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 $\mathsf{Proj}$-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.

LaTeX 2e, 30 pages; v.2: Remark 4.11 and Example 6.9 inserted; v.3: end of Section 0.2, proof of Theorem 4.2, and Example 6.9 expanded; v.4: small corrections

References in corpus (2)

• On exact categories and applications to triangulated adjoints and model structures
• Differential graded Koszul duality: an introductory survey

Cited by in corpus (7)

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• Locally coherent exact categories
• Local, colocal, and antilocal properties of modules and complexes over commutative rings
• Homological theory of representations having pure acyclic injective resolutions
• Generalized periodicity theorems
• A contramodule generalization of Neeman's flat and projective module theorem