Greenlees-May duality of formal schemes, in Studies in duality on noetherian formal schemes and nonnoetherian ordinary schemes

Abstract. For a closed subscheme Z of a noetherian separated schemeX, let ΓZ be the functor of sections with support in Z, taking OX-modules to OX-modules. Inspired by a theorem of Greenlees and May [GM] about duality between local cohomology and local homology for modules over a commutative ring, we gave in [AJL] the result that on quasi-coherent complexes F the \homology localization" functor RHom(RΓZOX; F) is a left-derived functor of Z:=completion along Z, the corresponding map to ZF being such that its composition with the natural map F! RHom(RΓZOX; F) is the completion map F! ZF. We also showed how this unies and generalizes several other recorded duality theorems. Here we extend the result to an arbitrary noetherian separated formal scheme X, with \quasi-coherent " replaced by \lim−! of coherent." Given an open OX-ideal I, with : X̂I! X the completion by I and ΓI the functor lim−! Hom(OX=In;−), we deduce for any OX-complex F having coherent homology sheaves a canonical duality isomorphism RHom(E; F) −! RHom(RΓIE; F) (E 2 D(X)): The special case where I is an ideal of denition|so that = identity|plays an important part in the duality theory of coherent sheaves on formal schemes