Generalised APR-Tilting Overview

• Generalised APR-tilting is a homological construction that generalizes classical APR tilting to more complex categorical and geometric settings.
• It employs mutations, translations, and duality strategies to construct tilting modules and establish derived equivalences across various algebraic contexts.
• This approach underpins classification schemes in module theory and connects representation theory with noncommutative geometry and higher Auslander–Reiten theory.

Generalised APR-tilting refers to a wide class of homological constructions generalising the celebrated Auslander–Platzeck–Reiten (APR) tilting procedure. Classically, APR-tilting produces a new tilting module by replacing a distinguished projective (typically at a source or sink in a quiver) with its Auslander–Reiten translate and thus constructs a derived equivalence between certain module categories. In its generalised forms, APR-tilting operates in more elaborate categorical settings—including higher homological algebra, functor categories, non-classical (e.g., extriangulated) environments, and geometric contexts—using analogous but more flexible mechanisms of mutation, translation, and homological duality.

1. Classical APR-Tilting and its Generalisation

The classical APR-tilting module over a finite dimensional algebra $A$ with a simple noninjective projective module $S$ is

$T = Q \oplus \tau^{-1}S$

where $Q$ is the direct sum of all indecomposable projectives other than $S$ and $\tau$ is the Auslander–Reiten translation. This module $T$ is tilting: $\operatorname{Ext}^1_A(T,T) = 0$, $T$ has $n$ non-isomorphic summands (for $n$ simple $A$-modules), and $\operatorname{pdim}_A T \leq 1$.

Generalised APR-tilting extends this paradigm by:

• Replacing $S$ with more general projective modules or summands (e.g., additively generated by a primitive idempotent $e$ such that $Ae = P_S$ has all composition factors isomorphic to a fixed simple $S$) (Li, 2012).
• Performing the operation in more elaborate settings like functor categories (Martínez-Villa et al., 2011), higher Auslander–Reiten theory (Mizuno, 2012), derived and dg-categories (Nicolas et al., 2012), weighted projective lines (Geng, 2019), or geometric contexts such as twisted flag varieties (Novaković, 2015).
• Allowing for "mutations" governed by combinatorial data from quivers with (graded) potentials, which generalises the classical reflection to more intricate situations (Mizuno, 2011, Guo et al., 2019).

2. Tilting in Functor Categories and Extensions of the Brenner–Butler Theorem

Classical tilting theory and the Brenner–Butler theorem establish equivalences between torsion-theoretic subcategories of module categories and their tilts. Generalising to functor categories $\operatorname{Mod}(C)$, with $C$ a small preadditive category, a full subcategory $\mathcal{T} \subset \operatorname{Mod}(C)$ is tilting if:

• Every $T\in\mathcal{T}$ is finitely presented and $\operatorname{pdim} T \leq 1$.
• $\operatorname{Ext}^1_C(T_1,T_2)=0$ for all $T_{1,2} \in \mathcal{T}$.
• Every representable functor $( -, C)$ admits a short $\mathcal{T}$-resolution:

$$0 \longrightarrow ( -, C) \longrightarrow T_0 \longrightarrow T_1 \longrightarrow 0,\qquad T_{0,1}\in \mathcal{T}.$$

This notion extends APR phenomena, enabling tilting-equivalences in categories such as $\operatorname{Mod}(\operatorname{mod}\Lambda)$ (Martínez-Villa et al., 2011).

Functorial equivalences $$F : \operatorname{Mod}(C)\to \operatorname{Mod}(\mathcal{T})$$ and its adjoint $G$ generalise the change-of-rings picture:

$$F(M) = \operatorname{Hom}_C(-,M)|_{\mathcal{T}}, \qquad G(N) = N\otimes_{\mathcal{T}} \mathcal{T}.$$

This allows for equivalence between torsion/perpendicular classes:

$$\mathcal{F} = \{M \mid \operatorname{Hom}_C(T, M) = 0~\forall~T\in\mathcal{T}\} \leftrightarrow \text{corresponding subcategory in}~\operatorname{Mod}(\mathcal{T}).$$

In this functorial setting, APR-tilting corresponds to taking perpendicularly-located subcategories—fundamental to the generalised framework and to modular representation theory (Martínez-Villa et al., 2011).

3. Higher-Dimensional and Cluster-Tilting Generalisations

In higher Auslander–Reiten theory and $n$-cluster tilting settings, APR-tilting is extended further:

• An $n$-APR tilting module for a finite-dimensional $A$ of $\operatorname{gl.dim}A\leq n$ is constructed via projective resolutions of higher AR translates:

$T_k = \tau_n^{-}P_k \oplus (DA/P_k)$

for a simple projective $P_k$ (sink), with $\tau_n^{-}$ the $n$-Auslander–Reiten translate (Mizuno, 2012, Mizuno et al., 2015).

