June 10, 2026
Let \(A\) be a finite-dimensional basic algebra over an algebraically closed field \(K\), \(T\) a finitely generated \(\tau\)-tilting right \(A\)-module and \(B={\rm End}_A T\). Denote by \({\rm Fac}T\) the subcategory of finitely generated right \(A\)-modules generated by \(T\). We define the depth relative to \(T\) and the delooping level relative to \(T\) and show that the finitistic dimension of the opposite algebra of \(B\) is bounded by the depth of \(\mathrm{Fac}T\) relative to \(T\) and the delooping level of \(\mathrm{Fac}T\) relative to \(T\). We give applications to the finitistic dimension conjecture. More precisely, we show that if \(A\) is a minimal representation infinite algebra or an algebra of finite representation type, then the finitistic dimension of \(B^{op}\) is finite.
In 2014, Adachi, Iyama and Reiten [1] introduced \(\tau\)-tilting theory as a generalization of tilting theory from the viewpoint of mutation. From then on \(\tau\)-tilting theory became popular in the representation theory of finite-dimensional algebras. In the \(\tau\)-tilting theory, \(\tau\)-tilting modules (resp. support \(\tau\)-tilting modules) are the most important objects. For \(\tau\)-tilting modules over special algebras, we refer to [2]–[5] and the reference therein. For the infinitely generated case, that is, the silting modules, we refer to the works of [6]. For the connection between \(\tau\)-tilting modules and homological conjectures, we refer to [7], [8].
For a finite-dimensional algebra \(A\), the finitistic dimension of \(A\) is the supremum of the finite projective dimension of finitely generated modules. It is conjectured that the finitistic dimension of \(A\) is always finite. This is the famous finitistic dimension conjecture which is still open now. For the study of the finitistic conjecture, we refer to [9]–[13] and the references therein. In 2022, Gelinas [14] introduced a new homological notion delooping level to study the finitistic dimension. And he proved the following theorem [14].
Theorem 1. For a finite-dimensional algebra \(A\), the finitistic dimension of \(A^{op}\) is bounded by the depth of \(A\) and the delooping level of \(A\). That is, the depth of \(A\) \(\leq\) the finitistic dimension of \(A^{op}\) \(\leq\) the delooping level of \(A\).
If the delooping level of \(A\) is finite for any algebra \(A\), then by Theorem 1 the finitistic dimension conjecture is proved. Unfortunately, Kershaw and Rickard [15] constructed a counter-example for this. Moreover, Sen [16] computed the delooping level for Nakayama algebras. In 2025, Guo and Igusa [17] introduced the definitions of \(k\)-delooping level and derived delooping level and give a more precise bound for the finitistic dimension. Later Guo [18] showed that the derived delooping level is left and right symmetric. Chen and Hu [19] studied delooping level and derived delooping level in terms of functions. Recently, Chen [20] showed that the delooping level is not preserved under derived equivalence.
Since \(A\) is always a \(\tau\)-tilting module in \(\text{mod}A\), it is natural to ask: Is there a generalized version of Theorem 1 in term of \(\tau\)-tilting modules? In this paper, we give a positive answer to the question and then apply our result to the finitistic dimension conjecture.
In what follows, let \(A\) be a finite-dimensional algebra over an algebraically closed field \(K\) and let \(T\in \text{mod} A\) be a \(\tau\)-tilting module with \(B=\text{End}_A T\). Denote by \(\mathcal{C}=\text{Fac}T=T(S)=\text{Filt(Fac} S)\), where \(S=S_1\bigoplus \cdots \bigoplus S_m\) is a semi-brick and \(S_i\) is a brick. We first define the depth of \(\mathcal{C}\) relative to \(T\) (See Definition 19 for details) and then prove the following result.
Theorem 2. Let \(A, T, B\) and \(\mathcal{C}\) be as above. Then the depth of \(\mathcal{C}\) relative to \(T\) \(\leq\) the finitistic dimension of \(B^{op}\).
Keep \(A, T, S, S_i, B\) as above. We give the following bijection between bricks in \(\mathcal{C}\) and simple modules in \(\text{Sub}\mathbb{D}T\), which is partially shown in [21].
Theorem 3. Let \(A, T, S, S_i\) and \(B\) be as above. Then the functor \[\mathrm{Hom}_A(T, -): \mathrm{Fac}T \to \mathrm{Sub}(\mathbb{D}T)_B\] induces a bijection between \(\{S_1, \dots, S_m\}\) and \(\{\text{simple modules in \mathrm{Sub}}(\mathbb{D}T)\}\).
By using Theorem 3, we are able to define the \(k\)-delooping level of \(\mathcal{C}\) (See Definition 28 for details) and give the following theorem.
Theorem 4. Let \(A, T, S, S_i\) and \(B\) be as above. Let \(k\) be a positive integer. Then
The \(k\)-delooping level of \(B\) \(=\) the \(k\)-delooping level of \(\mathcal{C}\) relative to \(T\).
The finitistic dimension of \(B^{op}\) \(\leq\) the delooping level of \(\mathcal{C}\) relative to \(T\).
As a corollary of Theorem 2 and Theorem 4, we get that a generalized version of Theorem 1 via \(\tau\)-tilting modules.
It is natural to ask whether our Theorem 4 can be used for the study of the finitistic dimension conjecture. That is, which class of \(\tau\)-tilting modules \(T\) with \(B={\rm End}_A T\) admits the property \({\rm findim} B^{op}<\infty\) ?
Denote by \({\rm Sub}T\) the subcategory of \({\rm mod}A\) cogenerated by \(T\). We give a positive answer to the question above and prove the following result.
Theorem 5. Let \(A\) be an algebra, \(T\in {\rm mod}A\) a \(\tau\)-tilting module and \(B={\rm End}_A T\). If \({\rm Fac}T\cap {\rm Sub}T\) is of finite representation type, then the finitistic dimension of \(B^{op}\) is finite.
As an application of Theorem 5, we can get the following corollary.
Corollary 6. Let \(A\) be an algebra of finite representation type or a minimal representation infinite algebra, \(T\in {\rm mod}A\) a \(\tau\)-tilting module and \(B={\rm End}_A T\). Then the finitistic dimension of \(B^{op}\) is finite.
