The Stacks project

Lemma 47.27.2. Let $R \to A$ be a flat ring map of finite presentation. Any two relative dualizing complexes for $R \to A$ are isomorphic.

Proof. Let $K$ and $L$ be two relative dualizing complexes for $R \to A$. Denote $K_1 = K \otimes _ A^\mathbf {L} (A \otimes _ R A)$ and $L_2 = (A \otimes _ R A) \otimes _ A^\mathbf {L} L$ the derived base changes via the first and second coprojections $A \to A \otimes _ R A$. By symmetry the assumption on $L_2$ implies that $R\mathop{\mathrm{Hom}}\nolimits _{A \otimes _ R A}(A, L_2)$ is isomorphic to $A$. By More on Algebra, Lemma 15.98.3 part (3) applied twice we have

\[ A \otimes _{A \otimes _ R A}^\mathbf {L} L_2 \cong R\mathop{\mathrm{Hom}}\nolimits _{A \otimes _ R A}(A, K_1 \otimes _{A \otimes _ R A}^\mathbf {L} L_2) \cong A \otimes _{A \otimes _ R A}^\mathbf {L} K_1 \]

Applying the restriction functor $D(A \otimes _ R A) \to D(A)$ for either coprojection we obtain the desired result. $\square$

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