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The phenomenon of periodicity, discovered by Benson and Goodearl, is linked to the behavior of the objects of cocycles in acyclic complexes. It is known that any flat Proj-periodic module is projective, any fp-injective Inj-periodic module is injective, and any Cot-periodic module is cotorsion. It is also known that any pure PProj-periodic module is pure-projective and any pure PInj-periodic module is pure-injective. Generalizing a result of Šaroch and Št'ovíček, we show that every FpProj-periodic module is weakly fp-projective. The proof is quite elementary, using only a strong form of the pure-projective periodicity and the Hill lemma. More generally, we prove that, in a locally finitely presentable Grothendieck category, every FpProj-periodic object is weakly fp-projective. In a locally coherent category, all weakly fp-projective objects are fp-projective. We also present counterexamples showing that a non-pure PProj …