AIR Tilting Subcategories in Tilting Theory

• AIR tilting subcategories are full subcategories in abelian and triangulated settings that extend support τ-tilting theory by analyzing factor modules of support τ-tilting modules.
• They establish bijections with functorially finite torsion classes and τ-cotorsion pairs, unifying tilting, silting, and cotorsion theoretic frameworks.
• Their mutation and extension properties enable combinatorial classification and systematic transfer of results across various homological algebra contexts.

An AIR tilting subcategory is a full subcategory of an abelian, exact, triangulated, or extriangulated category—often a module category over an algebra or the heart/extended heart of a t-structure—structured as an extension of the support $\tau$-tilting theory of Adachi–Iyama–Reiten. AIR tilting subcategories systematically generalize tilting and silting subcategories, connecting them to cotorsion theory, torsion classes, silting theory, and relative settings such as extended hearts and extriangulated contexts.

1. Fundamental Definitions and Core Properties

Let $\mathscr{A}$ be an abelian category with enough projectives, typical cases including $\mathscr{A} = \mathrm{mod}\,\Lambda$ for a finite-dimensional algebra $\Lambda$, or $\mathscr{A} = \mathrm{Mod}\,R$. The classical AIR–tilting setting is as follows.

• Support $\tau$-tilting module: A module $M$ is $\tau$-rigid if $\operatorname{Hom}_{\Lambda}(M, \tau M) = 0$; it is support $\tau$-tilting if, for some idempotent $e\in\Lambda$, $M$ is $\tau$-tilting over $\Lambda/(e)$, i.e., $M$ is $\tau$-rigid and $|M|+|P|=|\Lambda|$ for some projective $P$.
• AIR tilting subcategory $\mathcal{T}_M := \operatorname{Fac} M$, where $M$ is a basic support $\tau$-tilting module. Here, $\operatorname{Fac} M$ is the class of factor modules of finite direct sums of $M$.
• Generalization to subcategory level: In an abelian category with enough projectives, a full subcategory $\mathcal{T}$ is support $\tau$-tilting if:
  1. $$\operatorname{Ext}^1_{\mathscr{A}}(\mathcal{T},\mathcal{T}')=0$$ for all $\mathcal{T},\mathcal{T}'\in\mathcal{T}$.
  2. For each projective $P$, there is a short exact sequence $0\to K\to T_0\to P\to 0$ with $T_0\in\mathcal{T}$ and $T_0\to P$ a left $\mathcal{T}$-approximation.
  3. $\mathcal{T}$ is contravariantly finite in $\mathscr{A}$.

AIR tilting subcategories have the properties:

• Closed under extensions, factor modules, direct sums, and direct summands.

• For finite-dimensional algebras, every functorially finite torsion class arises as $\operatorname{Fac} M$ for a basic support $\tau$-tilting module $M$ (Adachi et al., 21 Oct 2024, Asadollahi et al., 2022).

2. Bijections and Correspondences

The core of AIR tilting theory consists of tight correspondences between various classes of objects and subcategories. The pivotal bijections include:

Category	Bijection with	Details/Reference
Support $\tau$-tilting subcats	Functorially finite torsion classes	$\mathcal{J}\mapsto \operatorname{Fac}\mathcal{J}$; [AIR], (Adachi et al., 21 Oct 2024)
Support $\tau$-tilting subcats	$\tau$-cotorsion pairs (with torsion class)	$$T\mapsto ({}^{\perp_1}\operatorname{Fac}T, \operatorname{Fac}T)$$ (Zhu et al., 6 Mar 2024)
Two-term silting subcats in $K^b(\mathrm{proj}\,\Lambda)$	Support $\tau$-tilting subcats in $\mathrm{mod}\,\Lambda$	$P\mapsto H^0(P)$; (Adachi et al., 21 Oct 2024, Iyama et al., 2013)
Tilting (n-tilting) subcategories	Coresolving, covariantly finite subcategories	Auslander–Reiten correspondence (Zhu et al., 2019)

The AIR bijection can be phrased as:

$\{\text{support $\tau$-tilting subcategories}\}\;\longleftrightarrow\;\{\text{functorially finite torsion classes}\}$

with the map $T\mapsto \operatorname{Fac} T$ invertible when restricting to functorially finite torsion classes (Adachi et al., 21 Oct 2024).

For extriangulated categories $(\mathscr{C},\mathbb{E},\mathfrak{s})$ with enough projectives and injectives, a support $\tau$-tilting subcategory $T$ satisfies:

• $T$ is a generator for $\operatorname{Defl}(T)$, the closure under deflations.
• $\mathbb{E}(T,\operatorname{Defl}(T))=0$.
• For each projective $P$, a right-exact $\mathbb{E}$-triangle $P\xrightarrow{f} T^0\to T^1$ with $T^i\in T$ and $f$ a left $T$-approximation (Zhu et al., 6 Mar 2024).

