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!



Comparison of Local and Global Duality
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.


