The Stacks project

Lemma 47.27.3. Let $\varphi : R \to A$ be a flat finite type ring map of Noetherian rings. Then the relative dualizing complex $\omega _{A/R}^\bullet = \varphi ^!(R)$ of Section 47.25 is a relative dualizing complex in the sense of Definition 47.27.1.

Proof. From Lemma 47.25.2 we see that $\varphi ^!(R)$ is $R$-perfect. Denote $\delta : A \otimes _ R A \to A$ the multiplication map and $p_1, p_2 : A \to A \otimes _ R A$ the coprojections. Then

\[ \varphi ^!(R) \otimes _ A^\mathbf {L} (A \otimes _ R A) = \varphi ^!(R) \otimes _{A, p_1}^\mathbf {L} (A \otimes _ R A) = p_2^!(A) \]

by Lemma 47.24.4. Recall that $ R\mathop{\mathrm{Hom}}\nolimits _{A \otimes _ R A}(A, \varphi ^!(R) \otimes _ A^\mathbf {L} (A \otimes _ R A)) $ is the image of $\delta ^!(\varphi ^!(R) \otimes _ A^\mathbf {L} (A \otimes _ R A))$ under the restriction map $\delta _* : D(A) \to D(A \otimes _ R A)$. Use the definition of $\delta ^!$ from Section 47.24 and Lemma 47.13.3. Since $\delta ^!(p_2^!(A)) \cong A$ by Lemma 47.24.7 we conclude. $\square$

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