Fp-projective periodicity

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 $\mathsf{P}\mathsf{r}\mathsf{o}\mathsf{j}$-periodic module is projective, any fp-injective $\mathsf{I}\mathsf{n}\mathsf{j}$-periodic module is injective, and any $\mathsf{C}\mathsf{o}\mathsf{t}$-periodic module is cotorsion. It is also known that any pure $\mathsf{P}\mathsf{P}\mathsf{r}\mathsf{o}\mathsf{j}$-periodic module is pure-projective and any pure $\mathsf{P}\mathsf{I}\mathsf{n}\mathsf{j}$-periodic module is pure-injective. Generalizing a result of Saroch and Stovicek, we show that every $\mathsf{F}\mathsf{p}\mathsf{P}\mathsf{r}\mathsf{o}\mathsf{j}$-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 $\mathsf{F}\mathsf{p}\mathsf{P}\mathsf{r}\mathsf{o}\mathsf{j}$-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 $\mathsf{P}\mathsf{P}\mathsf{r}\mathsf{o}\mathsf{j}$-periodic module over a regular finitely generated commutative algebra (or a hereditary finite-dimensional associative algebra) over a field need not be pure-projective.