Lemma 47.27.6. Let R \to A \to B be a ring maps which are flat and of finite presentation. Let K_{A/R} and K_{B/A} be relative dualizing complexes for R \to A and A \to B. Then K = K_{A/R} \otimes _ A^\mathbf {L} K_{B/A} is a relative dualizing complex for R \to B.
Proof. We will use reduction to the Noetherian case. Namely, by Algebra, Lemma 10.168.1 there exists a finite type \mathbf{Z} subalgebra R_0 \subset R and a flat finite type ring map R_0 \to A_0 such that A = A_0 \otimes _{R_0} R. After increasing R_0 and correspondingly replacing A_0 we may assume there is a flat finite type ring map A_0 \to B_0 such that B = B_0 \otimes _{R_0} R (use the same lemma). If we prove the lemma for R_0 \to A_0 \to B_0, then the lemma follows by Lemmas 47.27.2, 47.27.3, and 47.27.4. This reduces us to the situation discussed in the next paragraph.
Assume R is Noetherian and denote \varphi : R \to A and \psi : A \to B the given ring maps. Then K_{A/R} \cong \varphi ^!(R) and K_{B/A} \cong \psi ^!(A), see references given above. Then
K = K_{A/R} \otimes _ A^\mathbf {L} K_{B/A} \cong \varphi ^!(R) \otimes _ A^\mathbf {L} \psi ^!(A) \cong \psi ^!(\varphi ^!(R)) \cong (\psi \circ \varphi )^!(R)
by Lemmas 47.24.10 and 47.24.7. Thus K is a relative dualizing complex for R \to B. \square
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