In his paper Generalized Divisors and Biliaison Hartshorne states that if $X$ is a noetherian, embeddable (say projective) scheme of pure dimension which satisfies the conditions $S_2$ (Serre conditions) and $G_0$ (all of $X$'s local rings of dimension zero are Gorenstein rings), then the dualizing sheaf $\omega_X$ of $X$ is locally free of rank one at the generic points of $X$. The statement can be found in Definition-Remark 2.7 here Generalized Divisors and Biliaison from Hartshorne.
Whats the reason for that? And what can we say about the support of $\omega_X$ in terms of the generic points?
Some thoughts on this:
For the sake of simplicity assume that $X$ is of dimension one and defined over the field $k$ and thus finite over ${\mathbb{P}_k^1}$ with $f:X \to {\mathbb{P}_k^1}$. Then by 6.4.26 (b) from Qing Liu's book we have $\omega_X \cong \omega_f \otimes_{\mathcal{O}_X} f^*\omega_g$ where $g:{\mathbb{P}_k^1} \to \operatorname{Spec}(k)$ is the structure morphism and $\omega_g$ the one-dualizing sheaf of $g$ and $\omega_f$ the 0-dualizing sheaf of $f$. Since $\omega_g$ is invertible, the same is true for its pullback $f^*\omega_g$ and hence it is enough to prove that $\omega_f$ is locally free at the generic points of $X$. By 6.4.25 we have $$\omega_f \cong \mathcal{Hom}_{\mathcal{O}_{\mathbb{P}^1}}(f_*\mathcal{O}_X,\mathcal{O}_{\mathbb{P}^1})$$
considered as an $\mathcal{O}_X$-module via the multiplication into the argument. Therefor it would be enough to prove that $\mathcal{Hom}_{\mathcal{O}_{\mathbb{P}^1}}(f_*\mathcal{O}_X,\mathcal{O}_{\mathbb{P}^1})$ is locally free of rank one at every generic point of $X$. But I do not see why this is true only because $X$ is $S_1$ (which is enough here) and $G_0$.
Any help is appreciated! Thanks!
Best Answer
As requested, I am fleshing out the details from my comments. I have decided just to stick to the terminology and the conventions in the Stacks project. We start with some definitions in the local case.
Definition 1 [Stacks, Tag 0A7B]. Let $A$ be a noetherian ring. A dualizing complex $\omega_A^\bullet$ is a complex of $A$-modules such that
Definition 2 [Stacks, Tag 0A7M]. Let $(A,\mathfrak{m},k)$ be a noetherian local ring with a dualizing complex $\omega_A^\bullet$. We say that $\omega_A^\bullet$ is normalized if $\operatorname{\mathbf{R}Hom}_A(k,\omega_A^\bullet) = k[0]$.
By [Stacks, Tag 0A7F and Tag 0A7L], normalized dualizing complexes are unique up to isomorphism, and every dualizing complex is isomorphic to a shift of a normalized dualizing complex.
Definition 3 [Stacks, Tag 0DW3]. Let $(A,\mathfrak{m},k)$ be a noetherian local ring of dimension $d$ with a normalized dualizing complex $\omega_A^\bullet$. The module $$\omega_A := \mathbf{h}^{-d}(\omega_A^\bullet)$$ is called a dualizing module.
Dualizing modules are closely related to the following notion:
Definition 4 [Waldi, Definition 5.6]. Let $(A,\mathfrak{m})$ be a noetherian local ring of dimension $d$. A finitely generated $A$-module $K$ is a canonical module for $A$ if $$\operatorname{Hom}_A\bigl(K,E_A(A/\mathfrak{m})\bigr) \simeq H^d_{\mathfrak{m}}(A).$$
A canonical module, if it exists, is finitely generated and unique up to isomorphism [Waldi, Satz 5.16 and Bemerkung 5.7]. Canonical modules are also known as modules of dualizing differentials or dualizing modules; see [Aoyama, p. 85]. Dualizing modules in the sense of Definition 3 are indeed examples of canonical modules [Stacks, Tag 0DW3].
We now consider the global case.
