## 48.28 Relative dualizing complexes

For a proper, flat morphism of finite presentation we have a rigid relative dualizing complex, see Remark 48.12.5 and Lemma 48.12.8. For a separated and finite type morphism $f : X \to Y$ of Noetherian schemes, we can consider $f^!\mathcal{O}_ Y$. In this section we define relative dualizing complexes for morphisms which are flat and locally of finite presentation (but not necessarily quasi-separated or quasi-compact) between schemes (not necessarily locally Noetherian). We show such complexes exist, are unique up to unique isomorphism, and agree with the cases mentioned above. Before reading this section, please read Dualizing Complexes, Section 47.27.

Definition 48.28.1. Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $W \subset X \times _ S X$ be any open such that the diagonal $\Delta _{X/S} : X \to X \times _ S X$ factors through a closed immersion $\Delta : X \to W$. A relative dualizing complex is a pair $(K, \xi )$ consisting of an object $K \in D(\mathcal{O}_ X)$ and a map

$\xi : \Delta _*\mathcal{O}_ X \longrightarrow L\text{pr}_1^*K|_ W$

in $D(\mathcal{O}_ W)$ such that

1. $K$ is $S$-perfect (Derived Categories of Schemes, Definition 36.35.1), and

2. $\xi$ defines an isomorphism of $\Delta _*\mathcal{O}_ X$ with $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ W}( \Delta _*\mathcal{O}_ X, L\text{pr}_1^*K|_ W)$.

By Lemma 48.9.3 condition (2) is equivalent to the existence of an isomorphism

$\mathcal{O}_ X \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ X, L\text{pr}_1^*K|_ W)$

in $D(\mathcal{O}_ X)$ whose pushforward via $\Delta$ is equal to $\xi$. Since $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ X, L\text{pr}_1^*K|_ W)$ is independent of the choice of the open $W$, so is the category of pairs $(K, \xi )$. If $X \to S$ is separated, then we can choose $W = X \times _ S X$. We will reduce many of the arguments to the case of rings using the following lemma.

Lemma 48.28.2. Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $(K, \xi )$ be a relative dualizing complex. Then for any commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(A) \ar[d] \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(R) \ar[r] & S }$

whose horizontal arrows are open immersions, the restriction of $K$ to $\mathop{\mathrm{Spec}}(A)$ corresponds via Derived Categories of Schemes, Lemma 36.3.5 to a relative dualizing complex for $R \to A$ in the sense of Dualizing Complexes, Definition 47.27.1.

Proof. Since formation of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits$ commutes with restrictions to opens we may as well assume $X = \mathop{\mathrm{Spec}}(A)$ and $S = \mathop{\mathrm{Spec}}(R)$. Observe that relatively perfect objects of $D(\mathcal{O}_ X)$ are pseudo-coherent and hence are in $D_\mathit{QCoh}(\mathcal{O}_ X)$ (Derived Categories of Schemes, Lemma 36.10.1). Thus the statement makes sense. Observe that taking $\Delta _*$, $L\text{pr}_1^*$, and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits$ is compatible with what happens on the algebraic side by Derived Categories of Schemes, Lemmas 36.3.7, 36.3.8, 36.10.8. For the last one we observe that $L\text{pr}_1^*K$ is $S$-perfect (hence bounded below) and that $\Delta _*\mathcal{O}_ X$ is a pseudo-coherent object of $D(\mathcal{O}_ W)$; translated into algebra this means that $A$ is pseudo-coherent as an $A \otimes _ R A$-module which follows from More on Algebra, Lemma 15.82.8 applied to $R \to A \otimes _ R A \to A$. Thus we recover exactly the conditions in Dualizing Complexes, Definition 47.27.1. $\square$

Lemma 48.28.3. Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $(K, \xi )$ be a relative dualizing complex. Then $\mathcal{O}_ X \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, K)$ is an isomorphism.

Proof. Looking affine locally this reduces using Lemma 48.28.2 to the algebraic case which is Dualizing Complexes, Lemma 47.27.5. $\square$

Lemma 48.28.4. Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. If $(K, \xi )$ and $(L, \eta )$ are two relative dualizing complexes on $X/S$, then there is a unique isomorphism $K \to L$ sending $\xi$ to $\eta$.

