LOCAL HOMOLOGY AND COHOMOLOGY ON SCHEMES

We present a sheafified derived-category generalization of Greenlees-M ay duality (a far-reaching generalization of Grothendieck's local dual ity theorem). for a quasi-compact separated scheme X and a ''proregula r'' subscheme Z-for example, any separated noetherian scheme and any c losed subscheme-there is a sort of adjointness between local cohomolog y supported in Z and left-derived completion along Z. In particular, l eft-derived completion can be identified with local homology, i.e., th e homology of RHcm(.)(R Gamma(Z)O(X), -). Generalizations of a number of duality theorems scattered about the literature result: the Peskine -Szpiro duality sequence (generalizing local duality), the Warwick Dua lity theorem of Greenlees, the Affine Duality theorem of Hartshorne. U sing Grothendieck Duality, we also yet a generalization of a Formal Du ality theorem of Hartshorne, and of a related local-global duality the orem. In a sequel we will develop the latter results further, to study Grothendieck duality and residues on formal schemes.