The Stacks project

Lemma 47.25.4. Let $\varphi : R \to A$ be a local complete intersection homomorphism of Noetherian rings. Then $\omega _{A/R}^\bullet $ is an invertible object of $D(A)$ and $\varphi ^!(K) = K \otimes _ R^\mathbf {L} \omega _{A/R}^\bullet $ for all $K \in D(R)$.

Proof. Recall that a local complete intersection homomorphism is a perfect ring map by More on Algebra, Lemma 15.82.6. Hence the final statement holds by Lemma 47.24.10. By More on Algebra, Definition 15.33.2 we can write $A = R[x_1, \ldots , x_ n]/I$ where $I$ is a Koszul-regular ideal. The construction of $\varphi ^!$ in Section 47.24 shows that it suffices to show the lemma in case $A = R/I$ where $I \subset R$ is a Koszul-regular ideal. Checking $\omega _{A/R}^\bullet $ is invertible in $D(A)$ is local on $\mathop{\mathrm{Spec}}(A)$ by More on Algebra, Lemma 15.126.4. Moreover, formation of $\omega _{A/R}^\bullet $ commutes with localization on $R$ by Lemma 47.24.4. Combining More on Algebra, Definition 15.32.1 and Lemma 15.30.7 and Algebra, Lemma 10.68.6 we can find $g_1, \ldots , g_ r \in R$ generating the unit ideal in $A$ such that $I_{g_ j} \subset R_{g_ j}$ is generated by a regular sequence. Thus we may assume $A = R/(f_1, \ldots , f_ c)$ where $f_1, \ldots , f_ c$ is a regular sequence in $R$. Then we consider the ring maps

\[ R \to R/(f_1) \to R/(f_1, f_2) \to \ldots \to R/(f_1, \ldots , f_ c) = A \]

and we use Lemma 47.24.7 (and the final statement already proven) to see that it suffices to prove the lemma for each step. Finally, in case $A = R/(f)$ for some nonzerodivisor $f$ we see that the lemma is true since $\varphi ^!(R) = R\mathop{\mathrm{Hom}}\nolimits (A, R)$ is invertible by Lemma 47.13.10. $\square$

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