Lower bounds for levels of complexes by resolution dimensions

TL;DR

This paper establishes lower bounds for the levels of objects in the bounded derived category of an abelian category, extending previous results and providing new bounds relative to specific subcategories.

Contribution

It introduces new lower bounds for levels in derived categories with respect to (co)resolving subcategories, extending prior work and applying to broader subcategories.

Findings

Recovered previous results on levels in derived categories.

Extended bounds to additional subcategories in abelian categories.

Provided a framework for estimating levels using resolution dimensions.

Abstract

Let $\mathcal{A}$ be an abelian category. Denote by $\mathrm{D}^{b}(\mathcal{A})$ the bounded derived category of $\mathcal{A}$. In this paper, we investigate the lower bounds for the levels of objects in $\mathrm{D}^{b}(\mathcal{A})$ with respect to a (co)resolving subcategory satisfying a certain condition. As an application, we not only recover the results of Altmann--Grifo--Monta\~{n}o--Sanders--Vu, and Awadalla--Marley but also extend them to establish lower bounds for levels with respect to some other subcategories in an abelian category.