---
title: Support τ-Tilting Poset
url: https://www.emergentmind.com/topics/support-tilting-poset
type: topic
---

# Support τ-Tilting Poset

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* [2305.06031].

## 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* [2305.06031].

Source: https://www.emergentmind.com/topics/support-tilting-poset