# On the dualizing sheaf of a curve

Let $X$ be a smooth projective surface in $\mathbb{P}^n$ and $C$ be an effective curve. I know that the dualizing sheaf, $\omega_C$ of $C$ is $\mathcal{E}xt^{n-1}_{\mathbb{P}^n}(\mathcal{O}_C,K_{\mathbb{P}^n})$ where $K_{\mathbb{P}^n}$ is the canonical sheaf on $\mathbb{P}^n$. As far as I have read (from some articles) that $\omega_C \cong \mathcal{E}xt^1_{X}(\mathcal{O}_C,K_X)$. But I do not understand why this is true. Could somebody help?

• Well, this supposes some basic knowledge of duality theory. I suggest Altman and Kleiman Introduction to Grothendieck duality theory.
– abx
Feb 16, 2014 at 19:46
• A more general fact (which can be generalized further) is that if $j:Z \hookrightarrow X$ is a closed immersion between Cohen-Macaulay projective schemes over a field $k$, with respective pure dimensions $d \le n$ then $\omega_{Z/k}=\mathcal{Ext}^{n-d}_X(O_{Z}, \omega_{X/k})$. The key is duality for finite morphisms beyond the finite flat case. The derived category framework (as in Hartshorne's R&D book) illuminates these constructions tremendously (and allows one to study the dualizing sheaf locally, which is ill-suited to the viewpoint of characterization by just a "global" property). Feb 16, 2014 at 20:39
• @user76758: Thank you for the answer. This is what I am looking for. Feb 16, 2014 at 20:43
• From another point of view, you can think of this as a generalization of the classical adjunction formula $K_C=K_X©|_C$ when $C$ is smooth. Feb 16, 2014 at 20:51
• @Arapura: Thank you for the answer. I am most interested in the case when $C$ is not reduced. I have read the above mentioned result in the reduced case although they did not give any reference. So, I wanted to know how general the result actually is. Feb 16, 2014 at 21:01

This of course can be found in any reference on duality (e.g. Hartshorne "Algebraic Geometry" chap. 3) but in Hartshorne's proof it's somehow mysterious where the $\mathscr{E}xt^{n-1}_P(\mathscr{O}, \omega_{P^n})$ comes from. You can explain it by studying relative duality for morphisms (e.g. Hartshorne "Residues and Duality"), in particular for the closed immersion $i:C\to \mathbb{P}^n$. Here is another argument (in a way it's the proof from Hartshorne's AG done backwards, so that we can see how this $\mathscr{E}xt^{n-1}$ appears).

Recall that a dualizing sheaf is a sheaf $\omega$ which satisfies $H^1(C, F)^\vee = Hom_C(F, \omega)$ for every coherent $F$ on $C$.

A thing to start with is duality on $P:=\mathbb{P}^n$: $$H^1(C, F)^\vee = H^1(P, F)^\vee = Ext^{n-1}_P(F, \omega_P).$$

We now need a natural isomorphism between $Ext^i_P(F, \omega_P)$ and $Hom_C(F, \omega)$ for some $\omega$. A good thing to do to compare $Hom$'s and $Ext$'s on $P$ and $C$ is to find a spectral sequence comparing the two. Let us look at the functors $$Coh(P) \to Coh(C) \to Ab$$ where the first functor is $\mathscr{H}om_P(\mathscr{O}_C, -)$ and the second is $Hom_C(F, -)$, their composition $Hom_P(F, -)$. The spectral sequence of this composition is $$E^{ij}_2 = Ext^i_C(F, \mathscr{E}xt^j_P(\mathscr{O}_C, G)) \Rightarrow Ext^{i+j}_P(F, G).$$ For $i+j=n-1$ and $G=\omega_P$ we have our group $Ext^{n-1}_P(F, \omega)$ on the right hand side, and we have $E^{0,n-1}_2 = Hom_C(F, \mathscr{E}xt^{n-1}_P(\mathscr{O}_C, \omega_P)$, which suggests we should take $\omega = \mathscr{E}xt^{n-1}_P(\mathscr{O}_C, \omega_P)$.

In fact, because $C$ is a smooth curve, the above spectral sequence just boils down a to universal coefficients'' short exact sequence $$0 \to Ext^1_C(F, \mathscr{E}xt^{n-2}_P(\mathscr{O}_C, \omega_P)) \to Ext^{n-1}_P(F, \omega_P) \to Hom_C(F, \mathscr{E}xt^{n-1}_P(\mathscr{O}_C, \omega_P))\to 0.$$

To prove that the second map is an isomorphism (for any $F$), we need to show that $\mathscr{E}xt^{n-2}_P(\mathscr{O}_C, \omega_P)= 0$, which is done in Hartshorne "Algebraic Geometry", chap. 3.

• @Achinger: Thank you for the answer, but $C$ is not necessarily a smooth curve. Feb 16, 2014 at 20:39
• @user46578 Right, sorry I missed that. Then you don't have the short exact sequence above and you have to show that $\mathscr{E}xt_P^j(\mathscr{O}_C, \mathscr{O}_P) = 0$ for $j<n-1$ which is still okay as $C$ is Cohen-Macaulay. Feb 16, 2014 at 21:18
• @Achinger: Thank you very much for the elaborate answer and the help. Feb 17, 2014 at 16:50