AIR tilting subcategories of extended hearts
Jiaqun Wei, Yu Zhou
TL;DR
This work unifies and extends tilting theory across broad triangulated settings by introducing AIR tilting subcategories of the $d$-extended heart $d-\mathcal{H}$ and establishing a bijection with $(d+1)$-term silting subcategories of the ambient subcategory $\mathcal{K}$. Central to the approach is the $\mathcal{P}$-presentation framework and the $s$-torsion pair $(\mathcal{T}(\mathcal{S}),\mathcal{F}(\mathcal{S}))$, which links AIR tilting to torsion theory via $\operatorname{Fac}_d(-)$. The paper then defines quasi-tilting and tilting subcategories of $d-\mathcal{H}$, proving structural results such as closedness properties of $\operatorname{Fac}_d(\mathcal{M})$, and shows that tilting subcategories coincide with $(d+1)$-term silting subcategories of $\mathcal{K}$, providing multiple equivalent characterizations. Applications to both extended finitely generated modules over finite-dimensional algebras and extended large modules over rings demonstrate that the framework recovers and extends classical notions like $\tau_{[d]}$-tilting, support $\tau$-tilting, and silting modules, thereby offering a broad, cohesive infrastructure for tilting theory in $d$-extended and extriangulated settings.
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 $τ_{[d]}$-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.