The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

46.21 Glueing dualizing complexes

We will now use glueing of dualizing complexes to get a theory which works for all finite type schemes over $S$ given a pair $(S, \omega _ S^\bullet )$ as in Situation 46.21.1. This is similar to [Remark on page 310, RD].

Situation 46.21.1. Here $S$ is a Noetherian scheme and $\omega _ S^\bullet $ is a dualizing complex.

Let $X$ be a scheme of finite type over $S$. Let $\mathcal{U} : X = \bigcup _{i = 1, \ldots , n} U_ i$ be a finite open covering of $X$ by quasi-compact compactifyable schemes over $S$. Every affine scheme of finite type over $S$ is compactifyable over $S$ by Morphisms, Lemma 28.37.3 hence such open coverings certainly exist. For each $i, j, k \in \{ 1, \ldots , n\} $ the schemes $p_ i : U_ i \to S$, $p_{ij} : U_ i \cap U_ j \to S$, and $p_{ijk} : U_ i \cap U_ j \cap U_ k \to S$ are compactifyable. From such an open covering we obtain

  1. $\omega _ i^\bullet = p_ i^!\omega _ S^\bullet $ a dualizing complex on $U_ i$, see Section 46.20,

  2. for each $i, j$ a canonical isomorphism $\varphi _{ij} : \omega _ i^\bullet |_{U_ i \cap U_ j} \to \omega _ j^\bullet |_{U_ i \cap U_ j}$, and

  3. for each $i, j, k$ we have

    \[ \varphi _{ik}|_{U_ i \cap U_ j \cap U_ k} = \varphi _{jk}|_{U_ i \cap U_ j \cap U_ k} \circ \varphi _{ij}|_{U_ i \cap U_ j \cap U_ k} \]

    in $D(\mathcal{O}_{U_ i \cap U_ j \cap U_ k})$.

Here, in (2) we use that $(U_ i \cap U_ j \to U_ i)^!$ is given by restriction (Lemma 46.18.1) and that we have canonical isomorphisms

\[ (U_ i \cap U_ j \to U_ i)^! \circ p_ i^! = p_{ij}^! = (U_ i \cap U_ j \to U_ j)^! \circ p_ j^! \]

by Lemma 46.17.2 and to get (3) we use that the upper shriek functors form a pseudo functor by Lemma 46.17.3.

In the situation just described a dualizing complex normalized relative to $\omega _ S^\bullet $ and $\mathcal{U}$ is a pair $(K, \alpha _ i)$ where $K \in D(\mathcal{O}_ X)$ and $\alpha _ i : K|_{U_ i} \to \omega _ i^\bullet $ are isomorphisms such that $\varphi _{ij}$ is given by $\alpha _ j|_{U_ i \cap U_ j} \circ \alpha _ i^{-1}|_{U_ i \cap U_ j}$. Since being a dualizing complex on a scheme is a local property we see that dualizing complexes normalized relative to $\omega _ S^\bullet $ and $\mathcal{U}$ are indeed dualizing complexes.

Lemma 46.21.2. In Situation 46.21.1 let $X$ be a scheme of finite type over $S$ and let $\mathcal{U}$ be a finite open covering of $X$ by compactifyable schemes. If there exists a dualizing complex normalized relative to $\omega _ S^\bullet $ and $\mathcal{U}$, then it is unique up to unique isomorphism.

Proof. If $(K, \alpha _ i)$ and $(K', \alpha _ i')$ are two, then we consider $L = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, K')$. By Lemma 46.2.5 and its proof, this is an invertible object of $D(\mathcal{O}_ X)$. Using $\alpha _ i$ and $\alpha '_ i$ we obtain an isomorphism

\[ \alpha _ i^ t \otimes \alpha '_ i : L|_{U_ i} \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\omega _ i^\bullet , \omega _ i^\bullet ) = \mathcal{O}_{U_ i}[0] \]

This already implies that $L = H^0(L)[0]$ in $D(\mathcal{O}_ X)$. Moreover, $H^0(L)$ is an invertible sheaf with given trivializations on the opens $U_ i$ of $X$. Finally, the condition that $\alpha _ j|_{U_ i \cap U_ j} \circ \alpha _ i^{-1}|_{U_ i \cap U_ j}$ and $\alpha '_ j|_{U_ i \cap U_ j} \circ (\alpha '_ i)^{-1}|_{U_ i \cap U_ j}$ both give $\varphi _{ij}$ implies that the transition maps are $1$ and we get an isomorphism $H^0(L) = \mathcal{O}_ X$. $\square$

