Level Inequalities for Complexes
Zachary Nason
TL;DR
The paper establishes universal upper bounds for the level of homologically bounded complexes with respect to projective, injective, flat, and various Gorenstein module classes over noetherian rings, expressing these bounds in terms of the homology dimensions of the complex. The main approach combines Adams resolutions, thick subcategory theory, and Ghost/Coghost lemmas to relate levels to classical homological dimensions, yielding sharp inequalities such as $level_{R}^{\mathsf{P}}(M) \leq \operatorname{pd}_R(\mathrm{H}(M)^\oplus) + 1$, $level_{R}^{\mathsf{I}}(M) \leq \operatorname{id}_R(\mathrm{H}(M)^\oplus) + 1$, and $level_{R}^{\mathsf{C}}(M) \leq \max\{2, \mathsf{C}-\text{dim}(\mathrm{H}(M)^\oplus) + 1\}$ for $\mathsf{C} \in \{\mathsf{F}, \mathsf{GP}, \mathsf{GI}, \mathsf{GF}\}$. As applications, the authors derive Bass-type formulas for injective and Gorenstein injective levels and prove the optimality of all bounds, including universal bounds over regular local rings and Gorenstein local rings. The work leverages the derived-category framework to connect level, depth, and dimension in a cohesive, general setting.
Abstract
We prove that for all noetherian rings, the level of any homologically bounded complex $M$ with respect to the collection of projective or injective modules is bounded above by the projective dimension of $\bigoplus H(M)$ plus one or the injective dimemsion of $\bigoplus H(M)$ plus one, respectively. In addition, we also prove that if $\mathsf{C}$ is the collection of flat, Gorenstein projective, Gorenstein injective, or Gorenstein flat modules, then the level of any homologically bounded complex $M$ is bounded above by the maximum of 2 or the $\mathsf{C}$-dimension of $\bigoplus H(M)$ plus 1. These results give universal bounds for the projective, injective, and flat levels over regular local rings, and give universal bounds for the Gorenstein projective, Gorenstein injective, and Gorenstein flat levels over Gorenstein local rings. As an application of the above results, we prove a version of the Bass Formula for complexes with respect to injective level and Gorenstein injective level. We also show that the bounds achieved for each homological and Gorenstein homological level considered is optimal.