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Let X be a closed immersion of dimension n in P = *P*$^N_k$, where k is an algebraically closed field. Let $\omega_P$ denote the canonical bundle and A the local ring $\mathcal O_{P,x}$.

Then Hartshorne argues on p. 244 that the condition $$\mathcal {Ext}^i_P(\mathcal O_X, \omega_P) = 0 $$ for i>N-n implies that $$Ext^i_A(\mathcal O_{X,x}, A) = 0 $$

Why? I can see how he got from $\mathcal Ext$ to $Ext$. The former is the derived functor of the sheaf $\mathcal {Hom}$, while the latter is the derived functor of just Hom. Obviously, if $\mathcal Ext$ is zero so is any localization of Ext. But how does one get from $\omega_P$ to A?

Rodrigo
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1 Answers1

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$\omega_P$ is invertible, hence its stalks are free of rank $1$.

Martin Brandenburg
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  • Oh of course! For some reason I had an automorphism between top-forms and 1-forms in mind, which I thought would prove it but didn't seem right. But obviously top-forms are isomorphic to zero-forms, if one so will. – Rodrigo Jun 19 '13 at 07:12