Lemma 46.21.3. In Situation 46.21.1 let $X$ be a scheme of finite type over $S$ and let $\mathcal{U}$, $\mathcal{V}$ be two finite open coverings of $X$ by compactifyable schemes. If there exists a dualizing complex normalized relative to $\omega _ S^\bullet $ and $\mathcal{U}$, then there exists a dualizing complex normalized relative to $\omega _ S^\bullet $ and $\mathcal{V}$ and these complexes are canonically isomorphic.

Proof. It suffices to prove this when $\mathcal{U}$ is given by the opens $U_1, \ldots , U_ n$ and $\mathcal{V}$ by the opens $U_1, \ldots , U_{n + m}$. In fact, we may and do even assume $m = 1$. To go from a dualizing complex $(K, \alpha _ i)$ normalized relative to $\omega _ S^\bullet $ and $\mathcal{V}$ to a dualizing complex normalized relative to $\omega _ S^\bullet $ and $\mathcal{U}$ is achieved by forgetting about $\alpha _ i$ for $i = n + 1$. Conversely, let $(K, \alpha _ i)$ be a dualizing complex normalized relative to $\omega _ S^\bullet $ and $\mathcal{U}$. To finish the proof we need to construct a map $\alpha _{n + 1} : K|_{U_{n + 1}} \to \omega _{n + 1}^\bullet $ satisfying the desired conditions. To do this we observe that $U_{n + 1} = \bigcup U_ i \cap U_{n + 1}$ is an open covering. It is clear that $(K|_{U_{n + 1}}, \alpha _ i|_{U_ i \cap U_{n + 1}})$ is a dualizing complex normalized relative to $\omega _ S^\bullet $ and the covering $U_{n + 1} = \bigcup U_ i \cap U_{n + 1}$. On the other hand, by condition (3) the pair $(\omega _{n + 1}^\bullet |_{U_{n + 1}}, \varphi _{n + 1i})$ is another dualizing complex normalized relative to $\omega _ S^\bullet $ and the covering $U_{n + 1} = \bigcup U_ i \cap U_{n + 1}$. By Lemma 46.21.2 we obtain a unique isomorphism

\[ \alpha _{n + 1} : K|_{U_{n + 1}} \longrightarrow \omega _{n + 1}^\bullet \]

compatible with the given local isomorphisms. It is a pleasant exercise to show that this means it satisfies the required property. $\square$

Lemma 46.21.4. In Situation 46.21.1 let $X$ be a scheme of finite type over $S$ and let $\mathcal{U}$ be a finite open covering of $X$ by compactifyable schemes. Then there exists a dualizing complex normalized relative to $\omega _ S^\bullet $ and $\mathcal{U}$.

Proof. Say $\mathcal{U} : X = \bigcup _{i = 1, \ldots , n} U_ i$. We prove the lemma by induction on $n$. The base case $n = 1$ is immediate. Assume $n > 1$. Set $X' = U_1 \cup \ldots \cup U_{n - 1}$ and let $(K', \{ \alpha '_ i\} _{i = 1, \ldots , n - 1})$ be a dualizing complex normalized relative to $\omega _ S^\bullet $ and $\mathcal{U}' : X' = \bigcup _{i = 1, \ldots , n - 1} U_ i$. It is clear that $(K'|_{X' \cap U_ n}, \alpha '_ i|_{U_ i \cap U_ n})$ is a dualizing complex normalized relative to $\omega _ S^\bullet $ and the covering $X' \cap U_ n = \bigcup _{i = 1, \ldots , n - 1} U_ i \cap U_ n$. On the other hand, by condition (3) the pair $(\omega _ n^\bullet |_{X' \cap U_ n}, \varphi _{ni})$ is another dualizing complex normalized relative to $\omega _ S^\bullet $ and the covering $X' \cap U_ n = \bigcup _{i = 1, \ldots , n - 1} U_ i \cap U_ n$. By Lemma 46.21.2 we obtain a unique isomorphism

