Finiteness of homological dimensions in triangulated categories

TL;DR

This paper explores the relationships between various homological dimensions in triangulated categories, providing inequalities that connect the dimensions of a middle category with those of related categories, extending known results.

Contribution

It establishes explicit inequalities relating finitistic, big finitistic, and global dimensions in recollements of triangulated categories, generalizing classical ring results.

Findings

Derived inequalities connect dimensions across categories

Unified framework extends classical homological results

Provides tools for analyzing categorical obstructions

Abstract

In a general triangulated category, the finiteness of the finitistic dimension serves as a prerequisite for a categorical obstruction, via the singularity category, to the existence of bounded $t$-structures. In this paper, we investigate the finitistic, big finitistic, and global dimensions, and establish explicit inequalities that relate these dimensions of the middle category in a recollement of triangulated categories to those of the outer categories. This provides a unified framework for extending some known results on the homological dimensions of ordinary rings to weakly approximable triangulated categories.