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?

  • $\begingroup$ Well, this supposes some basic knowledge of duality theory. I suggest Altman and Kleiman Introduction to Grothendieck duality theory. $\endgroup$
    – abx
    Feb 16, 2014 at 19:46
  • 2
    $\begingroup$ 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). $\endgroup$
    – user76758
    Feb 16, 2014 at 20:39
  • $\begingroup$ @user76758: Thank you for the answer. This is what I am looking for. $\endgroup$
    – user46578
    Feb 16, 2014 at 20:43
  • 1
    $\begingroup$ 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. $\endgroup$ Feb 16, 2014 at 20:51
  • $\begingroup$ @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. $\endgroup$
    – user46578
    Feb 16, 2014 at 21:01

1 Answer 1


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.

  • $\begingroup$ @Achinger: Thank you for the answer, but $C$ is not necessarily a smooth curve. $\endgroup$
    – user46578
    Feb 16, 2014 at 20:39
  • $\begingroup$ @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. $\endgroup$ Feb 16, 2014 at 21:18
  • $\begingroup$ @Achinger: Thank you very much for the elaborate answer and the help. $\endgroup$
    – user46578
    Feb 17, 2014 at 16:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.