\[ \epsilon : K'|_{X' \cap U_ n} \longrightarrow \omega _ i^\bullet |_{X' \cap U_ n} \]

compatible with the given local isomorphisms. By Cohomology, Lemma 20.39.1 we obtain $K \in D(\mathcal{O}_ X)$ together with isomorphisms $\beta : K|_{X'} \to K'$ and $\gamma : K|_{U_ n} \to \omega _ n^\bullet $ such that $\epsilon = \gamma |_{X'\cap U_ n} \circ \beta |_{X' \cap U_ n}^{-1}$. Then we define

\[ \alpha _ i = \alpha '_ i \circ \beta |_{U_ i}, i = 1, \ldots , n - 1, \text{ and } \alpha _ n = \gamma \]

We still need to verify that $\varphi _{ij}$ is given by $\alpha _ j|_{U_ i \cap U_ j} \circ \alpha _ i^{-1}|_{U_ i \cap U_ j}$. For $i, j \leq n - 1$ this follows from the corresponding condition for $\alpha _ i'$. For $i = j = n$ it is clear as well. If $i < j = n$, then we get

\[ \alpha _ n|_{U_ i \cap U_ n} \circ \alpha _ i^{-1}|_{U_ i \cap U_ n} = \gamma |_{U_ i \cap U_ n} \circ \beta ^{-1}|_{U_ i \cap U_ n} \circ (\alpha '_ i)^{-1}|_{U_ i \cap U_ n} = \epsilon |_{U_ i \cap U_ n} \circ (\alpha '_ i)^{-1}|_{U_ i \cap U_ n} \]

This is equal to $\alpha _{in}$ exactly because $\epsilon $ is the unique map compatible with the maps $\alpha _ i'$ and $\alpha _{ni}$. $\square$

Let $(S, \omega _ S^\bullet )$ be as in Situation 46.21.1. The upshot of the lemmas above is that given any scheme $X$ of finite type over $S$, there is a pair $(K, \alpha _ U)$ given up to unique isomorphism, consisting of an object $K \in D(\mathcal{O}_ X)$ and isomorphisms $\alpha _ U : K|_ U \to \omega _ U^\bullet $ for every open subscheme $U \subset X$ which has a compactification over $S$. Here $\omega _ U^\bullet = (U \to S)^!\omega _ S^\bullet $ is a dualizing complex on $U$, see Section 46.20. Moreover, if $\mathcal{U} : X = \bigcup U_ i$ is a finite open covering by opens which are compactifyable over $S$, then $(K, \alpha _{U_ i})$ is a dualizing complex normalized relative to $\omega _ S^\bullet $ and $\mathcal{U}$. Namely, uniqueness up to unique isomorphism by Lemma 46.21.2, existence for one open covering by Lemma 46.21.4, and the fact that $K$ then works for all open coverings is Lemma 46.21.3.

Definition 46.21.5. Let $S$ be a Noetherian scheme and let $\omega _ S^\bullet $ be a dualizing complex on $S$. Let $X$ be a scheme of finite type over $S$. The complex $K$ constructed above is called the dualizing complex normalized relative to $\omega _ S^\bullet $ and is denoted $\omega _ X^\bullet $.

As the terminology suggest, a dualizing complex normalized relative to $\omega _ S^\bullet $ is not just an object of the derived category of $X$ but comes equipped with the local isomorphisms described above. This does not conflict with setting $\omega _ X^\bullet = p^!\omega _ S^\bullet $ where $p : X \to S$ is the structure morphism if $X$ has a compactification over $S$ (see Dualizing Complexes, Section 45.15). More generally we have the following sanity check.

Lemma 46.21.6. Let $(S, \omega _ S^\bullet )$ be as in Situation 46.21.1. Let $f : X \to Y$ be a morphism of finite type schemes over $S$. Let $\omega _ X^\bullet $ and $\omega _ Y^\bullet $ be dualizing complexes normalized relative to $\omega _ S^\bullet $. Then $\omega _ X^\bullet $ is a dualizing complex normalized relative to $\omega _ Y^\bullet $.

