

Proof. (The parenthetical statement follows from More on Algebra, Lemma 15.82.4.) We can choose a factorization $R \to P \to A$ where $P$ is a polynomial ring in $n$ variables over $R$ and then $A$ is a perfect $P$-module, see More on Algebra, Lemma 15.82.2. Recall that $\varphi ^!(K) = R\mathop{\mathrm{Hom}}\nolimits (A, K \otimes _ R^\mathbf {L} P[n])$. Thus the result follows from Lemma 47.13.9 and More on Algebra, Lemma 15.60.5. $\square$

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