Maximal τ₍d₎-Rigid Pair in Homological Algebra

• Maximal τ₍d₎-rigid pairs are defined as objects in d-cluster-tilting subcategories that satisfy strict orthogonality and rigidity conditions, ensuring no enlargements preserve τ₍d₎-rigidity.
• They exhibit a bijective correspondence with functorially finite d-torsion classes and provide the structure for canonical (d+1)-term silting complexes in higher homological algebra.
• They bridge higher homological theory with classical tilting and silting frameworks, influencing applications such as d-APR tilting, derived equivalences, and combinatorial classifications.

A maximal $τ_d$-rigid pair generalizes the classical concept of support $\tau$-tilting pairs to the higher homological context associated with $d$-cluster-tilting subcategories and the higher Auslander–Reiten translation $τ_d$. These pairs play a foundational role in relating categorical torsion theories, rigidity under higher Auslander–Reiten duality, and silting theory in both classical and higher representation theory. Maximal $τ_d$-rigid pairs are precisely characterized by their orthogonality conditions and maximality properties, encode bijections with functorially finite $d$-torsion classes, and yield canonical $(d+1)$-term silting complexes, unifying several core objects in higher homological algebra (August et al., 3 Feb 2026).

1. Definition and Core Properties

Let $A$ be a finite-dimensional algebra over a field, fix $d\geq1$, and denote the higher Auslander–Reiten translation by

$$τ_d: \underline{\mathrm{mod}}\,A \longrightarrow \overline{\mathrm{mod}}\,A,$$

with $τ_d M$ defined via a minimal projective resolution as the kernel of $\nu(f_{-d})$, where $\nu=-\otimes_A DA$ (August et al., 3 Feb 2026, Jacobsen et al., 2018). A pair $(M,P)$ with $M\in\mathrm{mod}\,A$ and $P$ projective is called:

• $τ_d$-rigid if

$\Hom_A(M,τ_d M)=0, \quad \Hom_A(P,M)=0.$

• Maximal $τ_d$-rigid (within a functorially finite $d$-cluster-tilting subcategory $\mathcal M\subseteq\mathrm{mod}\,A$) if $(M,P)$ is $τ_d$-rigid, and for every $N\in\mathcal M$ and every projective $Q$, the conditions

$N\in\add M \;\Longleftrightarrow\; \Hom_A(M,τ_d N)=0,\, \Hom_A(N,τ_d M)=0,\, \Hom_A(P,N)=0,$

$Q\in\add P \;\Longleftrightarrow\; \Hom_A(Q,M)=0$

are satisfied (August et al., 3 Feb 2026, Jacobsen et al., 2018, Rundsveen et al., 2024, Zhou et al., 2020).

Maximality ensures no further enlargement of $(M,P)$ preserves $τ_d$-rigidity; these pairs are thus maximal objects in the poset of $τ_d$-rigid pairs under direct sum inclusions.

2. Relationship with $d$-Torsion Classes and Classification

Maximal $τ_d$-rigid pairs exhibit a bijective correspondence with basic functorially finite $d$-torsion classes in $d$-cluster-tilting subcategories. Let $U\subseteq\mathcal M$ be such a $d$-torsion class (i.e., closed under minimal $d$-extensions and quotients [Jasso, Jørgensen axioms]). The correspondence (August et al., 3 Feb 2026):

$U \longmapsto (M_U, P_U)$

where

• $M_U$: the basic $\Ext_A^d$-projective generator of $U$.
• $P_U$: the maximal projective satisfying $\Hom_A(P_U, U)=0$.

This map is injective and $|M_U|+|P_U|=|A|$. The partial inverse recovers $U$ as $\Fac M\cap\mathcal M$ for a maximal $τ_d$-rigid pair $(M,P)$.

In the case $d=1$, these notions recover the Adachi–Iyama–Reiten bijection between support $\tau$-tilting pairs and functorially finite torsion classes (August et al., 3 Feb 2026, Jacobsen et al., 2018). For $d>1$, the theory is fundamentally different: classification requires higher homological structures and often combinatorial data tied to the specific algebra (e.g., diagonals in Nakayama or Auslander algebras) (Rundsveen et al., 2024).

