Compatibilities in spaces-dualizing

There are infinitely many of these

spaces-duality.tex

@@ -2045,6 +2045,79 @@ \section{Relative dualizing complexes for proper flat morphisms}
 and the proof is complete.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-base-change-relative-dualizing}
+Let $S$ be a scheme. Consider a cartesian square
+$$
+\xymatrix{
+X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\
+Y' \ar[r]^g & Y
+}
+$$
+of algebraic spaces over $S$. Assume $X \to Y$ is proper, flat, and
+of finite presentation. Let $(\omega_{X/Y}^\bullet, \tau)$ be a
+relative dualizing complex for $f$. Then
+$(L(g')^*\omega_{X/Y}^\bullet, Lg^*\tau)$ is a relative dualizing
+complex for $f'$.
+\end{lemma}
+
+\begin{proof}
+Observe that $L(g')^*\omega_{X/Y}^\bullet$ is $Y'$-perfect by
+More on Morphisms of Spaces, Lemma
+\ref{spaces-more-morphisms-lemma-base-change-relatively-perfect}.
+The other condition of
+Definition \ref{definition-relative-dualizing-proper-flat}
+holds by transitivity of fibre products.
+\end{proof}
+
+
+
+
+
+
+
+\section{Comparison with the case of schemes}
+\label{section-comparison}
+
+\noindent
+We should add a lot more in this section.
+
+\begin{lemma}
+\label{lemma-compare}
+Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of
+quasi-compact and quasi-separated algebraic spaces over $S$.
+Assume $X$ and $Y$ are representable and let $f_0 : X_0 \to Y_0$ be a
+morphism of schemes representing $f$ (awkward but temporary notation).
+Let $a : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)$
+be the right adjoint of $Rf_*$ from Lemma \ref{lemma-twisted-inverse-image}.
+Let $a_0 : D_\QCoh(\mathcal{O}_{Y_0}) \to D_\QCoh(\mathcal{O}_{X_0})$
+be the right adjoint of $Rf_*$ from
+Duality for Schemes, Lemma \ref{duality-lemma-twisted-inverse-image}.
+Then 
+$$
+\xymatrix{
+D_\QCoh(\mathcal{O}_{X_0})
+\ar@{=}[rrrrrr]_{\text{Derived Categories of Spaces, Lemma
+\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}}}
+& & & & & &
+D_\QCoh(\mathcal{O}_X) \\
+D_\QCoh(\mathcal{O}_{Y_0}) \ar[u]^{a_0}
+\ar@{=}[rrrrrr]^{\text{Derived Categories of Spaces, Lemma
+\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}}}
+& & & & & &
+D_\QCoh(\mathcal{O}_Y) \ar[u]_a
+}
+$$
+is commutative.
+\end{lemma}
+
+\begin{proof}
+Follows from uniqueness of adjoints and the compatibilities of
+Derived Categories of Spaces, Remark
+\ref{spaces-perfect-remark-match-total-direct-images}.
+\end{proof}
+
+