$\tau$-tilting modules, depth and delooping level

Technical Summary: $\tau$-Tilting Modules, Depth and Delooping Level

Problem Statement
The paper addresses the finitistic dimension conjecture, a central open problem in the representation theory of finite-dimensional algebras. The conjecture posits that for any finite-dimensional algebra $A$, the supremum of the projective dimensions of all finitely generated modules with finite projective dimension (the finitistic dimension, denoted $\text{findim} A$) is finite.

Recent work by Gelinas (2022) introduced the concept of delooping level to provide bounds for the finitistic dimension. Specifically, Gelinas proved that for a finite-dimensional algebra $A$, the finitistic dimension of its opposite algebra $A^{\text{op}}$ is bounded by the depth of $A$ and the delooping level of $A$ (Theorem 1.1). However, counter-examples exist regarding the finiteness of delooping levels for general algebras, and the relationship between these homological invariants and specific classes of modules remains an area of active investigation.

The authors ask whether Gelinas's theorem can be generalized within the framework of $\tau$-tilting theory. Since every algebra $A$ is naturally a $\tau$-tilting module over itself, the authors seek to define analogous notions of depth and delooping level relative to an arbitrary $\tau$-tilting module $T$ and determine if these relative invariants bound the finitistic dimension of the endomorphism algebra $B = \text{End}_A(T)^{\text{op}}$.

Methodology
The authors employ the machinery of $\tau$-tilting theory, torsion pairs, and the theory of bricks and semi-bricks developed by Adachi, Iyama, and Reiten. The methodology proceeds through the following steps:

1. Definitions of Relative Invariants:

   • Relative Grade and Depth: For a $\tau$-tilting module $T$, the authors define the grade of a module $M$ in the subcategory $\text{Fac}T$ (generated by $T$) relative to $T$ using $\text{Ext}^i_A(M, T)$. The depth of $\text{Fac}T$ relative to $T$ is defined as the supremum of these grades over a specific semi-brick $S$ associated with $T$.
   • Relative Delooping Level: Utilizing the bijection between bricks in $\text{Fac}T$ and simple modules in a specific subcategory of $\text{mod} B$, the authors define the $k$-delooping level of modules in $\text{Fac}T$ relative to $T$. This involves syzygies relative to $T$ ($\Omega_T$) and stable retracts.

2. Homological Equivalences:
   The paper relies heavily on the equivalence of categories induced by the functor $\text{Hom}_A(T, -)$ between $\text{Fac}T$ and a subcategory of $\text{mod} B$. Specifically, Theorem 1.3 establishes a bijection between the bricks in the semi-brick decomposition of $\text{Fac}T$ and the simple modules in the subcategory $\text{Sub}(DT)$.

3. Resolution Techniques:
   The authors utilize minimal $T$-presentations (exact sequences where terms are in $\text{add} T$) to construct projective resolutions in $\text{mod} B$. Lemma 3.3 and Proposition 4.3 demonstrate how minimal $T$-presentations in $\text{mod} A$ translate into minimal projective presentations in $\text{mod} B^{\text{op}}$.

Key Contributions and Results

1. Generalization of Gelinas's Theorem (Theorem 1.2 & 3.4):
   The authors prove that the depth of $\text{Fac}T$ relative to $T$ is a lower bound for the finitistic dimension of $B^{\text{op}}$:
   $$\text{depth}_T(\text{Fac}T) \leq \text{findim}(B^{\text{op}})$$
   This generalizes Gelinas's result (where $T=A$) to any $\tau$-tilting module.

2. Upper Bound via Delooping Level (Theorem 1.4 & 4.5):
   The paper establishes that the delooping level of $\text{Fac}T$ relative to $T$ serves as an upper bound for the finitistic dimension of $B^{\text{op}}$:
   $$\text{findim}(B^{\text{op}}) \leq \text{delooping level}_T(\text{Fac}T)$$
   Furthermore, they show that the $k$-delooping level of the algebra $B$ is equal to the $k$-delooping level of the subcategory $\text{Fac}T$ relative to $T$.

3. Finiteness Conditions (Theorem 1.5 & 5.3):
   The central application of these bounds is a new criterion for the finiteness of the finitistic dimension. The authors prove that if the intersection of the subcategories $\text{Fac}T$ and $\text{Sub}T$ (cogenerated by $T$) is of finite representation type, then $\text{findim}(B^{\text{op}})$ is finite.

   • This implies that if $A$ is of finite representation type, then $\text{findim}(B^{\text{op}})$ is finite for any $\tau$-tilting module $T$.
   • This also implies that if $A$ is a minimal representation infinite algebra (infinite type, but all proper factor algebras are finite type), then $\text{findim}(B^{\text{op}})$ is finite.

4. Derived Equivalence and Symmetry (Proposition 5.6):
   The paper notes that if $T$ is a $\tau$-tilting module, the algebra $A$ (modulo the annihilator of $T$) and $B^{\text{op}}$ are derived equivalent. Consequently, the finiteness of the finitistic dimension is preserved between them.

Significance and Claims
The paper claims to provide a generalized version of Gelinas's theorem (Theorem 1.1) specifically tailored to the context of $\tau$-tilting modules. By introducing "depth relative to $T$" and "delooping level relative to $T$," the authors create a framework to bound the finitistic dimension of endomorphism algebras of $\tau$-tilting modules.

The primary significance lies in the application to the finitistic dimension conjecture. The authors demonstrate that for two significant classes of algebras—those of finite representation type and minimal representation infinite algebras—the finitistic dimension of the opposite algebra of any $\tau$-tilting endomorphism algebra is guaranteed to be finite. This offers a positive partial answer to the conjecture within the specific context of $\tau$-tilting theory, extending known results from the case where $T=A$ to a broader class of modules.

The authors remain modest regarding the general conjecture, noting that while they provide bounds and specific cases where finiteness holds, the general finiteness of the delooping level for arbitrary algebras remains an open question, as evidenced by existing counter-examples in the literature. The work serves as a bridge connecting the structural properties of $\tau$-tilting modules (specifically the representation type of $\text{Fac}T \cap \text{Sub}T$) to the homological finiteness of their endomorphism rings.