Fix problem with nonseparated morphisms

The diagonal needn't be a closed immersion...

duality.tex

@@ -5837,25 +5837,30 @@ \section{Relative dualizing complexes}
 \begin{definition}
 \label{definition-relative-dualizing-complex}
 Let $X \to S$ be a morphism of schemes which is flat and
-locally of finite presentation.
+locally of finite presentation. Let $W \subset X \times_S X$
+be any open such that the diagonal $\Delta_{X/S} : X \times_S X$
+factors through a closed immersion $\Delta : X \to W$.
 A {\it relative dualizing complex} is a
 pair $(K, \xi)$ consisting of an object $K \in D(\mathcal{O}_X)$
 and a map
 $$
-\xi : \Delta_*\mathcal{O}_X \longrightarrow L\text{pr}_1^*K
+\xi : \Delta_*\mathcal{O}_X \longrightarrow L\text{pr}_1^*K|_W
 $$
-in $D(\mathcal{O}_{X \times_S X})$ such that
+in $D(\mathcal{O}_W)$ such that
 \begin{enumerate}
 \item $K$ is $S$-perfect (Derived Categories of Schemes, Definition
 \ref{perfect-definition-relatively-perfect}), and
 \item $\xi$ defines an isomorphism of $\Delta_*\mathcal{O}_X$
 with
-$R\SheafHom_{\mathcal{O}_{X \times_S X}}(
-\Delta_*\mathcal{O}_X, L\text{pr}_1^*K)$.
+$R\SheafHom_{\mathcal{O}_W}(
+\Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)$.
 \end{enumerate}
 \end{definition}
 
 \noindent
+Observe that the category of pairs $(K, \xi)$ as in the
+definition is independent of the choice of $W$. 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.
 
