Tilting pairs and Wakamatsu tilting pairs of subcategories over cleft extensions

TL;DR

This paper studies how tilting pairs of subcategories behave over cleft extensions of abelian categories, unifying known results and providing new characterizations in module and ring extensions.

Contribution

It proves that the functor in a cleft extension preserves and reflects tilting pairs, unifying existing results and extending them to new contexts.

Findings

Functor $l$ preserves and reflects tilting pairs under certain conditions

Characterizations of tilting pairs over $ heta$-extensions of rings

Recovery of earlier results with new conclusions

Abstract

Let $(\mathcal{B},\mathcal{A}, i, e, l)$ be a cleft extension of abelian categories. We prove that the functor $l$ preserves and reflects (Wakamatsu) tilting pairs of subcategories under certain conditions, unifying an abundance of known results. Then, we apply our results to the cleft extensions of module categories, and give characterizations of tilting pairs and Wakamatsu tilting pairs over $\theta$-extension of rings and tensor rings, which not only recover the earlier results in this direction, but also obtain some new conclusions.