Support τ-Tilting Poset

• Support τ-tilting posets are partially ordered sets constructed via Bongartz intervals from 2-term presilting complexes in finite-dimensional algebras.
• They leverage torsion classes and numerical g-vector characterizations to establish functorial behavior and cover relations essential for tilting theory.
• In τ-tilting finite cases, these posets form finite semidistributive lattices, revealing deep connections between combinatorial, algebraic, and stability properties.

A support $τ$-tilting poset arises from categorifying tilting theory within the framework of finite-dimensional algebras, particularly through the study of torsion classes and their interval structures. These posets are constructed using intervals defined by 2-term presilting complexes, termed “Bongartz intervals,” and a facial semistable order which generalizes the facial weak order in Coxeter groups. In $\tau$-tilting finite situations, the resulting facial semistable order forms a finite semidistributive lattice, revealing deep connections between representation theory, lattice theory, and stability conditions.

1. Torsion Classes and the Interval Structure

Let $A$ be a basic finite-dimensional algebra over a field. The set of torsion classes in $\mod A$, denoted $\tors A = \{\text{torsion classes }T\subseteq\mod A\}$, forms a complete lattice under inclusion. For any 2-term presilting complex $U \in K^{[-1,0]}(\proj A)$, one defines the Bongartz interval as follows: $\bigl[T^-,T^+\bigr] = \bigl[\Gen(H^0(U)),\,(H^{-1}(\nu U))^\perp\bigr] \subseteq \tors A$ where $T^- := \Gen(H^0(U))$ and $T^+ := (H^{-1}(\nu U))^\perp$, with $\nu$ the Nakayama functor. The $g$-vector $g(U) \in \mathbb{Z}^n \cong K_0(A)$ provides a numerical characterization: $T^- = T_{g(U)}$, $T^+ = \overline{T}_{g(U)}$, using King–Bridgeland–Asai’s numerical approach.

2. Bongartz Completion and Functorial Properties

For each presilting complex $U$, Proposition 4.8 guarantees a unique basic 2-term silting complex $B_1(U)$, the Bongartz completion, such that: $U \subseteq B_1(U) \qquad \text{and} \qquad \Gen(H^0(B_1(U))) = T^-$ Dually, the co-Bongartz completion $B_0(U)$ is characterized by $\Gen(H^0(B_0(U))) = T^+$. The assignment $U \mapsto [T^-,T^+]$ is functorial: if $U$ is a direct summand of $V$, then $[T^-,T^+] \subseteq [\Gen(H^0(V)), (H^{-1}(\nu V))^\perp]$. The endpoints of each interval are computed numerically for $\theta \in \mathbb{R}^n$ as

$$T_\theta = \{ X \in \mod A : \theta \cdot [X'] > 0 \ \forall\ \text{nonzero quotients } X'\}, \quad \overline{T}_\theta = \{X : \theta\cdot[X'] \ge 0\}$$

This numerical method allows recovery of the torsion classes bounding each Bongartz interval.

3. Facial Semistable Order and Cover Relations

Within the set of “semistable intervals”

$\{\,[T_\theta,\overline T_\theta] \mid \theta\in\mathbb{R}^n\,\} \subseteq \binuc(\tors A)$

the partial order is given by: $$[\theta] \le_{\text{fs}} [\eta] \iff T_\theta \subseteq T_\eta \ \text{and}\ \overline T_\theta \subseteq \overline T_\eta$$ For 2-term presilting cones, this is equivalently stated as: $C^+(U) \le_{\text{fs}} C^+(V) \iff \Gen(H^0(U)) \subseteq \Gen(H^0(V)),\ (H^{-1}(\nu U))^\perp \subseteq (H^{-1}(\nu V))^\perp$ Cover relations in the facial semistable order are characterized as follows: $U$ and $V$ yield adjacent intervals (Theorem 5.17) iff one is a direct summand of the other and they share either the same Bongartz completion or co-Bongartz completion. This cover-relation property is tightly coupled with changes in the rank of torsion-class strata by one, reflecting the addition or removal of indecomposable summands.

4. $\tau$-Tilting Finiteness, Lattice Properties, and Semidistributivity

Definition-Theorem 4.15 establishes that $A$ is $\tau$-tilting finite if all torsion classes are functorially finite, or equivalently if the $g$-vector fan is complete. Theorem 6.21 then asserts that in the $\tau$-tilting finite case, the facial semistable order $(TF(A),\le_{\rm fs})$ is a finite lattice. Central to the argument is the “BEZ Lemma” (Lemma 2.1), which reduces general meet computations to those for pairs beneath a common upper bound, resolved via the aforementioned cover-relation characterization.

Corollary 7.4 confirms that $(TF(A),\le_{\rm fs})$ is in fact a finite semidistributive lattice. General results (Theorem 7.7) show that full semidistributivity of $\tors A$ transfers to its binuclear interval order, yielding a highly structured combinatorial and algebraic object.

5. Explicit Example: Path Algebra of Type $A_2$

For $A=K(1\leftarrow2)$, the indecomposables are $S(1)$, $S(2)$, $P(2)$, with $P(2) \twoheadrightarrow S(1)$. The basic support $\tau$-rigid pairs (equivalently, 2-term presilting complexes) and their Bongartz intervals in $\tors A$ are:

Pair $(M,P)$	Bongartz Interval $[T^-,T^+]$
$(0,P(1)\oplus P(2))$	$[0,\,0]$
$(P(1),P(2))$	$[T(P(1)),\,T(P(1))]$
$(0,P(2))$	$[0,\,T(P(1))]$
$(S(2),P(1))$	$[T(S(2)),\,T(S(2))]$
$(P(2)\oplus S(2),0)$	$[T(P(2)),\,T(P(2))]$

Additionally, there are two “full-dimensional” intervals: $[0,\tors A]$ from $(0,0)$ and $[\tors A,\tors A]$ from $(P(1)\oplus P(2),0)$. These six intervals, under inclusion, reproduce the Hasse quiver of the five chambers and five rays in the $g$-vector fan of $A_2$, yielding an eight-element lattice—a semidistributive lattice of rank 2.

6. Partitioning and Binuclear Interval Orders in Abelian Length Categories

In any abelian length category, the facial semistable order constructed via Bongartz intervals can be partitioned into a set of completely semidistributive lattices, one of which coincides with the original lattice of torsion classes. This suggests a broader organizational principle for torsion-class partial orders in abelian settings, confirming the deep correspondence between numerical stability, interval orders, and lattice-theoretic properties (Hanson, 2023).

7. Research Connections and Generalizations

The framework generalizes the facial weak order for Coxeter groups to torsion-class intervals, advances facial semistable orders via stability conditions (King–Bridgeland–Asai), and interprets cover relations and Bongartz completions in terms of combinatorial lattice theory. A plausible implication is that this categorical and combinatorial methodology may extend to further representations of finite or abelian categories, facilitating new approaches to interval orders in algebraic and geometric contexts (Hanson, 2023).