  We prove a sheaf-theoretic derived-category generalization of Greenlees-May
duality (a far-reaching generalization of Grothendieck's local duality
theorem): for a quasi-compact separated scheme X and a "proregular" subscheme
Z---for example, any separated noetherian scheme and any closed
subscheme---there is a sort of sheafified adjointness between local cohomology
supported in Z and left-derived completion along Z. In particular, the i-th
left-derived completion functor is the "local homology" sheaf
$Ext^i(\R\Gamma_Z\O_X, -)$. Sheafified generalizations of a number of duality
theorems scattered about the literature result, e.g., the Peskine-Szpiro
duality sequence (generalizing local duality), the Warwick Duality theorem of
Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck
Duality, we also get a generalization of a Formal Duality theorem of
Hartshorne, and of a related local-global duality theorem. In a sequel we will
develop the latter results further, to study Grothendieck duality and residues
on formal schemes.
