Igusa-Todorov distances, extension and Rouquier dimensions under cleft extensions of abelian categories
Yajun Ma, Junling Zheng, Yu-Zhe Liu
TL;DR
This work studies how key homological invariants—$\mathop{\mathrm{IT.dist}}$, $\mathop{\mathrm{ext.dim}}$, and $\mathop{\mathrm{Rouq.dim}}$—behave under cleft extensions and coextensions of abelian categories. By introducing the Igusa-Todorov distance for abelian categories and exploiting endofunctors $\mathsf{F}$ and $\mathsf{F}'$ arising from cleft (co)extensions, it derives bounds under nilpotence and exactness assumptions, and extends these results to module categories with left-perfect, nilpotent $\mathsf{F}$. The paper applies the framework to Morita context rings, trivial and tensor extensions, and arrow removal, obtaining concrete upper bounds for the Rouquier dimension of finite-dimensional algebras and linking IT-distance with extension and derived dimensions. Overall, the results provide a unified mechanism to transfer and estimate representation-theoretic invariants across cleft-type constructions, with practical implications in common algebraic settings.
Abstract
In this paper, we investigate the behavior of Igusa-Todorov distances, extension and Rouquier dimensions under cleft extensions of abelian categories. Applications are given to Morita context rings, trivial extension rings, tensor rings and arrow removals. In particular, we provide a new characterization of the upper bound for the Rouquier dimension of finite-dimensional algebras.