• Quivers with relations of such $n$-APR tilts are described by modifications to the classical procedure, leading to additional arrows and relations determined by the higher projective resolutions. This generalises BGP reflection functors (Mizuno, 2012).

Moreover, mutations and tilts in this context can be interpreted via combinatorics of quivers with potentials (QPs), with explicit mutation rules and invariants for graded algebras encoding tilting phenomena (Mizuno, 2011, Guo et al., 2019). This links generalised APR-tilting to cluster theory, Calabi–Yau categories, and the roots of Euler forms.

4. Homological and Derived-Categorical Perspectives

The most general treatments view APR-tilting as a special case of derived equivalences induced by appropriate (dg-)tilting bimodules (Nicolas et al., 2012). If $T$ is a dg $B$–$A$-bimodule, equivalences of the derived categories $\mathcal{D}(A) \simeq \mathcal{D}(B)$ arise under homological balance conditions, generalising classical finite type algebraic contexts to differential graded and "large" module settings.

These abstract frameworks also clarify when such tilting objects induce semiorthogonal decompositions, recollements, or derived equivalences, and recover a swath of tilting and reflection phenomena—subsume APR-tilting, classical tilting, and even recollement phenomena from topology and geometry.

5. Geometric and Combinatorial Realisations

Generalised APR-tilting interacts with geometry, particularly via tilting bundles and associated derived equivalences between coherent sheaves and module categories (Novaković, 2015). Construction and mutation of tilting bundles—such as on weighted projective lines of tubular type (Geng, 2019) or on Geigle–Lenzing projective spaces (Chen et al., 20 Mar 2025)—leverages the APR-mutation paradigm for geometric objects. Here, iterated APR mutations enable the construction and connectedness of the tilting graph for such objects, connecting representation theory, noncommutative geometry, and algebraic K-theory.

These geometric generalisations further demonstrate that tilting phenomena transfer between module-theoretic and derived categories, and that mutation operations correspond to deep alterations at the level of derived and t-structure hearts (Chen et al., 20 Mar 2025, Monroy et al., 26 Mar 2025).

6. Applications to Stratified and Directed Algebras

In the context of finite directed categories and stratified or triangular matrix algebras, a generalised APR-tilting module takes the form

$T = Q \oplus \tau^{-1}P_S$

with $P_S$ an indecomposable projective all of whose composition factors are a fixed simple $S$ (Li, 2012). Stratification (i.e., it's possible to write the algebra as an upper-triangular matrix algebra) ensures rigidity and enables explicit computation and verification of the tilting properties via control over their homological invariants.

The existence of such tilting modules has implications for torsion theories and for the structure of module categories over stratified and directed algebras.

7. Impact, Connections, and Further Directions

Generalised APR-tilting has produced:

• Systematic classification schemes for derived equivalence classes of algebras (using tilts, mutations, and reflection functors).
• Explicit computations for endomorphism algebras (e.g., Coxeter–Dynkin algebras via generalised APR-tilting over squid algebras, connecting canonical algebras and singularity theory (Perniok, 22 Sep 2025)).
• Construction of derived equivalences in non-classical settings, including functor categories, derived and dg-categories, and geometric contexts.

The universality of generalised APR-tilting is further highlighted by its compatibility with tensor product operations for higher APR-tilting modules (Lu, 2022), and its formalisation in stable homotopy theory, where such tilting results are consequences of stable axiomatic structures (Groth et al., 2014).

Table: Comparison of Generalised APR-Tilting Settings

Context	Key Construction	Distinguishing Properties
Module categories	Replace projective $P$ with $\tau^{-1}(P)$	Yields tilting module, reflects at source/sink
Functor categories	Use tilting subcategory $\mathcal{T}\subset\operatorname{Mod}(C)$	Equivalence between torsion pairs
Higher AR theory	$n$-APR tilting via higher $\tau_n^{-}(P)$	Connects to cluster, CY, and preprojective
Derived/dg-context	Tilting bimodules $T$ between dg-categories	Full derived/tria equivalence
Geometry/Flags	Tilting bundles, mutations via shift/twist	Derived equivalence of $\mathcal{D}^b(X)$, connects to exceptional collections

Generalised APR-tilting has unified and extended methods from classical module theory, combinatorics, noncommutative geometry, and algebraic topology, providing a toolkit for construction, mutation, and classification of algebras, their derived categories, and associated categorical invariants. The theory continues to see further development in higher homological algebra, categorical representation theory, and the paper of derived categories of stacks, schemes, and singular spaces.