How to construct a closed subscheme, or a coherent subsheaf, with prescribed germs

Abstract

We show that a closed subscheme of a given locally noetherian scheme can be constructed by prescribing it germs at all points of the ambient scheme in a manner consistent with specialization of points, provided the resulting set of all associated points of all the germs is locally finite. More generally, we prove a similar result for constructing a coherent subsheaf of a coherent sheaf by prescribing its stalks at all points in a manner consistent with specializations of points, again provided the set of all associated points of all the corresponding local quotients is locally finite. On any locally noetherian scheme, we show that there exists a unique global section of any coherent sheaf which has a prescribed family of germs which is consistent with specialization of points. It is not clear how to formulate an analogous result for constructing a coherent sheaf in terms of prescribed stalks. Even when the set of all associated points of all the prescribed stalks is locally finite, such a construction need not succeed as we show with an example. This raises the question of how to set up effective descent data for coherent sheaves purely in terms of germs and their specializations.