The paper is organized as follows: In Section 2, we recall the preliminaries on tilting modules, \(\tau\)-tilting modules, torsion pairs and bricks. In Section 3, we introduce the depth of modules relative to a \(\tau\)-tilting module \(T\) and prove Theorem 2. In Section 4, we first prove Theorem 3 and then introduce the definition of \(k\)-delooping level of \(\mathcal{C}\) relative to a \(\tau\)-tilting module \(T\) and prove Theorem 4. Then we give examples to illustrate Theorem 4. In Section 5, we give applications of Theorem 4 to the finitistic dimension conjecture and show Theorem 5 and Corollary 6.
Throughout the paper, we assume that all algebras are finite-dimensional basic algebras over an algebraically closed field \(K\) and all modules are finitely generated right modules. Denote by \(\mathbb{D}\) the ordinary duality and \(\tau\) the Auslander-Reiten translation functor.
In this section we recall from [1], [14], [21]–[23] basic facts on tilting modules, \(\tau\)-tilting modules and bricks for later use.
In the following we recall the definition of torsion pairs from [22].
Definition 7. Let \(A\) be an algebra. A pair \((\mathcal{T}, \mathcal{F})\) of full subcategories of \(\mathrm{mod}A\) is called a torsion pair if the following conditions are satisfied:
\(\mathrm{Hom}_A(M, N)=0\) for all \(M \in \mathcal{T}\), \(N \in \mathcal{F}\).
\(\mathrm{Hom}_A(M, -)|_\mathcal{F}=0\) implies \(M \in \mathcal{T}\).
\(\mathrm{Hom}_A(-, N)|_\mathcal{T}=0\) implies \(N \in \mathcal{F}\).
For a module \(M\), denote by \(|M|\) the number of indecomposable direct summands up to isomorphisms. Denote by \({\rm pd}_A M\) the projective dimension of \(M\). We recall the definition of tilting modules from [24].
Definition 8. Let \(A\) be an algebra. A module \(T\in{\rm mod} A\) is called a tilting module if the following two conditions are satisfied:
\({\rm pd}_A T\leq1\).
\(\text{Ext}^1_A(T,T)=0\).
\(|T|=|A|\).
We also need the definition of a torsion pair from [22].
Proposition 9. Let \(A\) be an algebra and \(T\in\mathrm{mod}A\). Then any tilting module \(T\) induces a torsion pair \((\mathcal{X}(T_A), \mathcal{Y}(T_A))\) in the category modB, where \(B = \mathrm{End}_A T\) and \[\mathcal{X}(T_A)=\{X_B|\mathrm{Hom}_B(X,\mathbb{D}T)=0\}=\{X_B|X \otimes_B T=0\},\] \[\mathcal{Y}(T_A)=\{Y_B|\mathrm{Ext}_B^1(Y,\mathbb{D}T)=0\}=\{Y_B|\mathrm{Tor}_1^B(Y,T)=0\}.\]
Now we recall the tilting theorem from [22] .
Theorem 10. Let \(A\) be an algebra, \(T\in \mathrm{mod}A\) a tilting module, \(B =\mathrm{End}_A T\), and \((\mathcal{T}(T_A), \mathcal{F}(T_A))\), \((\mathcal{X}(T_A), \mathcal{Y}(T_A))\) be the induced torsion pairs in mod\(A\) and mod\(B\), respectively. Then \(T\) has the following properties:
The functors \(\mathrm{Hom}_A(T,-)\) and \(-\otimes_B T\) induce quasi-inverse equivalences between \(\mathcal{T}(T_A)\) and \(\mathcal{Y}(T_A)\).
The functors \(\mathrm{Ext}^1_A(T,-)\) and \(\mathrm{Tor}^B_1(-,T)\) induce quasi-inverse equivalences between \(\mathcal{F}(T_A)\) and \(\mathcal{X}(T_A)\).
We also need the definition of \(\tau\)-tilting modules from [1].
Definition 11. Let \(A\) be an algebra, \(M\in{\text{mod}A}\) and \(P\in\text{mod}A\) be projective.
We call \(M\) \(\tau\)-rigid if \(\text{Hom}_A(M,\tau M)=0\).
We call \(M\) \(\tau\)-tilting if \(M\) is \(\tau\)-rigid and \(|M|= |A|\).
We call \(M\) support \(\tau\)-tilting if there exists an idempotent \(e\) of \(A\) such that \(M\) is a \(\tau\)-tilting \((A/\langle e\rangle)\)- module.
\((M, P)\) is a \(\tau\)-rigid pair if \(M\) is \(\tau\)-rigid and \(\text{Hom}_A(P, M)=0\).
\((M, P)\) is a support \(\tau\)-tilting pair (resp. an almost support \(\tau\)-tilting pair) if \((M,P)\) is a \(\tau\)-rigid pair and \(|M|+|P|=|A|\) (resp. \(|M|+|P|=|A|-1\) ).
The following result on the relations of \(\tau\)-tilting modules and tilting modules in [1] is essential in this paper.
Proposition 12. Let \(A\) be an algebra, \(M\in{\mathrm{mod}A}\) with right annihilator \(I=\mathrm{ann} M\) and \(\overline{A}=A/I\). Then \(M\) is a support \(\tau\)-tilting module in \(\mathrm{mod}A\) if and only if it is a tilting module in \(\mathrm{mod} \overline{A}\).
We also need the following result in [8].
Proposition 13. Let \(A\) be an algebra and let \(T\in\mathrm{mod}A\) be a \(\tau\)-tilting module. For any \(M\in\mathrm{Fac}T\), there is an exact sequence \(\cdots\rightarrow T_1\stackrel{f_1}{\rightarrow}T_0\stackrel{f_0}{\rightarrow}M\rightarrow 0\) with \(T_i\in \mathrm{add}T\) and \(\mathrm{Ker} f_i\in \mathrm{Fac}T\).
We also need the following definition of mutations for support \(\tau\)-tilting modules from [1].
