Lemma 48.20.9. Let $(S, \omega _ S^\bullet )$ be as in Situation 48.20.1. With $f^!_{new}$ and $\omega _ X^\bullet $ defined for all (morphisms of) schemes of finite type over $S$ as above:

the functors $f^!_{new}$ and the arrows $(g \circ f)^!_{new} \to f^!_{new} \circ g^!_{new}$ turn $D_{\textit{Coh}}^+$ into a pseudo functor from the category of schemes of finite type over $S$ into the $2$-category of categories,

$\omega _ X^\bullet = (X \to S)^!_{new} \omega _ S^\bullet $,

the functor $D_ X$ defines an involution of $D_{\textit{Coh}}(\mathcal{O}_ X)$ switching $D_{\textit{Coh}}^+(\mathcal{O}_ X)$ and $D_{\textit{Coh}}^-(\mathcal{O}_ X)$ and fixing $D_{\textit{Coh}}^ b(\mathcal{O}_ X)$,

$\omega _ X^\bullet = f^!_{new}\omega _ Y^\bullet $ for $f : X \to Y$ a morphism of finite type schemes over $S$,

$f^!_{new}M = D_ X(Lf^*D_ Y(M))$ for $M \in D_{\textit{Coh}}^+(\mathcal{O}_ Y)$, and

if in addition $f$ is proper, then $f^!_{new}$ is isomorphic to the restriction of the right adjoint of $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ to $D_{\textit{Coh}}^+(\mathcal{O}_ Y)$ and there is a canonical isomorphism

\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, f^!_{new}M) \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*K, M) \]for $K \in D^-_{\textit{Coh}}(\mathcal{O}_ X)$ and $M \in D_{\textit{Coh}}^+(\mathcal{O}_ Y)$, and

\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, \omega _ X^\bullet ) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*K, \omega _ Y^\bullet ) \]for $K \in D^-_{\textit{Coh}}(\mathcal{O}_ X)$ and

If $X$ is separated over $S$, then $\omega _ X^\bullet $ is canonically isomorphic to $(X \to S)^!\omega _ S^\bullet $ and if $f$ is a morphism between schemes separated over $S$, then there is a canonical isomorphism^{1} $f_{new}^!K = f^!K$ for $K$ in $D_{\textit{Coh}}^+$.

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