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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 [RD, Remark on page 310].

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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\section{Glueing dualizing complexes}
\label{section-glue}

\noindent
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 \ref{situation-dualizing}. This is similar to
\cite[Remark on page 310]{RD}.

\begin{situation}
\label{situation-dualizing}
Here $S$ is a Noetherian scheme and $\omega_S^\bullet$ is a dualizing
complex.
\end{situation}

\noindent
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 \ref{morphisms-lemma-quasi-projective-finite-type-over-S}
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
\begin{enumerate}
\item $\omega_i^\bullet = p_i^!\omega_S^\bullet$
a dualizing complex on $U_i$, see Section \ref{section-duality},
\item 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
\item
\label{item-cocycle-glueing}
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 $D(\mathcal{O}_{U_i \cap U_j \cap U_k})$.
\end{enumerate}
Here, in (2) we use that $(U_i \cap U_j \to U_i)^!$
is given by restriction (Lemma \ref{lemma-shriek-open-immersion})
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 \ref{lemma-upper-shriek-composition} and to get (3) we use
that the upper shriek functors form a pseudo functor by
Lemma \ref{lemma-pseudo-functor}.

\medskip\noindent
In the situation just described a
{\it 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.

\begin{lemma}
\label{lemma-good-dualizing-unique}
In Situation \ref{situation-dualizing} 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.
\end{lemma}

\begin{proof}
If $(K, \alpha_i)$ and $(K', \alpha_i')$ are two, then we consider
$L = R\SheafHom_{\mathcal{O}_X}(K, K')$.
By Lemma \ref{lemma-dualizing-unique-schemes}
and its proof, this is an invertible object of $D(\mathcal{O}_X)$.
Using $\alpha_i$ and $\alpha'_i$ we obtain an isomorphism
$$
\alpha_i^t \otimes \alpha'_i :
L|_{U_i} \longrightarrow
R\SheafHom_{\mathcal{O}_X}(\omega_i^\bullet, \omega_i^\bullet) =
\mathcal{O}_{U_i}[0]
$$
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 $H^0(L) = \mathcal{O}_X$.
\end{proof}

\begin{lemma}
\label{lemma-good-dualizing-independence-covering}
In Situation \ref{situation-dualizing} 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.
\end{lemma}

\begin{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 (\ref{item-cocycle-glueing}) 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 \ref{lemma-good-dualizing-unique} 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.
\end{proof}

\begin{lemma}
\label{lemma-existence-good-dualizing}
In Situation \ref{situation-dualizing} 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
$\mathcal{U}$.
\end{lemma}

\begin{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 (\ref{item-cocycle-glueing}) 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 \ref{lemma-good-dualizing-unique} 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 \ref{cohomology-lemma-glue}
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 $\alpha_{ni}$.
\end{proof}

\noindent
Let $(S, \omega_S^\bullet)$ be as in Situation \ref{situation-dualizing}.
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 \ref{section-duality}. 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 \ref{lemma-good-dualizing-unique},
existence for one open covering by
Lemma \ref{lemma-existence-good-dualizing}, and
the fact that $K$ then works for all open coverings is
Lemma \ref{lemma-good-dualizing-independence-covering}.

\begin{definition}
\label{definition-good-dualizing}
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
{\it dualizing complex normalized relative to $\omega_S^\bullet$}
and is denoted $\omega_X^\bullet$.
\end{definition}

\noindent
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 \ref{dualizing-section-dualizing}). More generally
we have the following sanity check.

\begin{lemma}
\label{lemma-good-over-both}
Let $(S, \omega_S^\bullet)$ be as in Situation \ref{situation-dualizing}.
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 $\omega_Y^\bullet$.
\end{lemma}

\begin{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.
\end{proof}

\begin{lemma}
\label{lemma-open-immersion-good-dualizing-complex}
Let $(S, \omega_S^\bullet)$ be as in Situation \ref{situation-dualizing}.
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 $\omega_X^\bullet = \omega_Y^\bullet|_X$.
\end{lemma}

\begin{proof}
Immediate from the construction of normalized dualizing complexes
given just above
Definition \ref{definition-good-dualizing}.
\end{proof}

\begin{lemma}
\label{lemma-proper-map-good-dualizing-complex}
Let $(S, \omega_S^\bullet)$ be as in Situation \ref{situation-dualizing}.
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 \ref{lemma-twisted-inverse-image} for
$f$. Then there is a canonical isomorphism
$a(\omega_Y^\bullet) = \omega_X^\bullet$.
\end{lemma}

\begin{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 \ref{lemma-upper-shriek-composition}).
In the general case we first use Lemma \ref{lemma-good-over-both}
to reduce to the case $Y = S$. In this case $X$ and $Y$
are compactifyable over $S$ and we've just seen the result.
\end{proof}

\noindent
Let $(S, \omega_S^\bullet)$ be as in Situation \ref{situation-dualizing}.
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 $D_X(-) = R\SheafHom_{\mathcal{O}_X}(-, \omega_X^\bullet)$
as in Lemma \ref{lemma-dualizing-schemes}.
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 \ref{lemma-dualizing-schemes}.
We collect the results together in the following lemma.

