Lemma 47.25.5 (0E4C)—The Stacks project

Lemma 47.25.5. Let $\varphi : R \to A$ be a flat finite type homomorphism of Noetherian rings. The following are equivalent

the fibres $A \otimes _ R \kappa (\mathfrak p)$ are Gorenstein for all primes $\mathfrak p \subset R$ , and
$\omega _{A/R}^\bullet$ is an invertible object of $D(A)$ , see More on Algebra, Lemma 15.126.4.

Proof. If (2) holds, then the fibre rings $A \otimes _ R \kappa (\mathfrak p)$ have invertible dualizing complexes, and hence are Gorenstein. See Lemmas 47.25.2 and 47.21.4.

For the converse, assume (1). Observe that $\omega _{A/R}^\bullet$ is in $D^ b_{\textit{Coh}}(A)$ by Lemma 47.24.2 (since flat finite type homomorphisms of Noetherian rings are perfect, see More on Algebra, Lemma 15.82.4). Take a prime $\mathfrak q \subset A$ lying over $\mathfrak p \subset R$ . Then

$\omega _{A/R}^\bullet \otimes _ A^\mathbf {L} \kappa (\mathfrak q) = \omega _{A/R}^\bullet \otimes _ A^\mathbf {L} (A \otimes _ R \kappa (\mathfrak p)) \otimes _{(A \otimes _ R \kappa (\mathfrak p))}^\mathbf {L} \kappa (\mathfrak q)$

Applying Lemmas 47.25.2 and 47.21.4 and assumption (1) we find that this complex has $1$ nonzero cohomology group which is a $1$ -dimensional $\kappa (\mathfrak q)$ -vector space. By More on Algebra, Lemma 15.77.1 we conclude that $(\omega _{A/R}^\bullet )_ f$ is an invertible object of $D(A_ f)$ for some $f \in A$ , $f \not\in \mathfrak q$ . This proves (2) holds. $\square$

The Stacks project

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