Definition 14. Let \(A\) be an algebra and let \((M,P)\) and \((N,Q)\) be support \(\tau\)-tilting pair in \(\text{mod}A\). If there exists an almost support \(\tau\)-tilting pair that is a common direct summand of \((M,P)\) and \((N,Q)\), then we say that\((N,Q)\) is a mutation of \((M,P)\). For each direct indecomposable summand \(L\) of \(M\) or \(P\), there uniquely exists a mutation of \((M,P)\) at \(L\). If \(\text{Fac}M \subseteq \text{Fac}N\), then \(N\) is called a left mutation of \(M\). Dually, \(N\) is called a right mutation of \(M\).
Now we recall the following definitions of bricks and semi-bricks, see [21], [25] for details.
Definition 15. Let \(A\) be an algebra and \(S\in {\rm mod}A\).
\(S\) is called a brick if \(\text{End}_A (S)\) is a division \(K\)-algebra.
\(S\) is called a semi-brick if \(S\simeq \oplus_{i=1}^m S_i\) and \(\text{Hom}_A (S_i, S_j) =0\) holds for any \(i\neq j\), where \(S_i\) is a brick and \(m\) is a positive integer.
For each full subcategory \(\mathcal{C}\subseteq \text{mod}A\). Denote by \(\text{add}\mathcal{C}\) the additive closure of \(\mathcal{C}\), \(\text{Fac}\mathcal{C}\) the subcategory of the factor modules of objects in \(\text{add}\mathcal{C}\), and \(\text{Filt}(\mathcal{C})\) the subcategory of the objects \(M\) such that there exists a sequence \(0= M_0 \subseteq M_1 \subseteq M_2 \cdots \subseteq M_n = M\) with \(M_i/M _{i-1} \in\) add\(\mathcal{C}\). We use \(T(\mathcal{C})\) to denote the subcategory \(\text{Filt}(\text{Fac}\mathcal{C})\).
The following property on semi-bricks [21] is essential in this paper.
Proposition 16. Let \(A\) be an algebra and \(S \in \mathrm{mod}A\) a semi-brick. Let \(S_1\) be a brick which is a direct summand of \(S\). Every nonzero homomorphism \(f: M \to S_1\) with \(M \in T(S)\) is surjective. Moreover, we have \(\mathrm{Ker}f \in T(S)\).
We also need the following proposition on relations between support \(\tau\)-tilting modules and semi-bricks, see [21].
Theorem 17. Let \(A\) be an algebra and \(T\in\mathrm{mod}A\). If \(T\) is a support \(\tau\)-tilting module, then \(\mathrm{Fac}T = T(S)=\mathrm{Filt(Fac}S)\), where S is a semi-brick.
Now we recall the definition of label support \(\tau\)-tilting quivers with bricks from [21].
Definition 18. Let \(A\) be an algebra and let \(M\in\mathrm{mod}A\) be a support \(\tau\)-tilting module and decompose \(M\) as \(M =\oplus_{i=1}^t M_i\) with \(M_i\) indecomposable. Assume that \(M \to N\) is an arrow in the support \(\tau\)-tilting quiver of \(A\), and that \(N\) is the left mutation of \(M\) at \(M_i\). Then we label this arrow with a brick \(S_i=M_i/\Sigma_{f\in \text{rad}\text{Hom}_A(M, M_i)}{\text{Im}f}\).
In this section, we assume that \(A\) is an algebra and \(T\in \text{mod} A\) is a \(\tau\)-tilting module. Denote by \(\text{Fac}T=T(S)=\text{Filt(Fac} S)\), where \(S=S_1\bigoplus \cdots \bigoplus S_m\) is a semi-brick and \(S_i\) is a brick. Then we show the depth theorem relative to \(T\) which generalizes [14].
Let \(A\) be an algebra. Recall from [23] that the grade of \(M\in \text{mod}A\) denoted by \(\text{grade}M\) is defined as the infimum of the integer \(i\geq 0\) such that \(\text{Ext}^i_A(M,A)\neq 0\). Recall from [14] that the depth of \(A\) denoted by \(\text{depth}A\) is defined as the supremum of the grade of simple modules.
In the following we give the definition of grade and depth relative to \(T\).
Definition 19. Let \(A\) and \(T\) be as above. Denote by \(\mathcal{C}=\text{Fac}T\).
The grade of \(M\in \mathcal{C}\) is defined as \(\text{grade}_T M =\text{inf}\{i\geq 0 |\text{Ext}^i_\Lambda(M,T)\neq 0\}.\)
The depth of \(\mathcal{C}\) is defined as \(\text{depth}_T \mathcal{C}:={\sup\limits_{1\leq i\leq m}} \text{grade}_T S_i\).
It is easy to see that the grade of \(M\) relative to \(T\) we defined is the classical grade whenever \(T=A\). If \(T=A\), then depth\(_T \mathcal{C}\) is the classical one in [14].
We give the following example to illustrate Definition 19.
Example 20. Let \(A\) be an algebra given by the quiver \(\xymatrix{1\ar[r]^a&2\ar[r]^b&3}\) with the relation \(ab=0\). Then
\(T=\substack{ 1\\2\\ }\oplus1\oplus3\) is a \(\tau\)-tilting module but not a tilting module and \(\mathcal{C}=\mathrm{Fac} T=\mathrm{add}(1\oplus3\oplus\substack{1\\2})\).
Every indecomposable module in \(\mathrm{Fac}T\) has \(T\)-grade \(0\).
The semi-brick relative to \(T\) is \(S=3\oplus\substack{1\\2}\). So the \(\mathrm{depth}_T \mathcal{C}=0\).
To prove the main result we need the following lemma from [26].
Lemma 21. Let \(A\) be an algebra and \(T\in \mathrm{mod}A\). Denote by \(B=\mathrm{End}_A T\). If \(T_1\stackrel{f_1}{\rightarrow} T_0\stackrel{f_0}{\rightarrow}M\rightarrow 0\) is a minimal \(T\)-presentation of \(M\), then \(\mathrm{Hom}_A(T_0,T)\stackrel{{f_1}^T}{\rightarrow}\mathrm{Hom}_A(T_0,T)\rightarrow \mathrm{Coker} {f_1}^T\rightarrow 0\) is a minimal projective presentation of \(\mathrm{Coker} {f_1}^T\) in \(\mathrm{mod}B^{op}\).
