I'm currently reading Hartshorne, Algebraic Geometry, Chapter III.7 The Serre Duality Theorem.

I'm stuck at Corollary 7.7: If $X$ is a projective Cohen-Macaulay scheme of equidimension $n$ over $k$. Then for any locally free sheaf $\mathcal{F}$on X there are natural isomorphisms $H^i(X, \mathcal{F}) \cong H^{n-i}(X, \mathcal{F}^{\vee}\otimes \omega_X^\circ)'$

Now his "Proof" is just

Use $(6.3)$ and $(6.7)$.

Trying this I get the following:

  1. $H^i(X, \mathcal{F}) \cong \text{Ext}^i(\mathcal{O}_X, \mathcal{F})$ by $(6.3)$
  2. $H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X^\circ)' \cong \text{Ext}^i(\mathcal{F}^\vee \otimes \omega_X^\circ, \omega_X^\circ)$ by Theorem $7.6$
  3. $\text{Ext}^i(\mathcal{F}^\vee \otimes \omega_X^\circ, \omega_X^\circ) \cong \text{Ext}^i(\omega_X^\circ, \mathcal{F} \otimes \omega_X^\circ)$ by $(6.7)$

But how do I know conclude, that the first and the last term are the same?


So it seems I got it backwards: If you start with applying Thm 7.6 to $H^i(X, \mathcal{F})$ you get:

$$H^i(X, \mathcal{F}) = Ext^{n-i}(\mathcal{F}, \omega_X^\circ)' = Ext^{n-i}(\mathcal{O}_X, \mathcal{F}^\vee \otimes \omega_X^\circ)' \cong H^{n-i}(\mathcal{F}^\vee \otimes \omega_X^\circ)'$$


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