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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


Comments (2)

Comment #9261 by andy on

Should it read "the assumption on L2 implies that RHomA⊗RA(A,L2) is isomorphic to L" instead of "to A"?

Comment #9262 by andy on

sorry ignore the above comment


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