48.22 Dualizing modules
This section is a continuation of Dualizing Complexes, Section 47.19.
Let $X$ be a Noetherian scheme and let $\omega _ X^\bullet $ be a dualizing complex. Let $n \in \mathbf{Z}$ be the smallest integer such that $H^ n(\omega _ X^\bullet )$ is nonzero. In other words, $-n$ is the maximal value of the dimension function associated to $\omega _ X^\bullet $ (Lemma 48.2.7). Sometimes $H^ n(\omega _ X^\bullet )$ is called a dualizing module or dualizing sheaf for $X$ and then it is often denoted by $\omega _ X$. We will say “let $\omega _ X$ be a dualizing module” to indicate the above.
Care has to be taken when using dualizing modules $\omega _ X$ on Noetherian schemes $X$:
the integer $n$ may change when passing from $X$ to an open $U$ of $X$ and then it won't be true that $\omega _ X|_ U = \omega _ U$,
the dualizing complex isn't unique; the dualizing module is only unique up to tensoring by an invertible module.
The second problem will often be irrelevant because we will work with $X$ of finite type over a base change $S$ which is endowed with a fixed dualizing complex $\omega _ S^\bullet $ and $\omega _ X^\bullet $ will be the dualizing complex normalized relative to $\omega _ S^\bullet $. The first problem will not occur if $X$ is equidimensional, more precisely, if the dimension function associated to $\omega _ X^\bullet $ (Lemma 48.2.7) maps every generic point of $X$ to the same integer.
Example 48.22.1. Say $S = \mathop{\mathrm{Spec}}(A)$ with $(A, \mathfrak m, \kappa )$ a local Noetherian ring, and $\omega _ S^\bullet $ corresponds to a normalized dualizing complex $\omega _ A^\bullet $. Then if $f : X \to S$ is proper over $S$ and $\omega _ X^\bullet = f^!\omega _ S^\bullet $ the coherent sheaf
\[ \omega _ X = H^{-\dim (X)}(\omega _ X^\bullet ) \]
is a dualizing module and is often called the dualizing module of $X$ (with $S$ and $\omega _ S^\bullet $ being understood). We will see that this has good properties.
Example 48.22.2. Say $X$ is an equidimensional scheme of finite type over a field $k$. Then it is customary to take $\omega _ X^\bullet $ the dualizing complex normalized relative to $k[0]$ and to refer to
\[ \omega _ X = H^{-\dim (X)}(\omega _ X^\bullet ) \]
as the dualizing module of $X$. If $X$ is separated over $k$, then $\omega _ X^\bullet = f^!\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}$ where $f : X \to \mathop{\mathrm{Spec}}(k)$ is the structure morphism by Lemma 48.20.9. If $X$ is proper over $k$, then this is a special case of Example 48.22.1.
Lemma 48.22.3. Let $X$ be a connected Noetherian scheme and let $\omega _ X$ be a dualizing module on $X$. The support of $\omega _ X$ is the union of the irreducible components of maximal dimension with respect to any dimension function and $\omega _ X$ is a coherent $\mathcal{O}_ X$-module having property $(S_2)$.
Proof.
By our conventions discussed above there exists a dualizing complex $\omega _ X^\bullet $ such that $\omega _ X$ is the leftmost nonvanishing cohomology sheaf. Since $X$ is connected, any two dimension functions differ by a constant (Topology, Lemma 5.20.3). Hence we may use the dimension function associated to $\omega _ X^\bullet $ (Lemma 48.2.7). With these remarks in place, the lemma now follows from Dualizing Complexes, Lemma 47.17.5 and the definitions (in particular Cohomology of Schemes, Definition 30.11.1).
$\square$
Lemma 48.22.4. Let $X/A$ with $\omega _ X^\bullet $ and $\omega _ X$ be as in Example 48.22.1. Then
$H^ i(\omega _ X^\bullet ) \not= 0 \Rightarrow i \in \{ -\dim (X), \ldots , 0\} $,
the dimension of the support of $H^ i(\omega _ X^\bullet )$ is at most $-i$,
$\text{Supp}(\omega _ X)$ is the union of the components of dimension $\dim (X)$, and
$\omega _ X$ has property $(S_2)$.
Proof.
Let $\delta _ X$ and $\delta _ S$ be the dimension functions associated to $\omega _ X^\bullet $ and $\omega _ S^\bullet $ as in Lemma 48.21.2. As $X$ is proper over $A$, every closed subscheme of $X$ contains a closed point $x$ which maps to the closed point $s \in S$ and $\delta _ X(x) = \delta _ S(s) = 0$. Hence $\delta _ X(\xi ) = \dim (\overline{\{ \xi \} })$ for any point $\xi \in X$. Hence we can check each of the statements of the lemma by looking at what happens over $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ in which case the result follows from Dualizing Complexes, Lemmas 47.16.5 and 47.17.5. Some details omitted. The last two statements can also be deduced from Lemma 48.22.3.
$\square$
Lemma 48.22.5. Let $X/A$ with dualizing module $\omega _ X$ be as in Example 48.22.1. Let $d = \dim (X_ s)$ be the dimension of the closed fibre. If $\dim (X) = d + \dim (A)$, then the dualizing module $\omega _ X$ represents the functor
\[ \mathcal{F} \longmapsto \mathop{\mathrm{Hom}}\nolimits _ A(H^ d(X, \mathcal{F}), \omega _ A) \]
on the category of coherent $\mathcal{O}_ X$-modules.
Proof.
We have
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}, \omega _ X) & = \mathop{\mathrm{Ext}}\nolimits ^{-\dim (X)}_ X(\mathcal{F}, \omega _ X^\bullet ) \\ & = \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}[\dim (X)], \omega _ X^\bullet ) \\ & = \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}[\dim (X)], f^!(\omega _ A^\bullet )) \\ & = \mathop{\mathrm{Hom}}\nolimits _ S(Rf_*\mathcal{F}[\dim (X)], \omega _ A^\bullet ) \\ & = \mathop{\mathrm{Hom}}\nolimits _ A(H^ d(X, \mathcal{F}), \omega _ A) \end{align*}
The first equality because $H^ i(\omega _ X^\bullet ) = 0$ for $i < -\dim (X)$, see Lemma 48.22.4 and Derived Categories, Lemma 13.27.3. The second equality is follows from the definition of Ext groups. The third equality is our choice of $\omega _ X^\bullet $. The fourth equality holds because $f^!$ is the right adjoint of Lemma 48.3.1 for $f$, see Section 48.19. The final equality holds because $R^ if_*\mathcal{F}$ is zero for $i > d$ (Cohomology of Schemes, Lemma 30.20.9) and $H^ j(\omega _ A^\bullet )$ is zero for $j < -\dim (A)$.
$\square$
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