Proof. Let $U \subset X$ be an affine open mapping into an affine open of $S$. Then there is an isomorphism $K|_ U \to L|_ U$ by Lemma 48.28.2 and Dualizing Complexes, Lemma 47.27.2. The reader can reuse the argument of that lemma in the schemes case to obtain a proof in this case. We will instead use a glueing argument.

Suppose we have an isomorphism $\alpha : K \to L$. Then $\alpha (\xi ) = u \eta$ for some invertible section $u \in H^0(W, \Delta _*\mathcal{O}_ X) = H^0(X, \mathcal{O}_ X)$. (Because both $\eta$ and $\alpha (\xi )$ are generators of an invertible $\Delta _*\mathcal{O}_ X$-module by assumption.) Hence after replacing $\alpha$ by $u^{-1}\alpha$ we see that $\alpha (\xi ) = \eta$. Since the automorphism group of $K$ is $H^0(X, \mathcal{O}_ X^*)$ by Lemma 48.28.3 there is at most one such $\alpha$.

Let $\mathcal{B}$ be the collection of affine opens of $X$ which map into an affine open of $S$. For each $U \in \mathcal{B}$ we have a unique isomorphism $\alpha _ U : K|_ U \to L|_ U$ mapping $\xi$ to $\eta$ by the discussion in the previous two paragraphs. Observe that $\text{Ext}^ i(K|_ U, K|_ U) = 0$ for $i < 0$ and any open $U$ of $X$ by Lemma 48.28.3. By Cohomology, Lemma 20.45.2 applied to $\text{id} : X \to X$ we get a unique morphism $\alpha : K \to L$ agreeing with $\alpha _ U$ for all $U \in \mathcal{B}$. Then $\alpha$ sends $\xi$ to $\eta$ as this is true locally. $\square$

Lemma 48.28.5. Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. There exists a relative dualizing complex $(K, \xi )$.

Proof. Let $\mathcal{B}$ be the collection of affine opens of $X$ which map into an affine open of $S$. For each $U$ we have a relative dualizing complex $(K_ U, \xi _ U)$ for $U$ over $S$. Namely, choose an affine open $V \subset S$ such that $U \to X \to S$ factors through $V$. Write $U = \mathop{\mathrm{Spec}}(A)$ and $V = \mathop{\mathrm{Spec}}(R)$. By Dualizing Complexes, Lemma 47.27.4 there exists a relative dualizing complex $K_ A \in D(A)$ for $R \to A$. Arguing backwards through the proof of Lemma 48.28.2 this determines an $V$-perfect object $K_ U \in D(\mathcal{O}_ U)$ and a map

$\xi : \Delta _*\mathcal{O}_ U \to L\text{pr}_1^*K_ U$

in $D(\mathcal{O}_{U \times _ V U})$. Since being $V$-perfect is the same as being $S$-perfect and since $U \times _ V U = U \times _ S U$ we find that $(K_ U, \xi _ U)$ is as desired.

If $U' \subset U \subset X$ with $U', U \in \mathcal{B}$, then we have a unique isomorphism $\rho _{U'}^ U : K_ U|_{U'} \to K_{U'}$ in $D(\mathcal{O}_{U'})$ sending $\xi _ U|_{U' \times _ S U'}$ to $\xi _{U'}$ by Lemma 48.28.4 (note that trivially the restriction of a relative dualizing complex to an open is a relative dualizing complex). The uniqueness guarantees that $\rho ^ U_{U''} = \rho ^ V_{U''} \circ \rho ^ U_{U'}|_{U''}$ for $U'' \subset U' \subset U$ in $\mathcal{B}$. Observe that $\text{Ext}^ i(K_ U, K_ U) = 0$ for $i < 0$ for $U \in \mathcal{B}$ by Lemma 48.28.3 applied to $U/S$ and $K_ U$. Thus the BBD glueing lemma (Cohomology, Theorem 20.45.8) tells us there is a unique solution, namely, an object $K \in D(\mathcal{O}_ X)$ and isomorphisms $\rho _ U : K|_ U \to K_ U$ such that we have $\rho ^ U_{U'} \circ \rho _ U|_{U'} = \rho _{U'}$ for all $U' \subset U$, $U, U' \in \mathcal{B}$.