3. Connections to Cotorsion Theory, Torsion Classes, and Torsion Triples

• Cotorsion pairs: In both abelian and extriangulated contexts, $\tau$-cotorsion pairs $(U,V)$ define an object $U={}^{\perp_1}V$ and require approximation properties relative to projectives.
• In module categories, the triple $(U,V,\mathcal{F})$ (where $(U,V)$ is a $\tau$-cotorsion pair and $(V,\mathcal{F})$ a torsion pair) is in bijection with a support $\tau$-tilting subcategory $\mathcal{J}=U\cap V$ (Asadollahi et al., 2022, Zhu et al., 6 Mar 2024).
• The AIR framework thereby extends tilting–cotorsion correspondences and integrates the combinatorics of wide subcategories and universal localizations in finite settings (Marks et al., 2015).

4. AIR Tilting Subcategories in Broader Frameworks

AIR tilting subcategories have been extended and unified with several higher and relative structures:

• Relative and extended settings: AIR tilting subcategories are constructed for extended hearts in triangulated categories with respect to silting subcategories, generalizing to $d$-tilting and $\tau_{[d]}$-tilting pairs for $d$-extended module categories and derived categories of (dg-)algebras (Wei et al., 15 Dec 2025).
• Relative cluster-tilting and silting: Two-term (weak) relative cluster-tilting subcategories in triangulated categories correspond via the Yoneda functor to support $\tau$-tilting subcategories in functor categories (Zhou et al., 2018, Iyama et al., 2013, Yang et al., 2017).
• Extriangulated generality: The definitions and bijection mechanisms naturally extend to extriangulated categories, integrating exact, triangulated, and more general settings and inheriting their homological structures (Zhu et al., 6 Mar 2024, Zhu et al., 2019).

5. Explicit Examples and Applications

• Module categories: For $\mathrm{mod}\,\Lambda$ (finite-dimensional $\Lambda$), AIR-tilting subcategories are exactly the functorially finite torsion classes. For instance, in type $A_2$, every support $\tau$-tilting module determines a unique functorially finite torsion class, and vice versa (Adachi et al., 21 Oct 2024).
• Extended hearts and DG categories: In the bounded derived category $D^b(\mathrm{mod}\,A)$, with the $d$-extended heart $\mathcal{H}$, AIR tilting subcategories in $\mathcal{H}$ correspond bijectively to $(d+1)$-term silting subcategories in the thick subcategory generated by the silting generator $P$ (Wei et al., 15 Dec 2025).
• Restriction–extension across one-point extensions: For one-point extension algebras $A=B[P_0]$, restriction and extension functors transport tilting and support $\tau$-tilting subcategories between module categories of $A$ and $B$ (Asadollahi et al., 2022).

6. Structural Consequences, Combinatorics, and Mutation

• For representation-finite algebras, AIR tilting subcategories, functorially finite torsion classes, wide subcategories, and universal localizations are in bijection, allowing for classification via combinatorial invariants such as Hasse quivers and $g$-vectors (Marks et al., 2015).
• There is a theory of mutation for support $\tau$-tilting objects that generalizes tilting mutation, yielding a class of combinatorial moves in the poset of support $\tau$-tilting modules and their corresponding subcategories (Adachi et al., 21 Oct 2024).
• In extended settings, AIR tilting subcategories induce torsion pairs on extriangulated and extended heart categories, and their mutation is expected to parallel and extend classical silting and tilting mutation techniques (Wei et al., 15 Dec 2025).

7. Unification, Hierarchies, and Open Problems

AIR tilting subcategories sit at the intersection of classical tilting/silting, $\tau$-tilting, cotorsion theory, and cluster-tilting:

• Hierarchy: AIR tilting $\Rightarrow$ quasi-tilting $\Rightarrow$ (classical) tilting subcategories (Wei et al., 15 Dec 2025).
• Connections: The framework unifies and generalizes tilting theory, silting theory, cluster-tilting theory, support $\tau$-tilting theory, cotorsion pairs, and various relative or extended contexts (Wei et al., 15 Dec 2025, Zhou et al., 2018, Iyama et al., 2013).
• Open Problems: Current research addresses the structure theory of mutation in extended hearts, wall-and-chamber decompositions on $t$-structure spaces via AIR tilting subcategories, contravariant finiteness in more general triangulated or extriangulated categories, and the development of co-silting analogues (Wei et al., 15 Dec 2025).

AIR tilting subcategories thus serve as a central organizing concept for modern tilting theory, harmonizing combinatorial, homological, and categorical approaches across a broad swath of representation theory and homological algebra. The main technical instruments remain the compositions of functorial approximations, bifunctorial vanishing, closure properties under extensions and factors, and the equivalence of the bijection with cotorsion and torsion objects in the ambient category, allowing for a systematic transfer of structural results between different categorical frameworks (Zhu et al., 6 Mar 2024, Adachi et al., 21 Oct 2024, Wei et al., 15 Dec 2025, Asadollahi et al., 2022, Iyama et al., 2013).