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

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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)