To finish the proof we have to construct the map

$\xi : \Delta _*\mathcal{O}_ X \longrightarrow L\text{pr}_1^*K|_ W$

in $D(\mathcal{O}_ W)$ inducing an isomorphism from $\Delta _*\mathcal{O}_ X$ to $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ W}(\Delta _*\mathcal{O}_ X, L\text{pr}_1^*K|_ W)$. Since we may change $W$, we choose $W = \bigcup _{U \in \mathcal{B}} U \times _ S U$. We can use $\rho _ U$ to get isomorphisms

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ W}( \Delta _*\mathcal{O}_ X, L\text{pr}_1^*K|_ W)|_{U \times _ S U} \xrightarrow {\rho _ U} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{U \times _ S U}}( \Delta _*\mathcal{O}_ U, L\text{pr}_1^*K_ U)$

As $W$ is covered by the opens $U \times _ S U$ we conclude that the cohomology sheaves of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ W}(\Delta _*\mathcal{O}_ X, L\text{pr}_1^*K|_ W)$ are zero except in degree $0$. Moreover, we obtain isomorphisms

$H^0\left(U \times _ S U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ W}(\Delta _*\mathcal{O}_ X, L\text{pr}_1^*K|_ W)\right) \xrightarrow {\rho _ U} H^0\left((R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{U \times _ S U}}( \Delta _*\mathcal{O}_ U, L\text{pr}_1^*K_ U)\right)$

Let $\tau _ U$ in the LHS be an element mapping to $\xi _ U$ under this map. The compatibilities between $\rho ^ U_{U'}$, $\xi _ U$, $\xi _{U'}$, $\rho _ U$, and $\rho _{U'}$ for $U' \subset U \subset X$ open $U', U \in \mathcal{B}$ imply that $\tau _ U|_{U' \times _ S U'} = \tau _{U'}$. Thus we get a global section $\tau$ of the $0$th cohomology sheaf $H^0(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ W}(\Delta _*\mathcal{O}_ X, L\text{pr}_1^*K|_ W))$. Since the other cohomology sheaves of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ W}(\Delta _*\mathcal{O}_ X, L\text{pr}_1^*K|_ W)$ are zero, this global section $\tau$ determines a morphism $\xi$ as desired. Since the restriction of $\xi$ to $U \times _ S U$ gives $\xi _ U$, we see that it satisfies the final condition of Definition 48.28.1. $\square$

Lemma 48.28.6. Consider a cartesian square

$\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ S' \ar[r]^ g & S }$

of schemes. Assume $X \to S$ is flat and locally of finite presentation. Let $(K, \xi )$ be a relative dualizing complex for $f$. Set $K' = L(g')^*K$. Let $\xi '$ be the derived base change of $\xi$ (see proof). Then $(K', \xi ')$ is a relative dualizing complex for $f'$.

Proof. Consider the cartesian square

$\xymatrix{ X' \ar[d]_{\Delta _{X'/S'}} \ar[r] & X \ar[d]^{\Delta _{X/S}} \\ X' \times _{S'} X' \ar[r]^{g' \times g'} & X \times _ S X }$

Choose $W \subset X \times _ S X$ open such that $\Delta _{X/S}$ factors through a closed immersion $\Delta : X \to W$. Choose $W' \subset X' \times _{S'} X'$ open such that $\Delta _{X'/S'}$ factors through a closed immersion $\Delta ' : X \to W'$ and such that $(g' \times g')(W') \subset W$. Let us still denote $g' \times g' : W' \to W$ the induced morphism. We have

$L(g' \times g')^*\Delta _*\mathcal{O}_ X = \Delta '_*\mathcal{O}_{X'} \quad \text{and}\quad L(g' \times g')^*L\text{pr}_1^*K|_ W = L\text{pr}_1^*K'|_{W'}$

The first equality holds because $X$ and $X' \times _{S'} X'$ are tor independent over $X \times _ S X$ (see for example More on Morphisms, Lemma 37.69.1). The second holds by transitivity of derived pullback (Cohomology, Lemma 20.27.2). Thus $\xi ' = L(g' \times g')^*\xi$ can be viewed as a map

