# Comparison of Local and Global Duality

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!

• 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
• 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