τ-tilting Theory in Finite-Dimensional Algebras

• τ-tilting theory is a mutation-based extension of classical tilting that classifies modules and torsion classes in finite-dimensional algebras.
• It establishes bijections between support τ-tilting modules, functorially finite torsion classes, and bricks using g-vector combinatorics.
• Applications include geometric realizations via simplicial complexes and criteria for τ-tilting finiteness across various algebraic settings.

τ-tilting theory is a mutation-theoretic generalization of classical tilting theory for finite-dimensional associative algebras. It provides a conjugate framework for classifying torsion classes, g-vector combinatorics, and brick modules, and is the foundation of moduli-theoretic and lattice-theoretic approaches to representation theory.

1. Core Definitions and Structural Objects

Let $A$ be a finite-dimensional algebra over a field $K$, $\mathrm{mod}\,A$ the category of finite-dimensional right $A$-modules, $\mathrm{proj}\,A$ its subcategory of projectives, and $\tau$ the Auslander–Reiten translate.

τ-rigid module: $M \in \mathrm{mod}\,A$ is τ-rigid if $\mathrm{Hom}_A(M,\tau M) = 0$.

τ-tilting module: $M$ is τ-tilting if it is τ-rigid and $|M| = |A|$, with $|M|$ the number of non-isomorphic indecomposable summands.

Support τ-tilting pair: $(M,P)$, with $M \in \mathrm{mod}\,A$, $P \in \mathrm{proj}\,A$, is support τ-tilting if $M$ is τ-rigid, $\mathrm{Hom}_A(P,M) = 0$, and $|M| + |P| = |A|$; $M$ is then a support τ-tilting module.

τ-tilting finite algebra: $A$ is τ-tilting finite if there are finitely many isomorphism classes of basic support τ-tilting modules.

Brick: An $A$-module $S$ with $\mathrm{End}_A(S)$ a division algebra.

In the context of moduli, the class of support τ-tilting modules is designed as a completion of the tilting modules, capturing the full mutation structure (Li et al., 2015).

2. Classification via Torsion Classes and Finite Finiteness Criteria

A foundational correspondence in τ-tilting theory establishes a bijection between support τ-tilting modules and functorially finite torsion classes. The precise statement (Adachi–Iyama–Reiten):

Support τ-tilting modules $M$	Functorially finite torsion classes $T$
$M$	$T = \mathrm{Fac}(M)$
$T$	$P(T)$ = Ext-projective generator of $T$

Thus, $$\mathrm{s}\tau\text{-}\mathrm{tilt}\,A \longleftrightarrow \mathrm{f}\text{-}\mathrm{tors}\,A$$ via $M \mapsto \mathrm{Fac}(M)$ (Demonet et al., 2015).

A finite-dimensional algebra $A$ is τ-tilting finite if and only if every torsion class (or torsion-free class) in $\mathrm{mod}\,A$ is functorially finite; this is equivalent to the finiteness of the lattice of torsion classes, or to the finiteness of bricks (Demonet et al., 2015). In particular:

• $A$ is τ-tilting finite $\Leftrightarrow$ brick $A$ is finite.
• Every indecomposable τ-rigid $A$-module corresponds bijectively to a brick whose minimal containing torsion class is functorially finite.

This establishes τ-tilting finiteness as a combinatorial and lattice-theoretic property, with explicit reductions possible for non-sincere algebras via idempotent quotients (Wang, 2019).

3. g-vectors, Simplicial Complexes, and the Geometry of Mutation

The combinatorial structure of τ-tilting theory is governed by the g-vectors of modules. For $M \in \mathrm{mod}\,A$ with minimal projective resolution $0 \to P_1 \to P_0 \to M \to 0$, the g-vector is $$g(M) = [P_0] - [P_1] \in K_0(\mathrm{proj}\,A) \cong \mathbb{Z}^{|A|}$$. Distinct τ-rigid modules have distinct g-vectors.

