Lower bounds for levels of complexes by resolution dimensions
Yuki Mifune
TL;DR
The paper develops a general lower bound for the level of objects in the bounded derived category with respect to a resolving subcategory under a structural hypothesis. The key result shows $level_{D^b(A)}^X M \ge X-resol.dim M + inf M + 1$ for nonzero $M$, provided condition (A) holds. This theorem is then used to recover and extend known bounds for classical subcategories (Proj, GProj, Inj, GInj) and for semidualizing contexts (Add C, tref_C(R)) and to derive Foxby-compatible level equalities. The results also apply to contravariantly finite resolving subcategories and to orthogonal/self-orthogonal settings, broadening the scope of level-bounds in derived categories and providing tools for understanding constructions in homological algebra and representation theory.
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--Montaño--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.