Phantom covering ideals in categories without enough projective morphisms

Abstract

We give sufficient conditions to ensure that the ideal $\Phi (\mathcal{E})$ of $\mathcal{E}$-phantom maps in a locally $\lambda$-presentable exact category $(\mathcal{A},\mathcal{E})$ is (special) (pre)covering ideal, where $\mathcal{E}$ is an exact substructure of $(\mathcal{A},\mathcal{E})$. 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 [FGHT13], as the categories involved in most of the examples we are interested in do not have enough projective morphisms.

18pp