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
\xymatrix{ & & Y \ar@{=}[r] & Y \ar[d]^ f \\ Y \ar[r]_{\Delta _{Y/X}} & Y \times _ X Y \ar[d]_ m \ar[r]_\delta \ar[ru]_ q & Y \times _ S Y \ar[d]^{f \times f} \ar[ru]_ p & X\\ & X \ar[r]^{\Delta _{X/S}} & X \times _ S X \ar[ru]_ r }
where p, q, r are the first projections. By Lemma 48.9.4 we have
R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{Y \times _ S Y}}( \Delta _{Y/S, *}\mathcal{O}_ Y, Lp^*E) = R\delta _*\left(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{Y \times _ X Y}}( \Delta _{Y/X, *}\mathcal{O}_ Y, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_{Y \times _ X Y}, Lp^*E))\right)
By Lemma 48.10.3 we have
R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_{Y \times _ X Y}, Lp^*E) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_{Y \times _ X Y}, L(f \times f)^*Lr^*K) \otimes _{\mathcal{O}_{Y \times _ S Y}}^\mathbf {L} Lq^*M
By Lemma 48.10.2 we have
R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_{Y \times _ X Y}, L(f \times f)^*Lr^*K) = Lm^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ X, Lr^*K)
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
R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{Y \times _ S Y}}( \Delta _{Y/S, *}\mathcal{O}_ Y, Lp^*E) = R\delta _*\left(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{Y \times _ X Y}}( \Delta _{Y/X, *}\mathcal{O}_ Y, Lq^*M)\right)
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
\xymatrix{ & & Y \ar@{=}[r] & Y \ar[d]^ f \\ Y \ar[r]_{\Delta '} & V \ar[d]_ m \ar[r]_\delta \ar[ru]_ q & W' \ar[d]^{f \times f} \ar[ru]_ p & X\\ & X \ar[r]^\Delta & W \ar[ru]_ r }
and we use exactly the same argument as before. \square
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