Fp-projective periodicity

TL;DR

This paper investigates the properties of fp-projective periodic modules, showing they are weakly fp-projective in certain categories, and provides counterexamples illustrating limitations of pure-projectivity in specific algebraic contexts.

Contribution

It generalizes previous results by proving that all fp-projective periodic modules are weakly fp-projective in locally finitely presentable Grothendieck categories.

Findings

Fp-projective periodic modules are weakly fp-projective in certain categories.

Counterexamples show non-pure PProj-periodic modules may not be pure-projective.

In locally coherent categories, weakly fp-projective objects are fp-projective.

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 $\mathsf{Proj}$-periodic module is projective, any fp-injective $\mathsf{Inj}$-periodic module is injective, and any $\mathsf{Cot}$-periodic module is cotorsion. It is also known that any pure $\mathsf{PProj}$-periodic module is pure-projective and any pure $\mathsf{PInj}$-periodic module is pure-injective. Generalizing a result of Saroch and Stovicek, we show that every $\mathsf{FpProj}$-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 $\mathsf{FpProj}$-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 $\mathsf{PProj}$-periodic module over a regular finitely generated commutative algebra (or a hereditary finite-dimensional associative algebra) over a field need not be pure-projective.