Let \(A\) be an algebra. Recall from [23], the finitistic dimension of \(A\) is defined as: \[\text{findim}A := \sup\left\{\text{pd}_A M \mid M \in \text{mod}A,\;\text{pd}_A M < \infty \right\}.\]
Dually, we use findim\(A^{op}\) to denote the finitistic dimension of \(A^{op}\). Now we are in a position to show our main result in this section.
Theorem 22. Let \(A\) be an algebra and \(T\in \mathrm{mod}A\) be a \(\tau\)-tilting module with \(\mathrm{Fac}T= \mathrm{Filt(Fac}S)\), where \(S=S_1 \bigoplus\cdots \bigoplus S_m\) is a semi-brick and \(S_i\) is a brick. Denote by \(B = \mathrm{End}_AT\). Then the depth of \(\mathcal{C}\) relative to \(T\) \(\leq\) the finitistic dimension of \(B^{op}\), that is, \({\rm depth}_T \mathcal{C}\leq {\rm findim} B^{op}\).
Proof. Let \(T_n \to \cdots \to T_1 \to T_0 \to S_i \to 0\;\;(*)\) be a minimal \(T\)-presentation of the brick \(S_i\). We divide the proof into two cases.
If there exists a brick \(S_i \in \text{Fac}T\) such that \(\text{Ext}^j_\Lambda (S_i, T) =0\) for all \(j\geq 0\), then findim\(B^{op}=+\infty\).
Applying \(\text{Hom}_A(-, T)\) to \((*)\), we get the following long exact sequence \[0\to \text{Hom}_A(T_0, T)\to \cdots \to \text{Hom}_A(T_{n-1}, T) \xrightarrow{\delta_n^*} \text{Hom}_A(T_n, T)\to \text{coker} (\delta_n ^*) \to 0 \;\;(**)\]
Then by Lemma 21 one gets that \((**)\) is a minimal projective resolution of \(\text{coker} (\delta_n ^*)\) and hence \(\text{Ext} _B^n (\text{coker} (\delta_n^*), B) \neq 0\). Therefore the projective dimension of coker\(\delta_n^* =n\) and therefore findim\(B^{op}= +\infty.\)
If \(\prod _{j\geq 0}\text{Ext} _A^{j} (S_i, T) \neq 0\) holds for each brick \(S_i\), then there exists minimal \(t\in \mathbb{Z}^{\geq 0}\) such that \(\text{Ext}^t_A(S_i, T) \neq 0\). We show the projective dimension of \(\text{Ext}^t_A (S_i, T)= t\).
If \(t=0\), then we have \(\text{Ext}^0_A(S_i, T) \neq 0\), therefore the projective dimension of \(\text{Hom}_A(S_i, T) \geq 0\). In the following, we show the assertion holds for \(t\geq1\). Applying the functor \(\text{Hom}_A(-, T)\) to \((*)\), one gets the following long exact sequence \[0\to \text{Hom}_A(T_0, T)\to\cdots \to \text{Hom}_A(T_t, T)\to \mathrm{Ext}^t_{A}(S_i, T) \to 0,\] which is a minimal projective resolution of \(\mathrm{Ext}^t_{A}(S_i, T)\) by Lemma 21. So we get that the projective dimension of \(\text{Ext}^t_A (S_i, T)=t\). In this case, we have shown that depth\(_T \mathcal{C} \leq \text{findim}B^{op}\).
◻
If we take \(T=A\) and \(A\) is a finite-dimensional algebra, then we have the following corollary [14].
Corollary 23. Let \(A\) be an algebra. We have the inequality \[{\rm depth}A \leq {\rm findim}A^{op}.\]
We end this section with the following example.
Example 24. Let \(A\) be an algebra given by the quiver \[\xymatrix{ 1\ar@<2pt>[r]^{\alpha}&2\ar@<2pt>[l]^{\beta} }\] with the relations \(\alpha\beta\alpha=\beta\alpha\beta=0\). Then
\(T=\substack{1\\2\\1}\oplus1\) is a \(\tau\)-tilting module.
\(B=\mathrm{End}_A T\) is given by\[\xymatrix{ 1\ar@<2pt>[r]^{\alpha}&2\ar@<2pt>[l]^{\beta} }\] with \(\beta\alpha=0\) and \(\mathrm{gldim}B=2\).
\(\mathrm{Fac}T=\mathrm{FiltFac}\substack{1\\2}\), one gets that \(\mathrm{depth}_T \mathcal{C}=\mathrm{depth}_T\substack{1\\2}=0\leq \mathrm{findim}B^{op}=2\).
Throughout this section, \(T=\oplus_{i=1}^nT_i\) is a \(\tau\)-tilting module with \(T_i\) indecomposable and \(\text{Fac}T=T(S)=\text{Filt(Fac}S)\), where \(S\simeq S_1\bigoplus \cdots \bigoplus S_m\) and \(S_i\) is a brick. Denote by \(B=\text{End}_A T\). Following [7], in this section we define the delooping level relative to \(T\) for the subcategory \(\mathcal{C}=\text{Fac}T\) and then study the relation between the finitistic dimension of \(B^{op}\) and the delooping level of \(\mathcal{C}\).
To define the delooping level relative to \(T\), we give the following result which gives a bijection between bricks \(S_i\) and simple \(B\)-modules in \(\text{Sub}\mathbb{D}T\). We should remark that part of this result is first shown in the proof of [21].
Theorem 25. Let \(A, T, S, S_i\) and \(B\) be as above. Then the functor \[\mathrm{Hom}_A(T, -): \mathrm{Fac}T \to \mathrm{Sub}(\mathbb{D}T)_B\] induces a bijection between \(\{S_1, \dots, S_m\}\) and \(\{\text{simple modules in \mathrm{Sub}}(\mathbb{D}T)\}\).
Proof. We divide the proof into two steps.
(1) We show that \(\mathrm{Hom}_A(T, S_i)\) is simple. This is actually shown by Asai in [21]. Here we give a new proof.
