Local cohomology and quasicoherent sheaves

Abstract

Local cohomology is usually defined only with respect to locally closed subsets of a (locally) noetherian scheme, which is amply sufficient for most applications in algebraic geometry. When dealing with ring theoretic problems more general subsets are needed, however. As a matter of fact, if one wishes to investigate the structure of reflexive sheaves on a Krull scheme, say, then one is inevitably led to consider the subset of all codimension 1 points, which is not necessarily locally closed. What all of these different types of subsets have in common is that they are so-called “generically closed” subsets ( = closed under generization). The behaviour of quasicoherent and coherent sheaves on generically closed subsets has been amply studied in [VVZ, VV3, V], where we have also introduced the localization of quasicoherent sheaves with respect to these. On the other hand, in [Sul, K. Suominen has introduced local cohomology with respect to arbitrary subsets of a scheme (actually, of an arbitrary ringed space!), with the purpose of applying this to general duality theory. The local cohomology groups and sheaves he obtains are also constructed through some localization in the category of sheaves on X. We will briefly recall the essentials about this below. One may of course wonder whether the two theories are connected in some way. At first glance, this may seem rather unlikely, since, e.g., the former theory only deals with quasicoherent sheaves, whereas the latter uses resolutions by arbitrary, not necessarily quasicoherent, injective sheaves. In this note, we prove the (maybe surprising) fact that on a locally noetherian scheme X the two “local cohomologies” with respect to an arbitrary closed subset Y coincide for any quasicoherent sheaf of modules on X. Actually, we prove a somewhat stronger result: in fact it suffices to