Abstract

We give sufficient conditions to ensure that the ideal Φ(E′) of E′-phantom maps in a locally λ-presentable exact category (A,E) is a (special) (pre)covering ideal, where E′ is an exact substructure of (A,E). As a byproduct, we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation. In particular, we deal with a Zariski-local notion of phantom maps in categories of sheaves. We would like to point out that our approach is necessarily different from [18], as the categories involved in most of the examples we are interested in, do not have enough projective morphisms.