3. Connection to $(d+1)$-Term Silting Complexes

Every maximal $τ_d$-rigid pair $(M,P)$ yields a canonical $(d+1)$-term silting complex in $\mathrm{K}^b(\mathrm{proj}\,A)$:

$$P_\bullet^{(M,P)} = (P_{-d}\to P_{-d+1}\to\cdots\to P_{-1}\to P_0) \oplus P[d],$$

where $P_\bullet^M$ is the minimal projective $d$-presentation of $M$, and $P[d]$ is the stalk complex with $P$ in degree $-d$ (August et al., 3 Feb 2026, Rundsveen et al., 2024). This complex satisfies: $\Hom_{K^b}(P_\bullet^{(M_U,P_U)},\,P_\bullet^{(M_U,P_U)}[i])=0\quad \forall\, i>0,$ and generates the bounded homotopy category by iterated cones.

Thus, maximal $τ_d$-rigid pairs provide a direct bridge to higher silting theory, with bijections to basic $(d+1)$-term silting complexes and $d$-torsion classes for a broad class of finite-dimensional algebras.

4. Combinatorial Models and Explicit Classification

Explicit classification is possible for specific higher Auslander and higher Nakayama algebras, leveraging combinatorial invariants:

• Higher Auslander algebras of type $\mathbb{A}$: Indecomposables are indexed by non-decreasing $(d+1)$-tuples. A functorially finite $d$-torsion class corresponds to subsets $I$ satisfying interval and extension conditions. Indecomposables in $(M_U, P_U)$ are those $x\in I$ with special projective or orthogonality properties (August et al., 3 Feb 2026).
• Linear Nakayama algebras $\Lambda(n,l)$: For a $d$-cluster-tilting subcategory, maximal $τ_d$-rigid pairs $(M,P)$ are those where $|M|+|P|=n$ (number of simples), and combinatorially characterized by local admissibility rules between “diagonals” in the Auslander–Reiten quiver and forbidden intervals for projectives (Rundsveen et al., 2024). Mutation and enumeration are algorithmically tractable, and explicit graphical encodings can be constructed.

The table below summarizes the characterization framework in these main examples:

Algebra Type	Indecomposable Description	Maximal $τ_d$-rigid Criterion
$A_n^d$ (type $\mathbb{A}$)	Non-decreasing $(d+1)$-tuples	Subset $I$ with interval/extension, projective/orthogonality
$\Lambda(n,l)$ (Nakayama)	Intervals/diagonals/subsets	$|M|+|P|=n$, combinatorial rules (forbidden intervals)

5. Interpretation via Higher Angulated and Abelian Categories

In the context of $(d+2)$-angulated or $(n+2)$-angulated categories with a (higher) cluster tilting object $T$, functors of the form $\Hom(T,-)$ induce $d$-cluster-tilting (or $n$-abelian) subcategories in module categories. Here, maximal $τ_d$-rigid pairs correspond precisely to maximal $d$-self-perpendicular objects, and the theory is a natural generalization of cluster-tilting/tilting/silting correspondences seen in classical representation theory (Jacobsen et al., 2018, Zhou et al., 2020).

For $d=1$, this reduces to classical $\tau$-tilting. In $2n$-Calabi–Yau $(n+2)$-angulated categories, the correspondence extends the picture to maximal $τ_n$-rigid pairs, support $τ_n$-tilting, and $n$-rigid/self-perpendicular objects (Zhou et al., 2020). Thus, maximal $τ_d$-rigid theory unifies the higher homological generalizations of silting/tilting/torsion theory in both purely abelian and higher angulated settings.

6. Applications: $d$-APR Tilting, Slices, and Derived Equivalences

• $d$-APR Tilting: For a simple projective $P$, the $d$-APR tilt $T_d^P=τ_d^{-1}P\oplus Q$ arises as the $Ext_A^d$-projective generator of a faithful split $d$-torsion class, and corresponds to a maximal $τ_d$-rigid pair (August et al., 3 Feb 2026).
• Slices in $d$-Cluster Tilting Categories: Slices define split $d$-torsion classes whose $Ext_A^d$-projective generator gives a $d$-tilting module, yielding new derived equivalences even when global dimension exceeds $d$ (August et al., 3 Feb 2026).
• Unification with Silting and Tilting: Maximal $τ_d$-rigid pairs accumulate the structure of both classical tilting and modern silting theory, mediating key derived and homological equivalences across cluster-tilting, silting, and torsion-theoretic frameworks (August et al., 3 Feb 2026, Jacobsen et al., 2018).

These features highlight the centrality of maximal $τ_d$-rigid pairs in the architecture of higher representation theory and the ongoing expansion of the homological toolkit for finite-dimensional algebras.