Silting and tilting objects in cleft extensions of abelian categories

TL;DR

This paper explores the relationship between silting and tilting objects in abelian categories and their cleft extensions, providing methods to construct new objects and characterizing modules over ring extensions.

Contribution

It introduces a framework connecting silting and tilting objects in abelian categories with their cleft extensions, extending known results to ring extensions.

Findings

Established connections between silting and tilting objects in abelian categories and their cleft extensions.

Provided methods for constructing additional silting and tilting objects in extended categories.

Characterized silting and tilting modules over ring $ heta$-extensions, extending previous results.

Abstract

We establish connections between silting and tilting objects in an abelian category $\mathcal{B}$ and those in a cleft extension $\mathcal{A}$ of $\mathcal{B}$, which provides a method for constructing more silting and tilting objects. Then we apply our results to the cleft extensions of module categories, and characterize silting and tilting modules over $\theta$-extension of rings. Some known results over trivial extension of rings are extended and strengthened.