I would like some references for Serre duality for a proper noetherian scheme $X$ (over $\mathbb C$). I know that in this case there exists a dualizing sheaf. But can theorem 7.6, III, in Harshorne's "Algebraic Geometry" (Duality dor a Projective Scheme) be extended to the case of a proper noetherian Cohen-Macaulay scheme ? i.e are there natural functorial isomorphisms $${\rm Ext}^i({\cal F},\omega_X^0)\to H^{n-i}(X,{\cal F}) \ ?$$ (where $n=\dim(X)$ and $\cal F$ is any coherent sheaf on $X$).