Covers in finitely accessible categories

Abstract

Abstract. We show that in a finitely accessible additive category every class of objects closed under direct limits and pure epimorphic images is covering. In particular, the classes of flat objects in a locally finitely presented additive category and of absolutely pure objects in a locally coherent category are covering.

Relatively divisible and relatively flat objects in exact categories

“…Proof. Consider the exact structures on (S op , Ab) given by the classes D of all short exact sequences and E ⊆ D of pure exact sequences, and the D-perfect D-cotorsion pair (F, C) in (S op , Ab) [3, Theorem 3] (also see [15,Corollary 3.3]), where F is the class of flat objects and C is the class of cotorsion objects. Note that F coincides with the class of D-E-flat objects in (S op , Ab) and weakly absolutely pure objects are the same as D-E-A-divisible objects, where A = F. Then use [14,Theorem 4.9].…”

FP-Injective Complexes

“…Let P be a finitely generated projective R-module and S a finitely generated submodule of P. Then 0 → S → P → P/S → 0 is exact with P/S finitely presented. Consider the commutative diagram: (2) with the upper row exact. Then 0 → Hom R P/S C → Hom R P C → Hom R S C → 0 is exact.…”

Locally finitely presented categories with no flat objects

“…Despite of the Govorov-Lazard theorem, this class has homological significance as well. In [7,28] is proved, as a consequence, that F lat provides with minimal flat resolutions that are unique up to homotopy, so they can be used to compute right derived functors of the Hom functor. Also the existence of such minimal approximations with respect to F lat can be used to infer the existence of pure-injective envelopes in a locally finitely presented category (see [15]).…”