48.19 A duality theory
In this section we spell out what kind of a duality theory our very general results above give for finite type separated schemes over a fixed Noetherian base scheme.
Recall that a dualizing complex on a Noetherian scheme X, is an object of D(\mathcal{O}_ X) which affine locally gives a dualizing complex for the corresponding rings, see Definition 48.2.2.
Given a Noetherian scheme S denote \textit{FTS}_ S the category of schemes which are of finite type and separated over S. Then:
the functors f^! turn D_\mathit{QCoh}^+ into a pseudo functor on \textit{FTS}_ S,
if f : X \to Y is a proper morphism in \textit{FTS}_ S, then f^! is the restriction of the right adjoint of Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y) to D_\mathit{QCoh}^+(\mathcal{O}_ Y) and there is a canonical isomorphism
Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, f^!M) \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*K, M)for all K \in D_{\textit{Coh}}^-(\mathcal{O}_ X) and M \in D_\mathit{QCoh}^+(\mathcal{O}_ Y),
if an object X of \textit{FTS}_ S has a dualizing complex \omega _ X^\bullet , then the functor D_ X = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(-, \omega _ X^\bullet ) 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),
if f : X \to Y is a morphism of \textit{FTS}_ S and \omega _ Y^\bullet is a dualizing complex on Y, then
\omega _ X^\bullet = f^!\omega _ Y^\bullet is a dualizing complex for X,
f^!M = D_ X(Lf^*D_ Y(M)) canonically for M \in D_{\textit{Coh}}^+(\mathcal{O}_ Y), and
if in addition f is proper then
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 )for K in D^-_{\textit{Coh}}(\mathcal{O}_ X),
if f : X \to Y is a closed immersion in \textit{FTS}_ S, then f^!(-) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ X, -),
if f : Y \to X is a finite morphism in \textit{FTS}_ S, then f_*f^!(-) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f_*\mathcal{O}_ Y, -),
if f : X \to Y is the inclusion of an effective Cartier divisor into an object of \textit{FTS}_ S, then f^!(-) = Lf^*(-) \otimes _{\mathcal{O}_ X} f^*\mathcal{O}_ Y(X)[-1],
if f : X \to Y is a Koszul regular immersion of codimension c into an object of \textit{FTS}_ S, then f^!(-) \cong Lf^*(-) \otimes _{\mathcal{O}_ X} \wedge ^ c\mathcal{N}[-c], and
if f : X \to Y is a smooth proper morphism of relative dimension d in \textit{FTS}_ S, then f^!(-) \cong Lf^*(-) \otimes _{\mathcal{O}_ X} \Omega ^ d_{X/Y}[d].
This follows from Lemmas 48.2.5, 48.3.6, 48.9.7, 48.11.4, 48.14.2, 48.15.6, 48.15.7, 48.16.3, 48.16.4, 48.17.4, 48.17.7, 48.17.8, and 48.17.9 and Example 48.3.9. We have obtained our functors by a very abstract procedure which finally rests on invoking an existence theorem (Derived Categories, Proposition 13.38.2). 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^!.
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