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!