Remark 48.20.10. Let $S$ be a Noetherian scheme which has a dualizing complex. Let $f : X \to Y$ be a morphism of schemes of finite type over $S$. Then the functor

$f_{new}^! : D^+_{Coh}(\mathcal{O}_ Y) \to D^+_{Coh}(\mathcal{O}_ X)$

is independent of the choice of the dualizing complex $\omega _ S^\bullet$ up to canonical isomorphism. We sketch the proof. Any second dualizing complex is of the form $\omega _ S^\bullet \otimes _{\mathcal{O}_ S}^\mathbf {L} \mathcal{L}$ where $\mathcal{L}$ is an invertible object of $D(\mathcal{O}_ S)$, see Lemma 48.2.6. For any separated morphism $p : U \to S$ of finite type we have $p^!(\omega _ S^\bullet \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{L}) = p^!(\omega _ S^\bullet ) \otimes ^\mathbf {L}_{\mathcal{O}_ U} Lp^*\mathcal{L}$ by Lemma 48.8.1. Hence, if $\omega _ X^\bullet$ and $\omega _ Y^\bullet$ are the dualizing complexes normalized relative to $\omega _ S^\bullet$ we see that $\omega _ X^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} La^*\mathcal{L}$ and $\omega _ Y^\bullet \otimes _{\mathcal{O}_ Y}^\mathbf {L} Lb^*\mathcal{L}$ are the dualizing complexes normalized relative to $\omega _ S^\bullet \otimes _{\mathcal{O}_ S}^\mathbf {L} \mathcal{L}$ (where $a : X \to S$ and $b : Y \to S$ are the structure morphisms). Then the result follows as

\begin{align*} & R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(Lf^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(K, \omega _ Y^\bullet \otimes _{\mathcal{O}_ Y}^\mathbf {L} Lb^*\mathcal{L}), \omega _ X^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} La^*\mathcal{L}) \\ & = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(Lf^*R(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(K, \omega _ Y^\bullet ) \otimes _{\mathcal{O}_ Y}^\mathbf {L} Lb^*\mathcal{L}), \omega _ X^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} La^*\mathcal{L}) \\ & = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(Lf^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(K, \omega _ Y^\bullet ) \otimes _{\mathcal{O}_ X}^\mathbf {L} La^*\mathcal{L}, \omega _ X^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} La^*\mathcal{L}) \\ & = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(Lf^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(K, \omega _ Y^\bullet ), \omega _ X^\bullet ) \end{align*}

for $K \in D^+_{Coh}(\mathcal{O}_ Y)$. The last equality because $La^*\mathcal{L}$ is invertible in $D(\mathcal{O}_ X)$.

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