On the contrary, suppose that \(\mathrm{Hom}_A(T, S_i)\) is not simple. Since \(S_i\) is a brick and a direct summand of \(S\), one gets an exact sequence \[0 \to L_i \to T_i \to S_i \to 0 \;\;\;\;(*1)\] with \(T_i\) an indecomposable direct summand of \(T\) and \(L_i \in \mathrm{Fac}T\) by Proposition 16. Applying \(\mathrm{Hom}_A(T, -)\) to \((*1)\), one gets an exact sequence \[0 \to \mathrm{Hom}_A(T, L_i) \to \mathrm{Hom}_A(T, T_i) \to \mathrm{Hom}_A(T, S_i) \to 0.\] On the other hand, one has the following exact sequence \[0 \to R \to \mathrm{Hom}_A(T, T_i) \to S \to 0\] with \(S\) simple in \(\text{mod} B\). We have the following commutative diagram
By using the snake lemma, one gets that \(\theta\) is a monomorphism and \(\pi\) is an epimorphism with \(\mathrm{Coker}\theta \cong \mathrm{Ker} \pi = N\). So the exact sequence \[0 \to N \stackrel{a}{\to} \mathrm{Hom}_A(T, S_i) \to S \to 0\] implies that \(N \in \mathrm{Sub}(\mathbb{D}T)\). Since \(\mathrm{Hom}_A(T, -): \mathrm{Fac}\;T \to \mathrm{Sub}(\mathbb{D}T)\) is an equivalence, there exists \(M \in \mathrm{Fac}T\) such that \(N \cong \mathrm{Hom}_A(T, M)\). Then one gets an exact sequence \[0 \to \mathrm{Hom}_A(T,M) \stackrel{a}{\to} \mathrm{Hom}_A(T, S_i) \to S \to 0. \;\;\;(*2)\] So there exists \(b: M \to S_i \ne 0\) such that \(a= \mathrm{Hom}_A(T, b)\).
By Proposition 16, \(b\) is surjective and \(Q=\mathrm{Ker} b\in \mathrm{Fac}T\), then one gets an exact sequence \[0 \to Q \to M \stackrel{b}{\to} S_i \to 0.\;\;\;\;\;\;(*3)\] Applying \(\mathrm{Hom}_A(T, -)\) to \((*3)\), we get the following exact sequence \[0 \to \mathrm{Hom}_A(T, Q) \to \mathrm{Hom}_A(T, M) \xrightarrow{a} \mathrm{Hom}_A(T, S_i) \to 0.\;\;\;\;\;(*4)\] Comparing \((*2)\) and \((*4)\), one gets that the map \(a\) is an isomorphism which implies that \(\mathrm{Hom}_A(T, Q) = 0\), and hence \(Q=0\). Then \(M \cong S_i\) implies \(S=0\) by using \((*2)\), contradiction.
(2) We show for any \(S'\) simple in \(\mathrm{Sub}(\mathbb{D}T)\), there is \(S_i \in \{S_1, \dots, S_m\}\) such that \(\mathrm{Hom}_A(T, S_i) \cong S'\).
Since \(\mathrm{Hom}_A(T, -): \mathrm{Fac}T \to \mathrm{Sub}(\mathbb{D}T)\) is an equivalence by Proposition 12 and Theorem 10, then there exists \(M \in \mathrm{Fac}T\) such that \(\mathrm{Hom}_A(T, M) \cong S'\). So the fact \[\mathrm{Hom}_A(M, M) \cong \mathrm{Hom}_B(\mathrm{Hom}_A(T, M), \mathrm{Hom}_A(T, M)) \cong \mathrm{Hom}_B(S', S') \cong K\] implies that \(M\) is a brick.
Since \(M \in \mathrm{Fac}T = \mathrm{Filt} (\mathrm{Fac}S)\), then there exists a brick \(S_i\) which is a direct summand of \(S\) such that \(S_i \xrightarrow{f} M \ne 0\). Since \(\mathrm{Hom}(T, -): \mathrm{Fac}\;T \to \mathrm{Sub}(\mathbb{D}T)\) is an equivalence, one gets \[\begin{align} \mathrm{Hom}_A(T, -): &\mathrm{Hom}_A(S_i, M) \to \mathrm{Hom}_B (\mathrm{Hom}_A(T, S_i), \mathrm{Hom}_A(T, M))\\ \;\;\;\;\;\;&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; f \to \mathrm{Hom}_A(T, f) \end{align}\] induces an isomorphism. So the map \(\mathrm{Hom}(T, S_i)\stackrel{\mathrm{Hom}(T, f)} {\rightarrow} \mathrm{Hom}_A(T, M)\) is not zero, which implies that \(\mathrm{Hom}_A(T, f)\) is an isomorphism since both \(\mathrm{Hom}_A(T, S_i)\) and \(\mathrm{Hom}_A(T, M)\) are simple. Using \(\mathrm{Hom}_A(T, -): \mathrm{Fac}T \to \mathrm{Sub}(\mathbb{D}T)\) is an equivalence again, one gets that \(S_i\simeq M\). ◻
By the proof of Theorem 25, it is not difficult to show that there is a same result for a support \(\tau\)-tilting module. Immediately, one has the following corollary.
Corollary 26. Let \(A, S, T\) and \(B\) be as above and \(|A|=n\). Then every simple \(B\)-module is in \(\mathrm{Sub}\mathbb{D}T\) if and only if \(T=A\).
Proof. If \(T=A\), there is nothing to show. Conversely, \(T\) is \(\tau\)-tilting implies that \(|T|=|A|=|B|=n\). Then by Theorem 25, \(S\) should have \(n\) indecomposable direct summands. We are done. ◻
Now we show some basic homological properties between \(\text{Fac}T\) and \(\text{mod}B\) for later use.
Proposition 27. Let \(A, T, S, S_i\) and \(B\) be as above.
For any \(N\) in \(\mathcal{C}\), if \(\to\cdots\to T_1\to T_0\to N\to 0\) is a minimal \(T\)-presentation, then \(\to\cdots\to \mathrm{Hom}_A(T,T_1)\to \mathrm{Hom}_A(T, T_0)\to\mathrm{Hom}_A(T, N)\to 0\) is a minimal projective resolution of \(\mathrm{Hom}_A(T, N)\).