@@ -5895,7 +5900,7 @@ \section{Relative dualizing complexes}
 \ref{perfect-lemma-quasi-coherence-internal-hom}.
 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 $D(\mathcal{O}_{X \times_S X})$;
+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
@@ -5943,7 +5948,7 @@ \section{Relative dualizing complexes}
 \medskip\noindent
 Suppose we have an isomorphism $\alpha : K \to L$.
 Then $\alpha(\xi) = u \eta$ for some invertible section
-$u \in H^0(X \times_S X, \Delta_*\mathcal{O}_X) = H^0(X, \mathcal{O}_X)$.
+$u \in H^0(W, \Delta_*\mathcal{O}_X) = H^0(X, \mathcal{O}_X)$.
 (Because both $\eta$ and $\alpha(\xi)$ are generators
 of an invertible $\Delta_*\mathcal{O}_X$-module by assumption.)
 Hence after replacing $\alpha$ by $u^{-1}\alpha$
@@ -6018,41 +6023,40 @@ \section{Relative dualizing complexes}
 \medskip\noindent
 To finish the proof we have to construct the map
 $$
-\xi : \Delta_*\mathcal{O}_X \longrightarrow L\text{pr}_1^*K
+\xi : \Delta_*\mathcal{O}_X \longrightarrow L\text{pr}_1^*K|_W
 $$
-in $D(\mathcal{O}_{X \times_S X})$ inducing an isomorphism from
-$\Delta_*\mathcal{O}_X$ to $R\SheafHom_{\mathcal{O}_{X \times_S X}}(
-\Delta_*\mathcal{O}_X, L\text{pr}_1^*K)$. We can use $\rho_U$ to
-get isomorphisms
+in $D(\mathcal{O}_W)$ inducing an isomorphism from $\Delta_*\mathcal{O}_X$ to
+$R\SheafHom_{\mathcal{O}_W}(\Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)$.
+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
 $$
-R\SheafHom_{\mathcal{O}_{X \times_S X}}(
-\Delta_*\mathcal{O}_X, L\text{pr}_1^*K)|_{U \times_S U}
+R\SheafHom_{\mathcal{O}_W}(
+\Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)|_{U \times_S U}
 \xrightarrow{\rho_U}
 R\SheafHom_{\mathcal{O}_{U \times_S U}}(
 \Delta_*\mathcal{O}_U, L\text{pr}_1^*K_U)
 $$
-As $R\SheafHom_{\mathcal{O}_{X \times_S X}}(
-\Delta_*\mathcal{O}_X, L\text{pr}_1^*K)$
-is supported on $\Delta(X)$ we conclude that its cohomology sheaves are zero
-except in degree $0$. Moreover, we obtain isomorphisms
+As $W$ is covered by the opens $U \times_S U$
+we conclude that the cohomology sheaves of
+$R\SheafHom_{\mathcal{O}_W}(\Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)$
+are zero except in degree $0$. Moreover, we obtain isomorphisms
 $$
+H^0\left(U \times_S U, R\SheafHom_{\mathcal{O}_W}(\Delta_*\mathcal{O}_X,
+L\text{pr}_1^*K|_W)\right)
+\xrightarrow{\rho_U}
 H^0\left((R\SheafHom_{\mathcal{O}_{U \times_S U}}(
 \Delta_*\mathcal{O}_U, L\text{pr}_1^*K_U)\right)
-\xrightarrow{\rho_U}
-H^0\left(R\SheafHom_{\mathcal{O}_{X \times_S X}}(\Delta_*\mathcal{O}_X,
-L\text{pr}_1^*K)\right)|_{U \times_S U}
 $$
-Let $\tau_U$ in the RHS be the image of $\xi_U$
-viewed as an element of the LHS under this map.
+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 $\tau_U|_{U' \times_S U'} = \tau_{U'}$. Since
-$H^0(R\SheafHom_{\mathcal{O}_{X \times_S X}}(\Delta_*\mathcal{O}_X,
-L\text{pr}_1^*K))$ is a sheaf we get a global section $\tau$.
+imply that $\tau_U|_{U' \times_S U'} = \tau_{U'}$.
+Thus we get a global section $\tau$ of the $0$th cohomology sheaf
+$H^0(R\SheafHom_{\mathcal{O}_W}(\Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W))$.
 Since the other cohomology sheaves of
-$R\SheafHom_{\mathcal{O}_{X \times_S X}}(\Delta_*\mathcal{O}_X,
-L\text{pr}_1^*K)$
+$R\SheafHom_{\mathcal{O}_W}(\Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)$
 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
@@ -6079,17 +6083,22 @@ \section{Relative dualizing complexes}
 Consider the cartesian square
 $$
 \xymatrix{
-X' \ar[d]_{\Delta'} \ar[r] & X \ar[d]^\Delta \\
+X' \ar[d]_{\Delta_{X'/S'}} \ar[r] & X \ar[d]^\Delta_{X/S} \\
 X' \times_{S'} X' \ar[r]^{g' \times g'} & X \times_S X
 }
 $$
-We have
+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
 $$
 L(g' \times g')^*\Delta_*\mathcal{O}_X =
 \Delta'_*\mathcal{O}_{X'}
 \quad\text{and}\quad
-L(g' \times g')^*L\text{pr}_1^*K =
-L\text{pr}_1^*K'
+L(g' \times g')^*L\text{pr}_1^*K|_W =
+L\text{pr}_1^*K'|_{W'}
 $$
 The first equality holds because $X$ and $X' \times_{S'} X'$
 are tor independent over $X \times_S X$ (see for example
@@ -6098,15 +6107,15 @@ \section{Relative dualizing complexes}
 (Cohomology, Lemma \ref{cohomology-lemma-derived-pullback-composition}).
 Thus $\xi' = L(g' \times g')^*\xi$ can be viewed as a map
 $$
-\xi' : \Delta'_*\mathcal{O}_{X'} \longrightarrow L\text{pr}_1^*K'
+\xi' : \Delta'_*\mathcal{O}_{X'} \longrightarrow L\text{pr}_1^*K'|_{W'}
 $$
 Having said this the proof of the lemma is straightforward.
 First, $K'$ is $S'$-perfect by Derived Categories of Schemes, Lemma
 \ref{perfect-lemma-base-change-relatively-perfect}.
 To check that $\xi'$ induces an isomorphism
 of $\Delta'_*\mathcal{O}_{X'}$ to
-$R\SheafHom_{\mathcal{O}_{X' \times_{S'} X'}}(
-\Delta'_*\mathcal{O}_{X'}, L\text{pr}_1^*K')$
+$R\SheafHom_{\mathcal{O}_{W'}}(
+\Delta'_*\mathcal{O}_{X'}, L\text{pr}_1^*K'|_{W'})$
 we may work affine locally. By
 Lemma \ref{lemma-relative-dualizing-complex-algebra}
 we reduce to the corresponding statement in algebra
@@ -6167,6 +6176,8 @@ \section{Relative dualizing complexes}
 
 
 
+
+
 \input{chapters}
 
 \bibliography{my}