The Stacks project

Lemma 48.28.2. Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $(K, \xi )$ be a relative dualizing complex. Then for any commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(A) \ar[d] \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(R) \ar[r] & S } \]

whose horizontal arrows are open immersions, the restriction of $K$ to $\mathop{\mathrm{Spec}}(A)$ corresponds via Derived Categories of Schemes, Lemma 36.3.5 to a relative dualizing complex for $R \to A$ in the sense of Dualizing Complexes, Definition 47.27.1.

Proof. Since formation of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ commutes with restrictions to opens we may as well assume $X = \mathop{\mathrm{Spec}}(A)$ and $S = \mathop{\mathrm{Spec}}(R)$. Observe that relatively perfect objects of $D(\mathcal{O}_ X)$ are pseudo-coherent and hence are in $D_\mathit{QCoh}(\mathcal{O}_ X)$ (Derived Categories of Schemes, Lemma 36.10.1). Thus the statement makes sense. Observe that taking $\Delta _*$, $L\text{pr}_1^*$, and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ is compatible with what happens on the algebraic side by Derived Categories of Schemes, Lemmas 36.3.7, 36.3.8, 36.10.8. For the last one we observe that $L\text{pr}_1^*K$ is $S$-perfect (hence bounded below) and that $\Delta _*\mathcal{O}_ X$ is a pseudo-coherent object of $D(\mathcal{O}_ W)$; translated into algebra this means that $A$ is pseudo-coherent as an $A \otimes _ R A$-module which follows from More on Algebra, Lemma 15.82.8 applied to $R \to A \otimes _ R A \to A$. Thus we recover exactly the conditions in Dualizing Complexes, Definition 47.27.1. $\square$

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