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:

1. 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,

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

3. 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 $D_{\textit{Coh}}^ b(\mathcal{O}_ X)$,

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

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

6. 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 isomorphism1 $f_{new}^!K = f^!K$ for $K$ in $D_{\textit{Coh}}^+$.

Proof. Let $f : X \to Y$, $g : Y \to Z$, $h : Z \to T$ be morphisms of schemes of finite type over $S$. We have to show that

$\xymatrix{ (h \circ g \circ f)^!_{new} \ar[r] \ar[d] & f^!_{new} \circ (h \circ g)^!_{new} \ar[d] \\ (g \circ f)^!_{new} \circ h^!_{new} \ar[r] & f^!_{new} \circ g^!_{new} \circ h^!_{new} }$

is commutative. Let $\eta _ Y : \text{id} \to D_ Y^2$ and $\eta _ Z : \text{id} \to D_ Z^2$ be the canonical isomorphisms of Lemma 48.2.5. Then, using Categories, Lemma 4.28.2, a computation (omitted) shows that both arrows $(h \circ g \circ f)^!_{new} \to f^!_{new} \circ g^!_{new} \circ h^!_{new}$ are given by

$1 \star \eta _ Y \star 1 \star \eta _ Z \star 1 : D_ X \circ Lf^* \circ Lg^* \circ Lh^* \circ D_ T \longrightarrow D_ X \circ Lf^* \circ D_ Y^2 \circ Lg^* \circ D_ Z^2 \circ Lh^* \circ D_ T$

This proves (1). Part (2) is immediate from the definition of $(X \to S)^!_{new}$ and the fact that $D_ S(\omega _ S^\bullet ) = \mathcal{O}_ S$. Part (3) is Lemma 48.2.5. Part (4) follows by the same argument as part (2). Part (5) is the definition of $f^!_{new}$.

Proof of (6). Let $a$ be the right adjoint of Lemma 48.3.1 for the proper morphism $f : X \to Y$ of schemes of finite type over $S$. The issue is that we do not know $X$ or $Y$ is separated over $S$ (and in general this won't be true) hence we cannot immediately apply Lemma 48.17.8 to $f$ over $S$. To get around this we use the canonical identification $\omega _ X^\bullet = a(\omega _ Y^\bullet )$ of Lemma 48.20.8. Hence $f^!_{new}$ is the restriction of $a$ to $D_{\textit{Coh}}^+(\mathcal{O}_ Y)$ by Lemma 48.17.8 applied to $f : X \to Y$ over the base scheme $Y$! The displayed equalities hold by Example 48.3.9.

The final assertions follow from the construction of normalized dualizing complexes and the already used Lemma 48.17.8. $\square$

[1] We haven't checked that these are compatible with the isomorphisms $(g \circ f)^! \to f^! \circ g^!$ and $(g \circ f)^!_{new} \to f^!_{new} \circ g^!_{new}$. We will do this here if we need this later.

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