A generic sheaf representation for rings

Abstract

The representation is considered to be interesting for a given ring R when the stalks of its ringed space have sufficiently rich properties: local rings, integral domains, fields, and so on ... Very often, a sheaf representation theorem holds for a wide class of rings, but gives pertinent results just for a restricted class of rings. On a given spectrum Sp(R) of the ring R, there are generally several sheaves representing R. As previously noticed by Hofmann [3] and Mulvey [5], one of those sheaves can be defined in a rather canonical way, and in particular, the stalks of this structural sheaf are quotients of the original ring. A sheaf can also be defined in terms of generators and relations. The generators are some elements x E F(U) for some open subsets U of Sp(R). The relation which exists between two elements x E F(U) and y E F(V) is the knowledge of the biggest open subset W where xlw = Ylw, A subset E IT F(U) (where U runs through the open subsets of Sp(R» is a set of generators for F when every element x E F(U) is completely characterized by its relations with the various generators. This yields the definition of a Sp(R)-set as being a set E together with an equality