Bongartz Intervals in τ-Tilting Theory

• Bongartz intervals are structural tools in τ-tilting theory that organize support τ-tilting pairs between a τ–rigid pair and its left Bongartz completion.
• They provide a framework to understand mutation operations and the organization of torsion classes in finite-dimensional module categories.
• Their study links combinatorial lattice structures and silting theory by enabling minimal left approximations and the construction of maximal green sequences.

A Bongartz interval is a structural object in the theory of $\tau$-tilting modules over finite-dimensional algebras, organizing the set of support $\tau$-tilting pairs lying between a given basic $\tau$-rigid pair and its relative left Bongartz completion. These intervals play a fundamental role in the combinatorial and categorical study of torsion classes, module mutations, and maximal green sequences in module categories and silting theory (Cao et al., 2022).

1. Preliminaries and Definition of Bongartz Intervals

Let $A$ be a basic finite-dimensional algebra over a field $K$, and work in the module category $\text{mod}\,A$. The Auslander–Reiten translate is denoted $\tau$. A pair $(U, Q)$ is called a basic $\tau$-rigid pair in $\text{mod}\,A$ if $U$ is a module with $\operatorname{Hom}_A(U, \tau U) = 0$ and $Q$ is a projective module satisfying $\operatorname{Hom}_A(Q, U) = 0$.

Given such a pair $(U, Q)$, the associated subcategory

$W = U^\perp \cap {}^\perp(\tau U) \cap Q^\perp$

is a wide subcategory of $\text{mod}\,A$, where for a class $X$, $X^\perp = \{ M \mid \operatorname{Hom}_A(X,M)=0 \}$ and $${}^\perp(X) = \{ M \mid \operatorname{Hom}_A(M,X)=0 \}$$.

For any basic $\tau$-tilting pair $(M, P)$ with $$\operatorname{Fac} M \subset {}^\perp(\tau U) \cap Q^\perp$$, the relative left Bongartz completion $B^-_{(U,Q)}(M,P) = (M^-, P^-)$ is the unique basic $\tau$-tilting pair whose torsion class satisfies $$\operatorname{Fac} M^- = \operatorname{Fac} U * (W \cap \operatorname{Fac} M)$$, where

$$X * Y = \{ \text{extensions of objects in } Y \text{ by objects in } X \}.$$

Given $(U, Q)$ and $(M, P)$ as above, the Bongartz interval is defined by

$[(U, Q), B^-_{(U,Q)}(M,P)] = \{ \text{support $\tau$-tilting pairs } (N,R) \mid (U,Q) \leq (N,R) \leq B^-_{(U,Q)}(M,P) \}$

with $(U,Q) \leq (N,R) \leq (M^-,P^-)$ iff $$\operatorname{Fac} U \subset \operatorname{Fac} N \subset \operatorname{Fac} M^-$$.

2. Torsion Classes, Mutation, and the $\tau$-Tilting Poset

A full subcategory $\mathcal{T} \subset \text{mod}\,A$ is a torsion class if it is closed under extensions and quotients. Torsion classes are called functorially finite if they are both covariantly and contravariantly finite. The assignment $(M,P) \mapsto \operatorname{Fac} M$ induces a bijection between basic $\tau$-tilting pairs and functorially finite torsion classes, ordered by inclusion of their $\operatorname{Fac}$-classes.

Every torsion class $\mathcal{T}$ with $$\operatorname{Fac} U \subset \mathcal{T} \subset {}^\perp(\tau U) \cap Q^\perp$$ corresponds to a unique basic $\tau$-tilting pair containing $(U,Q)$ as a direct summand.

3. Structure and Extremal Cases of Bongartz Intervals

Consider the following extremal examples:

• Absolute left Bongartz completion: When $(M,P) = (0,A)$, the relative completion $B^-_{(U,Q)}(0, A)$ is called the absolute left Bongartz completion (or Bongartz co-completion), with torsion class $\operatorname{Fac} U$.
• Classical completions: If $(U,Q) \preceq (A,0)$, then $W = \text{mod}\,A$, recovering classic completions for $U \subset \text{add}\,A$.

Generally, the Bongartz interval $[(U,Q), B^-_{(U,Q)}(M,P)]$ contains all support $\tau$-tilting pairs with torsion classes between $\operatorname{Fac} U$ and $$\operatorname{Fac} U * (W \cap \operatorname{Fac} M)$$.

