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Let $X$ be a smooth projective variety over $k$, let $\mathcal{O}_x$ be the skyscraper sheaf of the residue field $k(x)$ at a closed point $x$, and let $\omega_X$ be the canonical bundle. Assuming possibly that $\omega_X$ is ample, I believe there is an isomorphism between $\mathcal{O}_x$ and $\mathcal{O}_x \otimes \omega_X$ (at least in the derived category of coherent sheaves on $X$). Can someone provide a reference for this fact?

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The sheaf $O_x\otimes\omega_X$ is also skyscraper supported at $x$, and its stalk at $x$ is $k(x)\otimes \omega_X\simeq k(x)$ (because $\omega_X$ is locally free of rank $1$). So $O_x\otimes\omega_X$ is isomorphic to $O_x$. Am I wrong ?

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  • $\begingroup$ Ah, of course. Thank you! $\endgroup$ – user314 Nov 16 '12 at 15:01

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