$\xi ' : \Delta '_*\mathcal{O}_{X'} \longrightarrow L\text{pr}_1^*K'|_{W'}$

Having said this the proof of the lemma is straightforward. First, $K'$ is $S'$-perfect by Derived Categories of Schemes, Lemma 36.35.6. To check that $\xi '$ induces an isomorphism of $\Delta '_*\mathcal{O}_{X'}$ to $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{W'}}( \Delta '_*\mathcal{O}_{X'}, L\text{pr}_1^*K'|_{W'})$ we may work affine locally. By Lemma 48.28.2 we reduce to the corresponding statement in algebra which is proven in Dualizing Complexes, Lemma 47.27.4. $\square$

Lemma 48.28.7. Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a proper, flat morphism of finite presentation. The relative dualizing complex $\omega _{X/S}^\bullet$ of Remark 48.12.5 together with (48.12.8.1) is a relative dualizing complex in the sense of Definition 48.28.1.

Proof. In Lemma 48.12.7 we proved that $\omega _{X/S}^\bullet$ is $S$-perfect. Let $c$ be the right adjoint of Lemma 48.3.1 for the diagonal $\Delta : X \to X \times _ S X$. Then we can apply $\Delta _*$ to (48.12.8.1) to get an isomorphism

$\Delta _*\mathcal{O}_ X \to \Delta _*(c(L\text{pr}_1^*\omega _{X/S}^\bullet )) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X \times _ S X}}( \Delta _*\mathcal{O}_ X, L\text{pr}_1^*\omega _{X/S}^\bullet )$

The equality holds by Lemmas 48.9.7 and 48.9.3. This finishes the proof. $\square$

Remark 48.28.8. Let $X \to S$ be a morphism of schemes which is flat, proper, and of finite presentation. By Lemma 48.28.5 there exists a relative dualizing complex $(\omega _{X/S}^\bullet , \xi )$ in the sense of Definition 48.28.1. Consider any morphism $g : S' \to S$ where $S'$ is quasi-compact and quasi-separated (for example an affine open of $S$). By Lemma 48.28.6 we see that $(L(g')^*\omega _{X/S}^\bullet , L(g')^*\xi )$ is a relative dualizing complex for the base change $f' : X' \to S'$ in the sense of Definition 48.28.1. Let $\omega _{X'/S'}^\bullet$ be the relative dualizing complex for $X' \to S'$ in the sense of Remark 48.12.5. Combining Lemmas 48.28.7 and 48.28.4 we see that there is a unique isomorphism

$\omega _{X'/S'}^\bullet \longrightarrow L(g')^*\omega _{X/S}^\bullet$

compatible with (48.12.8.1) and $L(g')^*\xi$. These isomorphisms are compatible with morphisms between quasi-compact and quasi-separated schemes over $S$ and the base change isomorphisms of Lemma 48.12.4 (if we ever need this compatibility we will carefully state and prove it here).

Lemma 48.28.9. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. If $f$ is flat, then $f^!\mathcal{O}_ Y$ is (the first component of) a relative dualizing complex for $X$ over $Y$ in the sense of Definition 48.28.1.

Proof. By Lemma 48.17.10 we have that $f^!\mathcal{O}_ Y$ is $Y$-perfect. As $f$ is separated the diagonal $\Delta : X \to X \times _ Y X$ is a closed immersion and $\Delta _*\Delta ^!(-) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X \times _ Y X}}(\mathcal{O}_ X, -)$, see Lemmas 48.9.7 and 48.9.3. Hence to finish the proof it suffices to show $\Delta ^!(L\text{pr}_1^*f^!(\mathcal{O}_ Y)) \cong \mathcal{O}_ X$ where $\text{pr}_1 : X \times _ Y X \to X$ is the first projection. We have

$\mathcal{O}_ X = \Delta ^! \text{pr}_1^!\mathcal{O}_ X = \Delta ^! \text{pr}_1^! L\text{pr}_2^*\mathcal{O}_ Y = \Delta ^!(L\text{pr}_1^* f^!\mathcal{O}_ Y)$

where $\text{pr}_2 : X \times _ Y X \to X$ is the second projection and where we have used the base change isomorphism $\text{pr}_1^! \circ L\text{pr}_2^* = L\text{pr}_1^* \circ f^!$ of Lemma 48.18.1. $\square$

