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Hephaistos
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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 $${\rm Ext}^i({\cal F},\omega_X^0)\to H^{n-i}(X,{\cal F}) \ ?$$$${\rm Ext}^i({\cal F},\omega_X^0)\to H^{n-i}(X,{\cal F})^* \ ?$$ (where $n=\dim(X)$ and $\cal F$ is any coherent sheaf on $X$).

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 $X$).

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 $X$).

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Hephaistos
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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 $X$).

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 $n=\dim(X)$).

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 $X$).

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Hephaistos
  • 739
  • 3
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A reference for Serre duality

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 $n=\dim(X)$).