Kaplansky classes, finite character and ℵ1-projectivity

Kaplansky classes emerged in the context of Enochs' solution of the Flat Cover Conjecture. Their connection to abstract model theory goes back to Baldwin et al.: a class C of roots of Ext is a Kaplansky class closed under direct limits if and only if the pair (C, <=) is an abstract elementary class (AEC) in the sense of Shelah. We prove that this AEC has finite character in case C = C-perpendicular to' for a class C' of pure-injective modules. In particular, all AECs of roots of Ext over any right noetherian right hereditary ring R have finite character (but the case of general rings remains open). If (C, <=) is an AEC of roots of Ext, then C is known to be a covering class. However, Kaplansky classes need not even be precovering in general: We prove that the class D of all aleph(1)-projective modules (which is equal to the class of all flat Mittag-Leffler modules) is a Kaplansky class for any ring R, but it fails to be precovering in case R is not right perfect, the class (perpendicular to)(D-perpendicular to) equals the class of all flat modules and consists of modules of projective dimension <= 1. Assuming the Singular Cardinal Hypothesis, we prove that D is not precovering for each countable non-right perfect ring R.