[论文解读] Silting and tilting objects in cleft extensions of abelian categories

Guoqiang Zhao, Juxiang Sun|arXiv (Cornell University)|Feb 7, 2026

Algebraic structures and combinatorial models被引用 0

一句话总结

这篇论文在分裂扩张及其基础阿贝尔范畴之间，将沉淀对象与倾斜对象联系起来，提供提升与转移沉淀/倾斜数据的方法，并将其应用于 theta-扩张与张量环。

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.

研究动机与目标

• Motivate the study of cleft extensions as a unifying framework for transfer of homological properties.
• Develop methods to lift (partial) silting and tilting objects from a base category to its cleft extension.
• Characterize when lifted objects remain silting or tilting in the extension.
• Apply the transfer results to cleft extensions in module categories, including theta-extensions and tensor rings.

提出的方法

• Review foundational notions of (co)silting and tilting objects in abelian and derived categories.
• Establish adjoint-pair based criteria for lifting objects along cleft extensions (Theorems 3.3 and 3.5).
• Characterize when lifted objects are silting or n-tilting using conditions on associated functors (F, q, l).
• Provide transfer results from extension to base category (Theorem 3.8 and supporting lemmas).
• Derive corollaries for cleft extensions that are also cleft coextensions (Corollary 3.11).
• Apply framework to theta-extensions and tensor rings to recover and extend known results on trivial extensions and triangular matrix rings.

实验结果

研究问题

• RQ1 How can silting and tilting objects in a base abelian category B be lifted to a cleft extension A?
• RQ2 What conditions ensure that lifted objects l(B) are (partial) silting or n-tilting in A?
• RQ3 When can silting/tilting properties transfer back from A to B?
• RQ4 How do these transfers specialize in module categories, particularly for theta-extensions and tensor rings?
• RQ5 Do known results for trivial extensions extend to the broader setting of cleft extensions and their duals?

主要发现

• Lifting results: l(B) is partial silting in A with respect to l(σ) iff B is partial silting in B with respect to σ and F(B) ∈ D_σ (Theorem 3.3).
• Lifting results: l(B) is silting in A with respect to l(σ) iff B is silting in B with respect to σ and F(B) ∈ Gen(B) (Theorem 3.3).
• n-tilting transfer: l(X) is partial n-tilting iff X is partial n-tilting and F(X) ∈ X^⊥_n (Theorem 3.5).
• Bidirectional transfer: if Ker q = 0, l(X) is n-tilting iff X is n-tilting and F(X) ∈ X^⊥_n (Theorem 3.5).
• Transfer of silting objects: if A is (partial) silting with respect to δ, then q(A) is (partial) silting with respect to q(δ) (Theorem 3.8).
• Special case: corollaries recover known results for cleft extensions and theta-extensions, including corollaries for when i preserves coproduct (Corollary 3.9) and dual statements (Corollary 3.11).