fp-projective periodicity

0575080 - MÚ 2025 RIV NL eng J - Článek v odborném periodiku
Bazzoni, S. - Hrbek, Michal - Positselski, Leonid
fp-projective periodicity.
Journal of Pure and Applied Algebra. Roč. 228, č. 3 (2024), č. článku 107497. ISSN 0022-4049. E-ISSN 1873-1376
Grant CEP: GA ČR(CZ) GA20-13778S
Institucionální podpora: RVO:67985840
Klíčová slova: fp-injective modules * fp-projective modules * locally finitely presentable abelian categories
Obor OECD: Pure mathematics
Impakt faktor: 0.8, rok: 2022
Způsob publikování: Omezený přístup
https://doi.org/10.1016/j.jpaa.2023.107497

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.
Trvalý link: https://hdl.handle.net/11104/0344935