Flat complexes, pure periodicity and pure acyclic complexes

The paper can be viewed as an addition and extension of the recent paper of Emmanouil (2016) [5]. Among others, an alternative functor category approach to Emmanouil's results is presented. By applying an idea of Neeman (2008) [16] and the main results obtained there, we prove that a chain complex F in a locally finitely presented Grothendieck category A is pure acyclic if and only if any chain map f:P→F from a complex P of pure-projective objects in A to F is null-homotopic. As a consequence we prove that any pure periodic object in A is pure-projective. Moreover, we show that A is pure semisimple if and only if A has the pure QF-property, that is, every pure-injective object in A is pure-projective.