τ-tilting fan and simplicial complex $\Delta(A)$: The cones generated by the g-vectors of the indecomposable summands of each support τ-tilting module define a finite, pure $(n-1)$-dimensional simplicial complex $\Delta(A)$, where $n = |A|$ (Demonet et al., 2015). If $A$ is τ-tilting finite:

• $\Delta(A)$ is shellable and its realization $|\Delta(A)|$ is homeomorphic to the sphere $S^{n-1}$.
• The interiors of these cones decompose the sphere bijectively, directly encoding the partial order and Hasse quiver of support τ-tilting modules through geometric adjacency.

Maximal cones in $\Delta(A)$	Basic support τ-tilting modules
Cones of $g$-vectors from indecomposables	Correspondence via the silting bijection

Mutations between support τ-tilting modules correspond to adjacency of cones sharing codimension-one faces, interpretable as hyperplane-phased transitions (Demonet et al., 2015).

4. Applications: Module Classification, Example Calculations, and Explicit Bijections

Explicit computations in representation-finite or hereditary settings showcase the combinatorics:

• In $K[x]/(x^2)$, $|A| = 1$: only the regular and zero module are support τ-tilting, corresponding to an $S^0$ sphere (Demonet et al., 2015).
• For $KQ$ with $Q: 1 \to 2$, $|A| = 2$, the support τ-tilting modules and their g-vectors partition $S^1$ into three arcs, creating a cyclic chain in the Hasse diagram.

In the hereditary or cluster-tilted cases, support τ-tilting modules coincide with tilting modules, and mutations recover the classical and cluster-tilting structures.

The brick–τ-rigid correspondence enables enumeration and stratification of module categories: indecomposable τ-rigids correspond to functorially finite bricks, and their minimal generating torsion classes capture the entire stratified lattice of module varieties (Demonet et al., 2015, Mendoza et al., 2019).

5. Extensions: Theory in Abelian Categories and Generalizations

τ-tilting theory admits generalizations to Hom-finite abelian categories with enough projectives. Covariantly finite τ-rigid subcategories can be completed to support τ-tilting subcategories, which are in bijection with finitely generated torsion classes. Explicit completions and reduction procedures (e.g., Bongartz completion analogues) are provable in this context (Liu et al., 2020).

In the classical hereditary case, partial tilting subcategories admit tilting completions, and mutation phenomena restrict to the existence of exactly two distinct tilting subcategories containing an almost-complete partial tilting, recovering the classical tilting results (Liu et al., 2020).

6. Example Table: τ-tilting Finite Criteria Summary

Property	Equivalent Characterization	Reference
τ-tilting finite algebra	All torsion classes functorially finite	(Demonet et al., 2015)
τ-tilting finite algebra	Finite number of bricks	(Demonet et al., 2015)
τ-tilting finite algebra	Simplicial complex $\Delta(A)$ is sphere	(Demonet et al., 2015)
τ-tilting finite for hereditary	Representation-finite	(Wang, 2019)

These equivalences provide a geometric, combinatorial, and homological toolkit for classifying and understanding the structure of finite-dimensional algebras within the τ-tilting paradigm.

7. Summary: Significance and Theoretical Integration

τ-tilting theory unifies tilting modules, torsion class lattices, silting complexes, and cluster-tilting objects through the machinery of mutations, g-vectors, and combinatorial fans. The structure of support τ-tilting modules encodes the representation type of the algebra, the geometry of the associated g-vector sphere, the finiteness of brick modules, and the exact stratification of the module category.

The geometric realization of the mutation order, the shellability of $\Delta(A)$, and the brick–τ-rigid bijection are cornerstones for leveraging τ-tilting finite criteria in both module-theoretic and algebraic settings, enabling explicit descriptions of Hasse quivers, stratifications, and classification results in broad classes of finite-dimensional algebras (Demonet et al., 2015).