Spell out duality theory a bit more

Direct the reader to the results computing what f^! is like
in specific cases. One missing case is where one has a smooth
morphism (and not just a smooth and proper morphism). It could
easily be deduced from the properties already esthablished
for the f^! functors in the same way we do for the smooth and
proper case. In fact, one could do this more generally for
locally complete intersection morphisms

TODO: Elucidate relative dualizing complex for lci morphisms

dualizing.tex

@@ -8555,23 +8555,41 @@ \section{A duality theory}
 R\SheafHom_{\mathcal{O}_Y}(Rf_*K, \omega_Y^\bullet)
 $$
 \end{enumerate}
+\item if $f : X \to Y$ is a closed immersion of compactifyable schemes
+over $S$, then $f^!(-) = R\SheafHom(\mathcal{O}_X, -)$,
+\item if $f : Y \to X$ is a finite morphism of compactifyable schemes
+over $S$, then $f_*f^!(-) = R\SheafHom(f_*\mathcal{O}_X, -)$,
+\item if $f : X \to Y$ is the inclusion of an effective Cartier divisor
+into a compactifyable scheme over $S$, then
+$f^!(-) = Lf^*(-) \otimes_{\mathcal{O}_X} \mathcal{O}_Y(-X)[-1]$,
+\item if $f : X \to Y$ is a Koszul regular immersion of codimension $c$
+into a compactifyable scheme over $S$, then
+$f^!(-) \cong Lf^*(-) \otimes_{\mathcal{O}_X} \wedge^c\mathcal{N}[-c]$, and
+\item if $f : X \to Y$ is a smooth proper morphism of relative dimension $d$
+of compactifyable schemes over $S$, then
+$f^!(-) \cong Lf^*  \otimes_{\mathcal{O}_X} \Omega^d_{X/Y}[d]$.
 \end{enumerate}
 See Lemmas
-\ref{lemma-pseudo-functor},
-\ref{lemma-shriek-dualizing},
 \ref{lemma-dualizing-schemes},
+\ref{lemma-proper-noetherian-relative},
+\ref{lemma-twisted-inverse-image-closed},
+\ref{lemma-finite-twisted},
+\ref{lemma-sheaf-with-exact-support-effective-Cartier},
+\ref{lemma-regular-immersion},
+\ref{lemma-smooth-proper},
 \ref{lemma-upper-shriek-composition},
+\ref{lemma-pseudo-functor},
+\ref{lemma-shriek-closed-immersion},
+\ref{lemma-shriek-dualizing},
 \ref{lemma-shriek-via-duality}, and
-\ref{lemma-proper-noetherian-relative}. 
-
-\medskip\noindent
+\ref{lemma-perfect-comparison-shriek}.
 We have obtained our functors by a very abstract procedure
 which finally rests on invoking an existence theorem
 (Derived Categories, Proposition \ref{derived-proposition-brown}).
-This means we have no explicit description of the functors $f^!$.
-This can sometimes be a problem. However, as we will see,
-often it is enough to know the existence of a dualizing complex
-and the duality isomorphism to pin down what it is more exactly.
+This means we have, in general, no explicit description of the functors $f^!$.
+This can sometimes be a problem. But in fact, it is often enough to know
+the existence of a dualizing complex and the duality isomorphism
+to pin down $f^!$.