Phantom covering ideals in categories without enough projective morphisms
Sergio Estrada, Pedro A. Guil Asensio, Sinem Odabasi

TL;DR
This paper establishes conditions under which phantom map ideals in certain exact categories are covering, with applications to sheaf categories and a new approach for categories lacking enough projective morphisms.
Contribution
It introduces new criteria for phantom map ideals to be covering in locally presentable exact categories without relying on enough projective morphisms.
Findings
Identifies conditions for phantom map ideals to be covering
Applies results to categories of sheaves with geometric motivation
Develops a new approach for categories without enough projective morphisms
Abstract
We give sufficient conditions to ensure that the ideal of -phantom maps in a locally -presentable exact category is (special) (pre)covering ideal, where is an exact substructure of . 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.
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