## Mathematics, Department of

#### Date of this Version

11-13-2021

#### Citation

Published in Journal of Algebra vol 609, Pages 606 - 618, 1 November 2022

DOI:10.1016/j.jalgebra.2022.06.012

http://arxiv.org/abs/2109.05548v2

#### Abstract

We investigate the relationship between the level of a bounded complex over a commutative ring with respect to the class of Gorenstein projective modules and other invariants of the complex or ring, such as projective dimension, Gorenstein projective dimension, and Krull dimension. The results build upon work done by J. Christensen [7], H. Altmann et al. [1], and Avramov et al. [4] for levels with respect to the class of finitely generated projective modules.

The concept of level in a triangulated category, first defined by Avramov, Buch- weitz, Iyengar, and Miller [4], is a measure of how many mapping cones (equiva- lently, extensions) are needed to build an object from a collection of other objects, up to suspensions, finite sums, and retractions. This concept has its origins in the works of Beilinson, Bernstein, and Deligne [5], J. Christensen [7], Bondal and Van den Bergh [6], Rouquier [16], and others. In particular, the concept of level is implicit in Rouquier’s definition of dimension of a triangulated category.