The Stacks project

Lemma 47.26.2. Let $\varphi : A \to B$ be a homomorphism of Noetherian rings. Assume

  1. $A \to B$ is syntomic and induces a surjective map on spectra, or

  2. $A \to B$ is a faithfully flat local complete intersection, or

  3. $A \to B$ is faithfully flat of finite type with Gorenstein fibres.

Then $K \in D(A)$ is a dualizing complex for $A$ if and only if $K \otimes _ A^\mathbf {L} B$ is a dualizing complex for $B$.

Proof. Observe that $A \to B$ satisfies (1) if and only if $A \to B$ satisfies (2) by More on Algebra, Lemma 15.33.5. Observe that in both (2) and (3) the relative dualzing complex $\varphi ^!(A) = \omega _{B/A}^\bullet $ is an invertible object of $D(B)$, see Lemmas 47.25.4 and 47.25.5. Moreover we have $\varphi ^!(K) = K \otimes _ A^\mathbf {L} \omega _{B/A}^\bullet $ in both cases, see Lemma 47.24.10 for case (3). Thus $\varphi ^!(K)$ is the same as $K \otimes _ A^\mathbf {L} B$ up to tensoring with an invertible object of $D(B)$. Hence $\varphi ^!(K)$ is a dualizing complex for $B$ if and only if $K \otimes _ A^\mathbf {L} B$ is (as being a dualizing complex is local and invariant under shifts). Thus we see that if $K$ is dualizing for $A$, then $K \otimes _ A^\mathbf {L} B$ is dualizing for $B$ by Lemma 47.24.3. To descend the property, see Lemma 47.26.1. $\square$

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