Definition 5 [Stacks, Tag 0A87]. Let $X$ be a locally noetherian scheme. An object $\omega_X^\bullet$ of $D(\mathcal{O}_X)$ is called a dualizing complex if, for every affine open subset $U = \operatorname{Spec}(A) \subseteq X$, there exists a dualizing complex $\omega_A^\bullet$ for $A$ such that $\omega_X^\bullet\rvert_U$ is isomorphic to the image of $\omega_A^\bullet$ under $\widetilde{(\,\cdot\,)}\colon D(A) \to D(\mathcal{O}_U)$.
Definition 6 [Stacks, Tag 0AWH]. Let $X$ be a noetherian scheme with a dualizing complex $\omega_X^\bullet$. Let $n \in \mathbf{Z}$ be the smallest integer such that $\mathbf{h}^n(\omega_X^\bullet)$ is nonzero. The coherent sheaf $$\omega_X := \mathbf{h}^n(\omega_X^\bullet)$$ is called a dualizing sheaf. Note that $\omega_X$ depends on the choice of $\omega_X^\bullet$.
When $X$ is proper over a field, then the sheaf $\omega_X$ satisfies the definition of a dualizing sheaf in [Hartshorne, p. 241]; see [Stacks, Tag 0AWP]. On the other hand, under the same assumptions on $X$, $\omega_X$ does not satisfy Serre duality unless $X$ is Cohen–Macaulay and equidimensional by [Hartshorne, Theorem III.7.6].
Since a dualizing sheaf is not the same thing as a dualizing complex that happens to be a sheaf (in the sense that it is concentrated in one degree), the sheaf $\omega_X$ is also called a canonical sheaf. This terminology comes from the fact that if $X$ is a geometrically normal variety of dimension $d$, then $$\omega_X \simeq \mathcal{O}_X(K_X) \simeq j_*\Bigl(\bigwedge^d\Omega_{X_\mathrm{sm}}\bigr),$$ where $K_X$ is a canonical divisor on $X$, and where $X_\mathrm{sm}$ is the smooth locus of $X$ and $j\colon X_\mathrm{sm} \hookrightarrow X$ is the natural open embedding.
Another reason for the terminology canonical sheaf comes from the fact that the sheaf $\omega_X$ is the global version of a canonical module of a noetherian local ring, which was your original question.
Proposition 7. Let $X$ be a connected noetherian scheme of finite Krull dimension $d$ with a dualizing complex $\omega_X^\bullet$. For every point $x \in X$ in the support of the canonical sheaf $\omega_X$ associated to $\omega_X^\bullet$, the module $\omega_{X,x}$ is a canonical module for $\mathcal{O}_{X,x}$. In particular, if $X$ is equidimensional, then $\omega_{X,x}$ is a canonical module for $\mathcal{O}_{X,x}$ for every $x \in X$.
Proof. Let $n$ be the integer such that $\omega_X = \mathbf{h}^n(\omega_X^\bullet)$. After shifting $\omega_X^\bullet$, we may assume that $n = -d$. Then, for every point $x \in X$ such that $\dim(\mathcal{O}_{X,x}) = d$, we see that $\omega_{X,x}^\bullet$ is a normalized dualizing complex for $\mathcal{O}_{X,x}$ by [Stacks, Tag 0A7Z].
We now note that the support of $\omega_X$ is the union of irreducible components of $X$ of dimension $d$ by [Stacks, Tag 0AWK]. Thus, we have $\omega_{X,x} = \mathbf{h}^{-d}\omega_{X,x}^\bullet$ for every point $x \in X$ such that $\dim(\mathcal{O}_{X,x}) = d$, and hence $\omega_{X,x}$ is a dualizing module for $\mathcal{O}_{X,x}$, and is also a canonical module for $\mathcal{O}_{X,x}$ by [Stacks, Tag 0DW3]. Moreover, every point $y \in X$ contained in the support of $\omega_X$ specializes to a point $x \in X$ such that $\dim(\mathcal{O}_{X,x}) = d$, and $\omega_{X,y}$ is therefore a canonical module for $\mathcal{O}_{X,y}$ by the fact that canonical modules on local rings localize to localizations at every prime ideal in their support [Aoyama, Corollary 4.3].
Finally, the "in particular" statement follows from the fact that if $X$ is equidimensional, then the support of $\omega_X$ is all of $X$ by [Stacks, Tag 0AWK]. $\blacksquare$