AIR tilting subcategories of extended hearts (2512.13218v2)

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 extends $τ_{[d]}$-tilting pairs of extended finitely generated modules over finite-dimensional algebras to a broader framework, encompassing extended large modules over unitary rings as well as truncated subcategories of finite-dimensional derived categories of proper non-positive differential graded algebras. Within this framework, we establish a bijection between AIR tilting subcategories and silting subcategories. We further define quasi-tilting and tilting subcategories of extended hearts, generalizing the corresponding notions in module categories, and investigate their fundamental properties and the relationships among these three tilting-related concepts.

• The paper establishes a unified framework for AIR tilting subcategories by generalizing classical tilting and silting theory within extended hearts via (d+1)-term presentations.
• It introduces a categorical model using extriangulated structures to characterize d-factor classes and ensure closure and projectivity properties.
• The results pave the way for advances in higher homological algebra and representation theory, offering new insights into Auslander–Reiten theory and categorical dualities.

AIR Tilting Subcategories of Extended Hearts: An Expert Synopsis

Background and Motivations

This paper establishes a comprehensive new framework for AIR tilting subcategories within the extended hearts of $t$-structures in triangulated categories tied to silting theory. The author generalizes the deep interactions between torsion theory, silting subcategories, and tilting-type objects well beyond the classical scheme of finitely generated module categories over finite-dimensional algebras, extending to module categories over arbitrary unitary rings and truncated subcategories of derived categories associated to non-positive DG algebras.

Historically, tilting theory has been pivotal in representation theory for constructing equivalences between module and derived categories, with numerous generalizations (e.g., support $\tau$-tilting [AIR], silting modules [AMV], and higher extension functors). This work unifies these lines by formalizing AIR tilting subcategories, quasi-tilting subcategories, and tilting subcategories in the extended heart context. The theory here leverages the structure of extriangulated categories, thus enveloping broader cases than abelian or triangulated settings.

Silting Theory and Extended Hearts

The work develops within triangulated categories $D$ admitting a presilting (typically silting) subcategory $P$ such that the associated aisles/heart $H$ are well-defined. Let $K = thick(P)$ and, for integer $d>0$, work with the $d$-extended heart $\mathscr{H} := D^{[-d+1,0]} = H[d-1]*\cdots*H$, which inherits an extriangulated structure.

A foundational aspect is the characterization of silting subcategories in $K$ in terms of their closure properties and their ability to generate the entire category (analogously to the minimality in classical tilting objects). The extended heart $\mathscr{H}$ generalizes module categories and allows the introduction of higher structures such as $d$-factors and generalized torsion classes.

AIR Tilting Subcategories: Definitions and Main Results

AIR tilting subcategories are defined relative to a $(d+1)$-term $P$-presentation, generalizing the Adachi–Iyama–Reiten setup. The primary technical result is a bijection between:

• $(d+1)$-term silting subcategories of $K$ that are contravariantly finite in $D$;
• AIR tilting subcategories of the extended heart $\mathscr{H}$.

This correspondence, made explicit via the functor $H^{[-d+1,0]}$, subsumes previous bijections (e.g., between support $\tau$-tilting modules and two-term silting complexes) and demonstrates that in the extended heart, AIR tilting subcategories play the universal role analogous to tilting modules or silting objects in their respective settings. The uniqueness of $P$-presentations of AIR tilting subcategories is also established.

The paper further details the correspondence between $s$-torsion classes in the extriangulated setting and the AIR tilting subcategories, thus providing a categorical structure link parallel to the classical torsion theory in modules.

Quasi-Tilting and Tilting Subcategories

A quasi-tilting subcategory is defined via $\mathbb{E}$-projectivity within its $d$-factor class and a stability condition: $Fac_d(M) = Fac_{d+1}(M)$. AIR tilting subcategories are shown to be quasi-tilting, and the authors analyze closure properties (extensions, $d$-factors, direct summands) of the associated $d$-factor subcategories. This generalizes the relation between quasi-tilting and silting/tilting modules of [AMV, CDT].

Tilting subcategories in the extended heart (under suitable assumptions on projectives in $\mathscr{H}$) are characterized by:

• Having projective dimension at most $d$,
• Vanishing higher self-extensions,
• Generating all projectives via finite chains of extriangles.

This notion is shown to be equivalent to the classical concept in module categories and, crucially, equivalent to $(d+1)$-term silting subcategories in $K$. The work provides signals for the robust stability (via $s$-torsion classes containing all injectives) of such tilting subcategories, in line with classical observations (e.g., all injectives in the torsion class of a tilting module [Ba]).

Implications, Applications, and Further Directions

Practical and Structural Implications

The theoretical advances unify several standing frameworks—$\tau$-tilting, silting, and extended module categories—under a fully general, functorial model. This allows immediate generalization of known torsion-theoretic, homological, and categorical results to broader settings, including:

• Large module categories (arbitrary additive closure, e.g., $Add R$);
• Extriangulated contexts (hearts of $t$-structures, truncated derived subcategories);
• $d$-representation theoretic settings (higher Auslander–Reiten theory, higher extensions).

Strong formal consequences include the canonical closure and projectivity properties of the $d$-factor classes, explicit correspondence to $s$-torsion pairs, and conceptual clarity on the relationship between AIR tilting, quasi-tilting, and full tilting subcategories.

Theoretical Significance and Future Directions

The systematic bijection between silting subcategories and AIR tilting subcategories in arbitrary extended hearts creates a scaffolding for generalizing numerous results from classical tilting/silting theory (e.g., mutation, reduction, torsion-pair classification) to settings such as DG algebras, extriangulated categories, and derived categories with arbitrary coproducts.

The framework connects naturally to ongoing research in higher homological algebra, $d$-abelian categories, and categorifications in representation theory. Given the precise control over projective and injective objects, there is a clear path toward developing Auslander–Reiten theory in these extriangulated settings; connections to homological mirror symmetry and categorifications of cluster theory are also plausible.

Given the categorical flexibility, intrinsic in the reliance on extriangulated and extended heart structures, applications to AI, particularly in categorical semantics (e.g., categorical representations of neural architectures) and the formal study of higher symmetries in data, may become possible, although further work on concrete realizations is required.

Conclusion

This paper rigorously unifies and extends the landscape of tilting, silting, and torsion theory in module and derived categories to broad, functorial settings via the notion of AIR tilting subcategories in extended hearts. Its categorical results encapsulate and generalize key correspondences and closure properties that underpin modern tilting theory. The framework sets a robust stage for further explorations both in representation theory and in the extension and application of these ideas to broader categories and higher homological structures.