AIR tilting subcategories of extended hearts

TL;DR

This paper introduces AIR tilting subcategories in extended hearts of t-structures, generalizing classical tilting theory to a broader triangulated category framework, establishing bijections with silting subcategories and exploring related tilting notions.

Contribution

It generalizes $ au_{[d]}$-tilting theory to extended hearts of t-structures, connecting AIR tilting subcategories with silting subcategories in a broad triangulated setting.

Findings

Established a bijection between AIR tilting and silting subcategories.

Defined and analyzed quasi-tilting and tilting subcategories in extended hearts.

Extended classical tilting concepts to a more general triangulated category context.

Abstract

We introduce the notion of AIR tilting subcategories of extended hearts of $t$-structures on a triangulated category associated with silting subcategories. This notion generalizes $\tau_{[d]}$-tilting pairs of extended finitely generated modules over finite-dimensional algebras to a more general framework, which includes both extended large modules over unitary rings and truncated subcategories of finite-dimensional derived categories of proper non-positive differential graded algebras. Within this setting, we establish a bijection between AIR tilting subcategories and silting subcategories. Furthermore, we define quasi-tilting and tilting subcategories of extended hearts, extending the corresponding notions from module categories, and investigate their fundamental properties along with the relationships among these tilting-related classes.