Silting and tilting objects in cleft extensions of abelian categories

TL;DR

This work addresses constructing and transferring silting and tilting objects within cleft extensions of abelian categories, and applies the results to module categories over $\theta$-extensions and tensor rings. It develops lifting criteria that characterize when $l(B)$ is (partial) silting or silting in the extension in terms of $B$ and the endofunctor $F$, and establishes transfer results that descend silting properties from the extension back to the base category via the functor $q$, under precise hypotheses. The main contributions provide explicit equivalences and generation/closure conditions (involving $F(B)$ and $\mathcal{D}_{\sigma}$) that connect silting/tilting data across the cleft pair, and extend known results from trivial extensions to more general $\theta$-extensions and tensor rings, including connections to $\tau$-tilting theory. Collectively, these results offer a systematic approach to constructing and recognizing silting/tilting objects in complex module categories arising from ring- and category-theoretic extensions, with potential broad implications for homological and representation-theoretic problems.

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 $θ$-extension of rings. Some known results over trivial extension of rings are extended and strengthened.

Theorems & Definitions (32)

• Lemma 2.1
• proof
• Definition 2.2
• Definition 2.3
• Lemma 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Lemma 3.4