Lemma 48.28.10. Let $f : Y \to X$ and $X \to S$ be morphisms of schemes which are flat and of finite presentation. Let $(K, \xi )$ and $(M, \eta )$ be a relative dualizing complex for $X \to S$ and $Y \to X$. Set $E = M \otimes _{\mathcal{O}_ Y}^\mathbf {L} Lf^*K$. Then $(E, \zeta )$ is a relative dualizing complex for $Y \to S$ for a suitable $\zeta $.

**Proof.**
Using Lemma 48.28.2 and the algebraic version of this lemma (Dualizing Complexes, Lemma 47.27.6) we see that $E$ is affine locally the first component of a relative dualizing complex. In particular we see that $E$ is $S$-perfect since this may be checked affine locally, see Derived Categories of Schemes, Lemma 36.35.3.

Let us first prove the existence of $\zeta $ in case the morphisms $X \to S$ and $Y \to X$ are separated so that $\Delta _{X/S}$, $\Delta _{Y/X}$, and $\Delta _{Y/S}$ are closed immersions. Consider the following diagram

where $p$, $q$, $r$ are the first projections. By Lemma 48.9.4 we have

By Lemma 48.10.3 we have

By Lemma 48.10.2 we have

The last expression is isomorphic (via $\xi $) to $Lm^*\mathcal{O}_ X = \mathcal{O}_{Y \times _ X Y}$. Hence the expression preceding is isomorphic to $Lq^*M$. Hence

The material inside the parentheses is isomorphic to $\Delta _{Y/X, *}*\mathcal{O}_ X$ via $\eta $. This finishes the proof in the separated case.

In the general case we choose an open $W \subset X \times _ S X$ such that $\Delta _{X/S}$ factors through a closed immersion $\Delta : X \to W$ and we choose an open $V \subset Y \times _ X Y$ such that $\Delta _{Y/X}$ factors through a closed immersion $\Delta ' : Y \to V$. Finally, choose an open $W' \subset Y \times _ S Y$ whose intersection with $Y \times _ X Y$ gives $V$ and which maps into $W$. Then we consider the diagram

and we use exactly the same argument as before. $\square$

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