Higher Cotorsion Pairs on Image-Nilpotent \(η\)-Extensions

Abstract

Let \(\cB\) be an abelian category with enough projective and injective objects, and let $ \cA=\cB\ltimes_η\sF $ be an \(η\)-extension induced by a right exact endofunctor \(\sF\). We introduce uniform image-nilpotence for \(η\)-extensions and characterize it in terms of the nilpotence of the associative natural transformation \(η:\sF^2\to\sF\). Using the associated \(η\)-Loewy filtration, we prove a lifting theorem for right \(n\)-cotorsion pairs over image-nilpotent \(η\)-extensions and establish a corresponding completeness theorem. As applications, our results recover and unify related constructions for split nilpotent ring extensions, comma categories, formal triangular matrix rings, Morita context rings and classical trivial extensions.

arXiv comment29pages