The Stacks project

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:

1. the functors $f^!$ turn $D_\mathit{QCoh}^+$ into a pseudo functor on $\textit{FTS}_ S$,

2. 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 $M \in D_\mathit{QCoh}^+(\mathcal{O}_ Y)$,

3. 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 $D_{\textit{Coh}}^ b(\mathcal{O}_ X)$,

4. if $f : X \to Y$ is a morphism of $\textit{FTS}_ S$ and $\omega _ Y^\bullet $ is a dualizing complex on $Y$, then

   1. $\omega _ X^\bullet = f^!\omega _ Y^\bullet $ is a dualizing complex for $X$,

   2. $f^!M = D_ X(Lf^*D_ Y(M))$ canonically for $M \in D_{\textit{Coh}}^+(\mathcal{O}_ Y)$, and

   3. 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 $D^-_{\textit{Coh}}(\mathcal{O}_ X)$,

5. if $f : X \to Y$ is a closed immersion in $\textit{FTS}_ S$, then $f^!(-) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ X, -)$,

6. if $f : Y \to X$ is a finite morphism in $\textit{FTS}_ S$, then $f_*f^!(-) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f_*\mathcal{O}_ Y, -)$,

7. if $f : X \to Y$ is the inclusion of an effective Cartier divisor into an object of $\textit{FTS}_ S$, then $f^!(-) = Lf^*(-) \otimes _{\mathcal{O}_ X} f^*\mathcal{O}_ Y(X)[-1]$,

8. 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

9. 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 $f^!$.