Definition 48.28.1. Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $W \subset X \times _ S X$ be any open such that the diagonal $\Delta _{X/S} : X \to X \times _ S X$ factors through a closed immersion $\Delta : X \to W$. A *relative dualizing complex* is a pair $(K, \xi )$ consisting of an object $K \in D(\mathcal{O}_ X)$ and a map

in $D(\mathcal{O}_ W)$ such that

$K$ is $S$-perfect (Derived Categories of Schemes, Definition 36.35.1), and

$\xi $ defines an isomorphism of $\Delta _*\mathcal{O}_ X$ with $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ W}( \Delta _*\mathcal{O}_ X, L\text{pr}_1^*K|_ W)$.

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