\begin{lemma}
\label{lemma-duality-bootstrap}
Let $(S, \omega_S^\bullet)$ be as in Situation \ref{situation-dualizing}.
With $f^!_{new}$ and $\omega_X^\bullet$ defined for all (morphisms of)
schemes of finite type over $S$ as above:
\begin{enumerate}
\item 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,
\item $\omega_X^\bullet = (X \to S)^!_{new} \omega_S^\bullet$,
\item 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
$D_{\textit{Coh}}^b(\mathcal{O}_X)$,
\item $\omega_X^\bullet = f^!_{new}\omega_Y^\bullet$ for
$f : X \to Y$ a morphism of finite type schemes over $S$,
\item $f^!_{new}M = D_X(Lf^*D_Y(M))$ for
$M \in D_{\textit{Coh}}^+(\mathcal{O}_Y)$, and
\item if in addition $f$ is proper, then $f^!_{new}$ is isomorphic
to the restriction of the right adjoint of
$Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ to
$D_{\textit{Coh}}^+(\mathcal{O}_Y)$ and there is a canonical isomorphism
$$
Rf_*R\SheafHom_{\mathcal{O}_X}(K, f^!_{new}M)
\to
R\SheafHom_{\mathcal{O}_Y}(Rf_*K, M)
$$
for all $K \in D_\QCoh(\mathcal{O}_X)$ and
$M \in D_{\textit{Coh}}^+(\mathcal{O}_Y)$, and most importantly
$$
Rf_*R\SheafHom_{\mathcal{O}_X}(K, \omega_X^\bullet) =
R\SheafHom_{\mathcal{O}_Y}(Rf_*K, \omega_Y^\bullet)
$$
\end{enumerate}
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 isomorphism\footnote{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.}
$f_{new}^!K = f^!K$ for $K$ in $D_{\textit{Coh}}^+$.
\end{lemma}

\begin{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 \ref{lemma-dualizing-schemes}. Then, using
Categories, Lemma \ref{categories-lemma-properties-2-cat-cats},
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 \ref{lemma-dualizing-schemes}.
Part (4) follows by the same argument as part (2).
Part (5) is the definition of $f^!_{new}$.

\medskip\noindent
Proof of (6). Let $a$ be the
right adjoint of Lemma \ref{lemma-twisted-inverse-image} 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 \ref{lemma-shriek-via-duality} to $f$ over $S$.
To get around this we use the canonical identification
$\omega_X^\bullet = a(\omega_Y^\bullet)$ of
Lemma \ref{lemma-proper-map-good-dualizing-complex}.
Hence $f^!_{new}$ is the restriction of $a$ to
$D_{\textit{Coh}}^+(\mathcal{O}_Y)$ by Lemma \ref{lemma-shriek-via-duality}
applied to $f : X \to Y$ over the base scheme $Y$!
Thus the result is true by Lemma \ref{lemma-iso-on-RSheafHom}.

\medskip\noindent
The final assertions follow from the construction of normalized
dualizing complexes and the already used Lemma \ref{lemma-shriek-via-duality}.
\end{proof}

\begin{remark}
\label{remark-independent-omega-S}
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 \ref{lemma-dualizing-unique-schemes}.
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 \ref{lemma-compare-with-pullback-perfect}.
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\SheafHom_{\mathcal{O}_X}(Lf^*R\SheafHom_{\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\SheafHom_{\mathcal{O}_X}(Lf^*R(\SheafHom_{\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\SheafHom_{\mathcal{O}_X}(Lf^*R\SheafHom_{\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\SheafHom_{\mathcal{O}_X}(Lf^*R\SheafHom_{\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
$D(\mathcal{O}_X)$.
\end{remark}


\begin{example}
\label{example-trace-proper}
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 \ref{lemma-proper-map-good-dualizing-complex})
and hence the trace map (Section \ref{section-trace}) is a canonical arrow
$$
\text{Tr}_f : Rf_*\omega_X^\bullet \longrightarrow \omega_Y^\bullet
$$
which produces the isomorphisms (Lemma \ref{lemma-duality-bootstrap})
$$
\Hom_X(L, \omega_X^\bullet) = \Hom_Y(Rf_*L, \omega_Y^\bullet)
$$
and
$$
Rf_*R\SheafHom_{\mathcal{O}_X}(L, \omega_X^\bullet) =
R\SheafHom_{\mathcal{O}_Y}(Rf_*L, \omega_Y^\bullet)
$$
for $L$ in $D_\QCoh(\mathcal{O}_X)$.
\end{example}

\begin{remark}
\label{remark-dualizing-finite}
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
$$
f_*\omega_X^\bullet = R\SheafHom(f_*\mathcal{O}_X, \omega_Y^\bullet)
$$
in $D_\QCoh^+(f_*\mathcal{O}_X)$ by Lemmas \ref{lemma-finite-twisted} and
\ref{lemma-proper-map-good-dualizing-complex}
and the trace map of Example \ref{example-trace-proper} is the map
$$
\text{Tr}_f : Rf_*\omega_X^\bullet = f_*\omega_X^\bullet =
R\SheafHom(f_*\mathcal{O}_X, \omega_Y^\bullet) \longrightarrow
\omega_Y^\bullet
$$
which often goes under the name ``evaluation at $1$''.
\end{remark}

\begin{remark}
\label{remark-relative-dualizing-complex-shriek}
Let $f : X \to Y$ be a flat proper morphism of finite type
schemes over a pair $(S, \omega_S^\bullet)$ as in
Situation \ref{situation-dualizing}. The relative dualizing complex
(Remark \ref{remark-relative-dualizing-complex}) is
$\omega_{X/Y}^\bullet = a(\mathcal{O}_Y)$. By
Lemma \ref{lemma-proper-map-good-dualizing-complex}
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 \ref{remark-relative-dualizing-complex}.
\end{remark}

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This captcha seems more appropriate than the usual illegible gibberish, right?