Lemma 48.28.10. Let $f : Y \to X$ and $X \to S$ be morphisms of schemes which are flat and of finite presentation. Let $(K, \xi )$ and $(M, \eta )$ be a relative dualizing complex for $X \to S$ and $Y \to X$. Set $E = M \otimes _{\mathcal{O}_ Y}^\mathbf {L} Lf^*K$. Then $(E, \zeta )$ is a relative dualizing complex for $Y \to S$ for a suitable $\zeta$.

Proof. Using Lemma 48.28.2 and the algebraic version of this lemma (Dualizing Complexes, Lemma 47.27.6) we see that $E$ is affine locally the first component of a relative dualizing complex. In particular we see that $E$ is $S$-perfect since this may be checked affine locally, see Derived Categories of Schemes, Lemma 36.35.3.

Let us first prove the existence of $\zeta$ in case the morphisms $X \to S$ and $Y \to X$ are separated so that $\Delta _{X/S}$, $\Delta _{Y/X}$, and $\Delta _{Y/S}$ are closed immersions. Consider the following diagram

$\xymatrix{ & & Y \ar@{=}[r] & Y \ar[d]^ f \\ Y \ar[r]_{\Delta _{Y/X}} & Y \times _ X Y \ar[d]_ m \ar[r]_\delta \ar[ru]_ q & Y \times _ S Y \ar[d]^{f \times f} \ar[ru]_ p & X\\ & X \ar[r]^{\Delta _{X/S}} & X \times _ S X \ar[ru]_ r }$

where $p$, $q$, $r$ are the first projections. By Lemma 48.9.4 we have

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{Y \times _ S Y}}( \Delta _{Y/S, *}\mathcal{O}_ Y, Lp^*E) = R\delta _*\left(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{Y \times _ X Y}}( \Delta _{Y/X, *}\mathcal{O}_ Y, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_{Y \times _ X Y}, Lp^*E))\right)$

By Lemma 48.10.3 we have

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_{Y \times _ X Y}, Lp^*E) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_{Y \times _ X Y}, L(f \times f)^*Lr^*K) \otimes _{\mathcal{O}_{Y \times _ S Y}}^\mathbf {L} Lq^*M$

By Lemma 48.10.2 we have

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_{Y \times _ X Y}, L(f \times f)^*Lr^*K) = Lm^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ X, Lr^*K)$

The last expression is isomorphic (via $\xi$) to $Lm^*\mathcal{O}_ X = \mathcal{O}_{Y \times _ X Y}$. Hence the expression preceding is isomorphic to $Lq^*M$. Hence

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{Y \times _ S Y}}( \Delta _{Y/S, *}\mathcal{O}_ Y, Lp^*E) = R\delta _*\left(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{Y \times _ X Y}}( \Delta _{Y/X, *}\mathcal{O}_ Y, Lq^*M)\right)$

The material inside the parentheses is isomorphic to $\Delta _{Y/X, *}*\mathcal{O}_ X$ via $\eta$. This finishes the proof in the separated case.

In the general case we choose an open $W \subset X \times _ S X$ such that $\Delta _{X/S}$ factors through a closed immersion $\Delta : X \to W$ and we choose an open $V \subset Y \times _ X Y$ such that $\Delta _{Y/X}$ factors through a closed immersion $\Delta ' : Y \to V$. Finally, choose an open $W' \subset Y \times _ S Y$ whose intersection with $Y \times _ X Y$ gives $V$ and which maps into $W$. Then we consider the diagram

$\xymatrix{ & & Y \ar@{=}[r] & Y \ar[d]^ f \\ Y \ar[r]_{\Delta '} & V \ar[d]_ m \ar[r]_\delta \ar[ru]_ q & W' \ar[d]^{f \times f} \ar[ru]_ p & X\\ & X \ar[r]^\Delta & W \ar[ru]_ r }$

and we use exactly the same argument as before. $\square$