Proof. This is just a matter of bookkeeping. Choose a finite affine open covering $\mathcal{V} : Y = \bigcup V_ j$. For each $j$ choose a finite affine open covering $f^{-1}(V_ j) = U_{ji}$. Set $\mathcal{U} : X = \bigcup U_{ji}$. The schemes $V_ j$ and $U_{ji}$ are compactifyable over $S$, hence we have the upper shriek functors for $q_ j : V_ j \to S$, $p_{ji} : U_{ji} \to S$ and $f_{ji} : U_{ji} \to V_ j$ and $f_{ji}' : U_{ji} \to Y$. Let $(L, \beta _ j)$ be a dualizing complex normalized relative to $\omega _ S^\bullet $ and $\mathcal{V}$. Let $(K, \gamma _{ji})$ be a dualizing complex normalized relative to $\omega _ S^\bullet $ and $\mathcal{U}$. (In other words, $L = \omega _ Y^\bullet $ and $K = \omega _ X^\bullet $.) We can define

\[ \alpha _{ji} : K|_{U_{ji}} \xrightarrow {\gamma _{ji}} p_{ji}^!\omega _ S^\bullet = f_{ji}^!q_ j^!\omega _ S^\bullet \xrightarrow {f_{ji}^!\beta _ j^{-1}} f_{ji}^!(L|_{V_ j}) = (f_{ji}')^!(L) \]

To finish the proof we have to show that $\alpha _{ji}|_{U_{ji} \cap U_{j'i'}} \circ \alpha _{j'i'}^{-1}|_{U_{ji} \cap U_{j'i'}}$ is the canonical isomorphism $(f_{ji}')^!(L)|_{U_{ji} \cap U_{j'i'}} \to (f_{j'i'}')^!(L)|_{U_{ji} \cap U_{j'i'}}$. This is formal and we omit the details. $\square$

Lemma 46.21.7. Let $(S, \omega _ S^\bullet )$ be as in Situation 46.21.1. Let $j : X \to Y$ be an open immersion of schemes of finite type over $S$. Let $\omega _ X^\bullet $ and $\omega _ Y^\bullet $ be dualizing complexes normalized relative to $\omega _ S^\bullet $. Then there is a canonical isomorphism $\omega _ X^\bullet = \omega _ Y^\bullet |_ X$.

Proof. Immediate from the construction of normalized dualizing complexes given just above Definition 46.21.5. $\square$

Lemma 46.21.8. Let $(S, \omega _ S^\bullet )$ be as in Situation 46.21.1. Let $f : X \to Y$ be a proper morphism of schemes of finite type over $S$. Let $\omega _ X^\bullet $ and $\omega _ Y^\bullet $ be dualizing complexes normalized relative to $\omega _ S^\bullet $. Let $a$ be the right adjoint of Lemma 46.3.1 for $f$. Then there is a canonical isomorphism $a(\omega _ Y^\bullet ) = \omega _ X^\bullet $.

Proof. Let $p : X \to S$ and $q : Y \to S$ be the structure morphisms. If $X$ and $Y$ are compactifyable over $S$, then this follows from the fact that $\omega _ X^\bullet = p^!\omega _ S^\bullet $, $\omega _ Y^\bullet = q^!\omega _ S^\bullet $, $f^! = a$, and $f^! \circ q^! = p^!$ (Lemma 46.17.2). In the general case we first use Lemma 46.21.6 to reduce to the case $Y = S$. In this case $X$ and $Y$ are compactifyable over $S$ and we've just seen the result. $\square$

Let $(S, \omega _ S^\bullet )$ be as in Situation 46.21.1. For a scheme $X$ of finite type over $S$ denote $\omega _ X^\bullet $ the dualizing complex for $X$ normalized relative to $\omega _ S^\bullet $. Define $D_ X(-) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(-, \omega _ X^\bullet )$ as in Lemma 46.2.4. Let $f : X \to Y$ be a morphism of finite type schemes over $S$. Define

\[ f_{new}^! = D_ X \circ Lf^* \circ D_ Y : D_{\textit{Coh}}^+(\mathcal{O}_ Y) \to D_{\textit{Coh}}^+(\mathcal{O}_ X) \]

If $f : X \to Y$ and $g : Y \to Z$ are composable morphisms between schemes of finite type over $S$, define

\begin{align*} (g \circ f)^!_{new} & = D_ X \circ L(g \circ f)^* \circ D_ Z \\ & = D_ X \circ Lf^* \circ Lg^* \circ D_ Z \\ & \to D_ X \circ Lf^* \circ D_ Y \circ D_ Y \circ Lg^* \circ D_ Z \\ & = f^!_{new} \circ g^!_{new} \end{align*}

where the arrow is defined in Lemma 46.2.4. We collect the results together in the following lemma.