For any \(n>d\geq1\) and \(M\in\mathrm{mod}B^{op}\), we have \(\mathrm{Tor}^B_n (\mathrm{Hom}_A(T, S_i), M)\cong \mathrm{Tor}^B_{n-d} (\mathrm{Hom}_A (T, \Omega^d_T S_i), M).\)
Proof. (1) By Proposition 13, one gets that every kernel in a minimal \(T\)-presentation is in \(\mathcal{C}\). So \(\to\cdots\to \mathrm{Hom}_A(T,T_1)\to \mathrm{Hom}_A(T, T_0)\to\mathrm{Hom}_A(T, N)\to 0\) is exact. It is a minimal projective resolution since \(\text{Hom}_A(T,-)\) is an equivalence.
(2) Denote \(L=\text{Hom}_A(T, S_i)\). Denote by \(T_* \xrightarrow{\simeq} S_i\) a \(T\)-resolution of a brick \(S_i\) by Proposition 13. Then one has the following exact sequence \[0\to \Omega_T^d S_i \to T_{d-1} \to \cdots \to T_1 \to T_0 \to S_i \to 0\;\;\;\;\;\;(*5)\] with \(\Omega_T^d S_i\in\text{Fac}T\). Applying \(\text{Hom}_A (T, -)\) to \((*5)\), we have the following long exact sequence \[0\to \text{Hom}_A(T,\Omega_T^d S_i) \to \text{Hom}_A(T,T_{d-1} )\to \cdots \to \text{Hom}_A(T, T_0) \to \text{Hom}_A(T, S_i) \to 0\;\;\;\;(*6),\] which is a projective dimension of \(L=\text{Hom}_A(T, S_i)\). Applying the functor \(~_-\bigotimes M\) to \((*6)\), one gets the following long exact sequence: \[\begin{align} &\cdots \mathrm{Tor}_2^B (P_0, M)\to \mathrm{Tor}_2^B (L, M) \to \mathrm{Tor}_1^B (\Omega^1 L, M) \to \mathrm{Tor}_1^B(P_0, M) \to \mathrm{Tor}_1^B(L, M)\\ &\to \Omega^1_L \otimes M \to P_1 \otimes M \to L \otimes M \to 0. \end{align}\] By using dimension shifting, we have \[\text{Tor}_n^B (L, M) \cong \text{Tor}_{n-1}^B (\Omega^1 L, M) \cong \text{Tor}_{n-2}^B (\Omega^2 L, M) \cong \text{Tor}_{n-d}^B (\Omega^d L, M).\] We are done. ◻
Let \(A\) be an algebra, \(M\in \text{mod}A\) and let \(k \in \mathbb{Z^+}\). Recall from [17] that the \(k\)-delooping level of \(M\) denoted by \(k\)-\(\text{dell}M\) is defined as the infimum of \(d\geq 1\) such that \(\Omega^d M\) is a stable retract of \(\Omega^{d+k}N\) for \(N\) in \(\text{\underline{mod}}A\). The \(k\)-delooping level of \(A\) denoted by \(k\)-\(A\) is defined as the supremum of the \(k\)-delooping level of simple \(A\)-modules. Putting \(k=1\), one gets the delooping level of \(M\) and \(A\) introduced in [14].
For a module \(M\) in \(\text{mod}A\), denote by \(\text{rad} M\) the radical of \(M\). By Proposition 13 and Theorem 25, now we are able to give the following definition.
Definition 28. Let \(A\), \(T\) and \(S_i\) be as above. Let \(k\) be a positive integer.
For any \(M\in \mathcal{C}=\text{Fac}T\), the \(k\)-delooping level of \(M\) relative to \(T\) is defined as the infimum of integer \(d\) such that \(\Omega_{T}^{d} M\) is a stable retract of \(\Omega_T^{d+k}N\) for some \(N\in \text{Fac}T\).
The \(k\)-delooping level of \(\mathcal{C}\) relative to \(T\) is defined as the supremum of the \(k\)-delooping level relative to \(T\) of \(S_i\) for \(1\leq i\leq m\) and the \((k+1)\)-delooping level of \(N_j\) \(+1\) for \(m+1\leq j\leq n\), where \(\text{Hom}_A(T,N_j)\cong \text{rad}\text{Hom}_A(T,T_j)\) .
The global \(k\)-delooping level of \(\mathcal{C}\) relative to \(T\) is defined as the supremum of the \(k\)-delooping level relative to \(T\) of \(M\) for all \(M\in\mathcal{C}\).
If \(T\)=\(A\), then one gets that the semi-brick \(S\) should be the direct sum of simple \(A\)-modules. Therefore the \(k\)-delooping level of \(\mathcal{C}\) is the classical case introduced by [17]. It is easy to see that the \(k\)-delooping level of \(\mathcal{C}\) relative to \(T\) is no more than the global \(k\)-delooping level of \(\mathcal{C}\) relative to \(T\). From now on, we use \(k\)-\(\text{dell}_TM\), \(k\)-\(\text{dell}_T\mathcal{C}\) and \(global\)-\(k\)-\(\text{dell}_T\mathcal{C}\) to denote the three notations above.
Now we are in a position to show the main result in this section.
Theorem 29. Let \(A, T, S, S_i\) and \(B\) be as above. Let \(k\) be a positive integer. Then
The \(k\)-delooping level of \(B\) \(=\) the \(k\)-delooping level of \(\mathcal{C}\) relative to \(T\), that is, \(k\)-\({\rm dell}B\)=\(k\)-\({\rm dell}_T\mathcal{C}\).
The finitistic dimension of \(B^{op}\) \(\leq\) the delooping level of \(\mathcal{C}\) relative to \(T\), that is, \({\rm findim}B^{op}\leq {\rm dell}_T \mathcal{C}\).
Proof. (1) By Theorem 25 and Proposition 29(1), one gets that the \(k\)-delooping level of \(S_i\) is equal to that of \(\text{Hom}_A(T, S_i)\) for \(1\leq i\leq m\). For any \(T_j\) with \(m+1\leq j\leq n\), one gets that \(\text{Hom}_A(T,T_j)\) is an indecomposable and projective \(B\)-module. So one has the following exact sequence: \[0\rightarrow \text{rad}\text{Hom}_A(T,T_j)\to \text{Hom}_A(T,T_j)\to S_j\to 0,\] where \(S_j\) is a simple \(B\)-module not in \(\text{Sub}\mathbb{D}T\).
