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

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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