Lemma 46.21.9. Let $(S, \omega _ S^\bullet )$ be as in Situation 46.21.1. 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 all $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and $M \in D_{\textit{Coh}}^+(\mathcal{O}_ Y)$, and most importantly

    \[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, \omega _ X^\bullet ) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*K, \omega _ Y^\bullet ) \]

If $X$ is compactifyable over $S$, then $\omega _ X^\bullet $ is canonically isomorphic to $(X \to S)^!\omega _ S^\bullet $ and if $f$ is a morphism between compactifyable schemes 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 46.2.4. Then, using Categories, Lemma 4.27.2, 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 46.2.4. 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 46.3.1 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 compactifyable over $S$ (and in general this won't be true) hence we cannot immediately apply Lemma 46.18.7 to $f$ over $S$. To get around this we use the canonical identification $\omega _ X^\bullet = a(\omega _ Y^\bullet )$ of Lemma 46.21.8. Hence $f^!_{new}$ is the restriction of $a$ to $D_{\textit{Coh}}^+(\mathcal{O}_ Y)$ by Lemma 46.18.7 applied to $f : X \to Y$ over the base scheme $Y$! Thus the result is true by Lemma 46.3.6.

The final assertions follow from the construction of normalized dualizing complexes and the already used Lemma 46.18.7. $\square$

Remark 46.21.10. 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 46.2.5. For any compactifyable $p : U \to S$ 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 46.8.1. 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 $D(\mathcal{O}_ X)$.

Example 46.21.11. Let $S$ be a Noetherian scheme and let $\omega _ S^\bullet $ be a dualizing complex. Let $f : X \to Y$ be a proper morphism of finite type schemes over $S$. Let $\omega _ X^\bullet $ and $\omega _ Y^\bullet $ be dualizing complexes normalized relative to $\omega _ S^\bullet $. In this situation we have $a(\omega _ Y^\bullet ) = \omega _ X^\bullet $ (Lemma 46.21.8) and hence the trace map (Section 46.7) is a canonical arrow

\[ \text{Tr}_ f : Rf_*\omega _ X^\bullet \longrightarrow \omega _ Y^\bullet \]

which produces the isomorphisms (Lemma 46.21.9)

\[ \mathop{\mathrm{Hom}}\nolimits _ X(L, \omega _ X^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _ Y(Rf_*L, \omega _ Y^\bullet ) \]

and

\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, \omega _ X^\bullet ) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*L, \omega _ Y^\bullet ) \]

for $L$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$.

Remark 46.21.12. Let $S$ be a Noetherian scheme and let $\omega _ S^\bullet $ be a dualizing complex. Let $f : X \to Y$ be a finite morphism between schemes of finite type over $S$. Let $\omega _ X^\bullet $ and $\omega _ Y^\bullet $ be dualizing complexes normalized relative to $\omega _ S^\bullet $. Then we have

\[ f_*\omega _ X^\bullet = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ X, \omega _ Y^\bullet ) \]

in $D_\mathit{QCoh}^+(f_*\mathcal{O}_ X)$ by Lemmas 46.11.4 and 46.21.8 and the trace map of Example 46.21.11 is the map

\[ \text{Tr}_ f : Rf_*\omega _ X^\bullet = f_*\omega _ X^\bullet = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ X, \omega _ Y^\bullet ) \longrightarrow \omega _ Y^\bullet \]

which often goes under the name “evaluation at $1$”.

Remark 46.21.13. Let $f : X \to Y$ be a flat proper morphism of finite type schemes over a pair $(S, \omega _ S^\bullet )$ as in Situation 46.21.1. The relative dualizing complex (Remark 46.12.5) is $\omega _{X/Y}^\bullet = a(\mathcal{O}_ Y)$. By Lemma 46.21.8 we have the first canonical isomorphism in

\[ \omega _ X^\bullet = a(\omega _ Y^\bullet ) = Lf^*\omega _ Y^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} \omega _{X/Y}^\bullet \]

in $D(\mathcal{O}_ X)$. The second canonical isomorphism follows from the discussion in Remark 46.12.5.

[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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