Now we show that the \(k\)-delooping level of \(S_j\) is equal to the \((k+1)\)-delooping level of \(\text{rad}\text{Hom}_A(T,T_j)+1\).
We only have to show the finite case. For \(M, N\in \text{mod}B\) with \(N\cong\Omega^1M\). If the \(k\)-delooping level of \(M\) is \(d\), then one gets that \(\Omega^d M\) is a direct summand of \(\Omega^{d+k}L\), that is, \(\Omega^{d-1}\Omega^1 M\) is a direct summand of \(\Omega^{d-1+k+1}L\). So the \((k+1)\)-delooping level of \(N\) is equal to \(d-1\). Conversely, the fact that the \((k+1)\)-delooping level of \(N\) is equal to \(d-1\) implies that \(\Omega^{d-1}\Omega^1 M\) is a direct summand of \(\Omega^{d-1+k+1}L\), that is, \(\Omega^d M\) is a direct summand of \(\Omega^{d+k}L\). The assertion holds.
Since \(\text{Hom}_A(T,-): \text{Fac}T\to\text{Sub}\mathbb{D}T\) is an equivalence and \(\text{rad}\text{Hom}_A(T,T_j)\) is in \(\text{Sub}\mathbb{D}T\), one gets that there is a module \(N_j\) such that \(\text{Hom}_A(T, N_j)\cong\text{rad}\text{Hom}_A(T,T_j)\). By a similar argument above, one gets that the \((k+1)\)-delooping level of \(\text{rad}\text{Hom}_A(T,T_j)+1\) is equal to the \((k+1)\)-delooping level relative to \(T\) of \(N_j\). We are done.
(2) It is a straight result of (1) and [14]. ◻
We have the following straight corollary which is the \(\tau\)-tilting version of Theorem 1.
Corollary 30. Let \(A\) be a finite-dimensional algebra, \(T\in{\rm mod}A\) a \(\tau\)-tilting module with \(\mathcal{C}={\rm Fac}T\) and \(B={\rm End}_A T\). Then the depth of \(\mathcal{C}\) relative to \(T\) \(\leq\) the finitistic dimension of \(B^{op}\) \(\leq\) the delooping level of \(\mathcal{C}\) relative to \(T\).
We also have the following corollary.
Corollary 31. Let \(A,T, B\) and \(\mathcal{C}\) be as above.
\(\mathrm{depth}_T \mathcal{C} = 0\).
\(\mathrm{findim}B^{op} = 0\).
\(\mathrm{dell}_T \mathcal{C} = 0\).
Then we have \((3)\Rightarrow(2)\Rightarrow (1)\).
Proof. This is the straight result of Corollary 30. ◻
In Theorem 29, we show that the finitistic dimension of \(B^{op}\) is bounded by the delooping level of \(\mathcal{C}\). But the delooping level of \(\mathcal{C}\) is not easy to compute by using Definition 28. It is natural to ask when the finitistic dimension of \(B^{op}\) can be bounded by the supremum of the delooping level of \(S_i\) ?
Theorem 32. Let \(A, T, S, S_i\) and \(B\) be as above. Let \(m\geq 1\), \(S_i\neq 0\) for \(1\leq i\leq m\) and \(S_j=0\) for \(m+1\leq j\leq n\). If the projective dimension of the top of \(\mathrm{Hom}_A(T,T_j)\) is \(l_j\) for \(m+1\leq j\leq n\), then \(\mathrm{findim}B^{op}\leq \mathrm{max}\{l_j|m+1\leq j\leq n\;\mathrm{and}\;\mathrm{dell}_T S_i|1\leq i\leq m\}\).
Proof. Assume that \(\mathrm{max}\{l_j|m+1\leq j\leq n\;\mathrm{and}\;\mathrm{dell}_T S_i|1\leq i\leq m\}=d\), we show \(\mathrm{findim}\Gamma^{op} \leq d\). For any \(M \in \mathrm{mod}\Gamma^{op}\) with the projective dimension of \(M\) \(t\geq m+1\), there exists \(S \in \mathrm{mod}\Gamma\) such that \(\mathrm{Tor}^\Gamma_t (S, M) \neq 0\). By the assumption, one gets that \(S\) is in \(\text{Sub}\mathbb{D}T\). Then by Theorem 25, one gets a brick \(S_i\in \mathcal{C}\) such that \(\text{Hom}_A(T,S_i)\cong S\). In the following, we show that \(t \leq d.\) On the contrary, suppose that \(n>d\).
\[\begin{align} \text{Tor}_{t}^\Gamma (S, M) = \text{Tor}_{t}^\Gamma (\mathrm{Hom}_\Lambda(T,S_i), M) \cong \text{Tor}_{t-d}^\Gamma (\Omega^d \mathrm{Hom}_\Lambda(T,S_i), M) \\ \cong \text{Tor}_{t-d}^\Gamma (\mathrm{Hom}_\Lambda(T,\Omega_T^d S_i), M) \underset{\oplus}{<} \text{Tor}_{t-d}^\Gamma (\mathrm{Hom}_\Lambda(T,\Omega_T^{d+k} N), M) \cong \\ \text{Tor}_{t-d}^\Gamma (\Omega^{d+k}\mathrm{Hom}(T, N), M) \cong \text{Tor}_{t-d+d+k}^\Gamma (\mathrm{Hom}(T, N), M)=0 \end{align}\] This is a contradiction. In all, the assertion holds. ◻
Immediately we have the following corollaries.
