Dimensions of triangulated categories with respect to subcategories

TL;DR

This paper defines a new notion of dimension for triangulated categories relative to subcategories, providing bounds and extending known results to broader algebraic contexts.

Contribution

It introduces the concept of dimension with respect to subcategories and applies it to derive bounds for derived categories over various rings.

Findings

Bounds for the dimension with respect to contravariantly finite subcategories

Bounds for the dimension with respect to resolving subcategories

Extension of known results to non-commutative rings

Abstract

This paper introduces the concept of the dimension of a triangulated category with respect to a fixed full subcategory. For the bounded derived category of an abelian category, upper bounds of the dimension with respect to a contravariantly finite subcategory and a resolving subcategory are given. Our methods not only recover some known results on the dimensions of derived categories in the sense of Rouquier, but also apply to various commutative and non-commutative noetherian rings.