Fp-Projective periodicity

Abstract

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-periodic module over a regular finitely generated commutative algebra (or a hereditary finite-dimensional associative algebra) over a field need not be pure-projective.