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Hello!

I'd like to understand the relation between the following two theorems:

  • The "global" duality for projective schemes, as explained in [Hartshorne]: If $X$ is an equidimensional projective Cohen-Macaulay scheme of dimension $n$ over an algebraically closed field with dualizing sheaf $\omega_X$, then for all $i$ there is a natural isomorphism $Ext^i({\mathcal F},\omega_X)\cong H^{n-i}(X,{\mathcal F})^{\ast}$.

  • The "local" duality theorem Cohen-Macaulay rings, as explained in [Bruns, Herzog]: If $(R,{\mathfrak m},k)$ is a complete local ring of dimension $d$, then for all finite $R$-modules $M$ and all $i$ there is a natural isomorphism $\text{Ext}_R^i(M,\omega_R)\cong\text{Hom}_R(H_{\mathfrak m}^{d-i}(M),E(k))$, where $\omega_R$ is the canonical module of $R$ and $E(k)$ the injective hull of the residue field $k$.

The isomorpisms are strikingly similar, but I don't know if there is a rigorous way to deduce, say, the global duality from the local one. Can somebody explain this to me or give references?

Thank you!

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Local doesn't imply global. For noetherian $X$ with finite Krull dimension one can define "dualizing complex" $\omega_ X$ (which may or may not exist, a priori) in bounded derived category of sheaves of modules on $X$, and for proper Cohen-Macaulay $X$ over regular base (like field) it enjoys features of "global duality". For local $X$ it meshes well with constructions in commutative algebra (such as "local duality"), and its formation respects localization. So key point is the unifying etale-local notion of "dualizing complex". See early parts of Hartshorne's book "residues and duality". –  BCnrd May 26 '10 at 22:28
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A projective variety $X$ gives you a local ring, namely the localization of the affine cone over $X$ at zero. Thus it is tempting to speculate, that a G_m equivatiant version of loca dulaity implies the global one. –  Heinrich Hartmann May 27 '10 at 15:11
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up vote 3 down vote accepted

In the case of varieties over a perfect field this question is explained with great detail in the book by J. Lipman:

Dualizing sheaves, differentials and residues on algebraic varieties. (French summary) Astérisque No. 117 (1984)

known by some people as "Lipman's blue book". If you want to get rid of the base field then you should look at Greenlees May duality for schemes and Grothedieck duality for formal schemes, see joint work by Alonso, Jeremías & Lipman.

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