4. Compatibility with $\tau$-Tilting Mutation

Relative left Bongartz completions are compatible with the mutation structure on $\tau$-tilting pairs. An irreducible left mutation of a $\tau$-tilting pair $(M,P)$—replacing an indecomposable summand—corresponds to advancing from $\operatorname{Fac} M$ to the unique covering torsion class $\operatorname{Fac} N \subset \operatorname{Fac} M$. The following dichotomy holds for mutations:

• Either $N^- = M^-$ (the Bongartz completion does not change), or
• $N^-$ is a left mutation of $M^-$.

This is summarized in the following commutative diagram, where the horizontal arrows denote mutations and the vertical arrows denote left Bongartz completions:

(N, R) ──→ (M, P) │ │ ↓ ↓ (N^-, R^-) ──→ (M^-, P^-)

The proof relies on transferring the covering relations on torsion classes to those in the wide subcategory $W$, establishing that the brick labeling remains invariant except when the brick lies outside $U^\perp$ (Cao et al., 2022).

5. Illustrative Example

Let $A$ be the bound quiver algebra on the quiver $1 \to 3 \to 2 \to 1$ with relations forcing length-$2$ paths to zero. Denote by $S_i$ (resp. $P_i$) the simple (resp. projective) $A$-modules. Consider the chain of left mutations:

$$(0,A) \to (S_3,\,P_2 \oplus P_1) \to (S_3 \oplus P_3,\,P_2) \to (S_3 \oplus P_2 \oplus P_3,\,0) \to (A,0)$$

This sequence realizes the maximal green sequence of torsion classes

$$0 \subset \operatorname{Fac} S_3 \subset \operatorname{Fac}(S_3 \oplus P_3) \subset \operatorname{Fac}(S_3 \oplus P_2 \oplus P_3) \subset \text{mod}\,A.$$

Fixing $U = P_1$, $Q = 0$, and noting ${}^\perp(\tau U) \cap Q^\perp = \text{mod}\,A$, the left Bongartz completions $B^-_{(P_1,0)}$ at each step preserve $P_1$ as a direct summand. The associated reduced algebra $A_{(P_1,0)} \cong K(3'\to2')$ possesses its own maximal green sequence $$0 \subset \operatorname{Fac} S_{3'} \subset \text{mod}\,A_{(P_1,0)}$$.

6. Applications to Maximal Green Sequences and Silting Theory

A maximal green sequence for a torsion class $\mathcal{T} \subset \text{mod}\,A$ is a chain of torsion classes

$$0 = \mathcal{T}_0 \subset \mathcal{T}_1 \subset \dots \subset \mathcal{T}_m = \mathcal{T}$$

corresponding bijectively to a sequence of left mutations from $(0,A)$ to the $\tau$-tilting pair for $\mathcal{T}$. Cao–Wang–Zhang established that if ${}^\perp(\tau U) \cap Q^\perp$ has a maximal green sequence, then the reduction algebra

$$A_{(U,Q)} := \operatorname{End}_A(M) / \operatorname{End}_A(M) e_U \operatorname{End}_A(M)$$

(where $(M,Q)$ is the absolute right Bongartz completion) also admits such a sequence.

In silting theory, under the standard bijection between two-term silting objects in $K^{\text{b}}(\text{proj}\,A)$ and support $\tau$-tilting pairs in $\text{mod}\,A$, relative left Bongartz completions correspond to minimal left approximations in the derived category. The compatibility of Bongartz intervals and mutation phenomena extends fully to this setting, preserving the combinatorial and categorical structures (Cao et al., 2022).

7. Significance and Combinatorial Structure

The Bongartz interval $[(U,Q), B^-_{(U,Q)}(M,P)]$ organizes all support $\tau$-tilting pairs containing $(U,Q)$ and contained in a specified completion, structuring the support $\tau$-tilting poset into subintervals with mutual compatibility under mutation. These intervals behave analogously to intervals in a lattice and facilitate applications to the analysis and construction of maximal green sequences under reduction. The approach provides a unified perspective on support $\tau$-tilting theory, mutation combinatorics, wide subcategories, and connections with silting objects in triangulated categories (Cao et al., 2022).