Extension dimensions under singular equivalences and recollements

TL;DR

The paper investigates how the extension dimension $\operatorname{ext.dim}$ of Artin algebras behaves under recollements of derived categories and under singular equivalences of Morita type with level. By introducing homological widths and the invariant $\Omega\text{-}ext.dim$, the authors derive concrete inequalities linking $\operatorname{ext.dim}$ of a middle algebra to those of its recollement components and establish invariance results for $\Omega\text{-}ext.dim$ and $\operatorname{ext.dim}(\Omega^{\infty}(-))$ under level-based singular equivalences. Key contributions include explicit bounds when recollements extend downward (and upward), as well as equalities in special finite-dimension scenarios, and a sharp invariance result under singular equivalences of Morita type with level. These results enhance understanding of how representation-theoretic complexity, as measured by extension dimensions, transfers through standard algebraic constructions and provide tools for assessing proximity to representation-finite behavior.

Abstract

The extension dimensions of an Artin algebra give a reasonable way of measuring how far an algebra is from being representation-finite. In this paper we mainly study extension dimensions linked by recollements of derived module categories and singular equivalences of Morita type with level, and establish a series of new inequalities and relationships among their extension dimensions.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Lemma 2.6
• Definition 2.7