Corollary 33. Let \(A\) be an algebra and \(T=P(1)\oplus P(2)\oplus\cdots\oplus T_j\oplus\cdots\oplus P(n)\) be a \(\tau\)-tilting module given by a left mutation from \(A\). Let \(S, S_i\) be as in Definition 28. If the top of \(\text{Hom}_A(T, T_j)\) is of projective dimension \(l\), then \(\mathrm{findim}B^{op}\leq \mathrm{max}\{l\;\mathrm{and}\;\mathrm{dell}_T S_i|1\leq i\leq n, i\neq j\}.\)
Proof. Since \(P(i)\) is not in \(\text{Fac} T/P(i)\), by Definition 14, one gets that there is a left mutation of \(T\) at \(P(i)\) . Then by Definition 18, one gets that the corresponding brick \(S_i\) is non-zero. Then the assertion holds. ◻
Corollary 34. Let \(A\) be an algebra and \(T=P(1)\oplus P(2)\oplus\cdots\oplus T_j\oplus\cdots\oplus P(n)\) be an APR-tilting module. Let \(S_i\) be as in Definition 28. If the top of \(\text{Hom}_A(T, T_j)\) is of projective dimension \(l\), then \(\mathrm{findim}B^{op}\leq \mathrm{max}\{l\;\mathrm{and}\;\mathrm{dell}_T S_i|1\leq i\leq n, i\neq j\}.\)
We give the following example to illustrate Theorem 29 and 32, which also shows why Definition 28(2) is reasonable.
Example 35. Let \(A\) be an algebra given by the quiver \[\xymatrix{ 1\ar@<2pt>[r]^{\alpha}&2\ar@<2pt>[l]^{\beta} }\] with the relations \(\alpha\beta\alpha=\beta\alpha\beta=0\). Then
\(T=\substack{1\\2\\1}\oplus1\) is a \(\tau\)-tilting module.
\(B=\mathrm{End}_A T\) is given by\[\xymatrix{ 1\ar@<2pt>[r]^{\alpha}&2\ar@<2pt>[l]^{\beta} }\] with \(\beta\alpha=0\) and \(\mathrm{gldim}B=2\).
Since \(\mathcal{C}=\mathrm{Fac}T=\mathrm{FiltFac}\substack{1\\2}\), \(\mathrm{dell}_T\substack{1\\2}=1\) and \(2-\mathrm{dell}\;\mathrm{radHom}_A(T,T_2)=1\), one gets that \(\text{findim}B^{op}\leq\mathrm{dell}_T \mathcal{C}={\rm sup}\{\mathrm{dell}_T\substack{1\\2}=1, 2-\mathrm{dell}\;\mathrm{radHom}_A(T,T_2)+1\}=2\).
The projective dimension of \(S(2)(={\rm top}\;\mathrm{Hom}_A(T, T_2))\in\mathrm{mod}B\) is \(2\) and \(\text{findim}B^{op}\leq \mathrm{max}\{2\;\mathrm{and}\;\mathrm{dell}_T\substack{1\\2}\}=2\)
We end this section with the following example which shows the finite projective dimension of the top in Theorem 32 is not necessary.
Example 36. Let \(A\) be a preprojective algebra of type \(A_3\). Then
\(T=\substack{1\\2\\3}\oplus\substack{1\\2}\oplus\substack{2\\1}\) is a \(\tau\)-tilting module with \(B=\mathrm{End}_A T\) given by the quiver \[\xymatrix{3\ar@<2pt>[r]^b & 2\ar@<2pt>[l]^c \ar[r]^a & 1 }\] with the relations \(bc=cb=0\). The finitistic dimension of \(B\) is \(1\).
\(\mathcal{C}=\mathrm{Fac}T=T(S)=\mathrm{Filt}(\mathrm{Fac}(S_1\oplus S_3))\) with \(S_1=\substack{1\\2\\3}\) and \(S_3=\substack{\;\\2}\), \(\mathrm{dell}_T S_1=\mathrm{dell}_T S_3=0\). The projective dimension of the top of \(\mathrm{Hom}_A(T, T_2 )=S(2)\) is \(\infty\) with the \(2\)-\(\mathrm{dell}\Omega^1 S(2)\)=\(2\)-\(\mathrm{dell}S(3)=0\) since \(S(3)\cong \Omega^2 S(3)\). Then \(\mathrm{dell}_T\mathcal{C}=1\).
\(U=\substack{2\\3}\oplus\substack{2\\1\;\; 3\\2}\oplus\substack{2\\1}\) is a \(\tau\)-tilting module with \(C=\mathrm{End}_A U\) given by\[\xymatrix{ 1\ar@<2pt>[r]^{a} & 2\ar@<2pt>[l]^{d} \ar@<2pt>[r]^{b} & 3\ar@<2pt>[l]^c }\] with \(ab=cd=0, cb=ad=0, da=bc\). The finitistic dimension of \(C\) is \(1\).
\(\mathcal{C'}=\mathrm{Fac}U=T(S')=\mathrm{Filt}(\mathrm{Fac}S'_2)\) with \(S'_2=\substack{2\\1\;\;3}\). One gets that \(\mathrm{dell}_U S'_2=0\), the projective dimension of the top of \(\mathrm{Hom}_A(U, U_i )=S(i)\) is \(\infty\) for \(i=1, 3\), and the \(2\)-\(\mathrm{dell}\Omega^1 S(i)\)=\(2\)-\(\mathrm{dell}S(2)=0\) since \(S(2)\cong \Omega^2 S(2)\). Then \(\mathrm{dell}_U\mathcal{C'}=1\).
In this section, we apply Theorem 29 to the finitistic dimension conjecture. Throughout this section, \(T=\oplus_{i=1}^nT_i\) is a \(\tau\)-tilting module with \(T_i\) indecomposable and \(\mathcal{C}=\text{Fac}T=T(S)=\text{Filt(Fac}S)\), where \(S\simeq S_1\bigoplus \cdots \bigoplus S_m\) and \(S_i\) is a brick. Denote by \(B=\text{End}_A T\).
For a positive integer \(i\), denote by \(\Omega^i_T(\mathcal{C})\) the subcategory of the \(i\)-th syzygy of modules relative to \(T\) in \(\mathcal{C}\) and denote by \(L_i\) the set of indecomposable modules in \(\Omega^i_T(\mathcal{C})\) and indecomposable direct summands of \(T\). We have the following lemma.
Lemma 37. Let \(A\) be an algebra, \(T\in {\rm mod}A\) a \(\tau\)-tilting module and \(\mathcal{C}=\mathrm{Fac}T\). If \(L_1\) is a finite set, then global \(2\)-delooping level of \(\mathcal{C}\) relative to \(T\) is finite.
Proof. It is obvious that \(L_1\supseteq L_2\supseteq\cdots\supseteq L_n\supseteq\cdots\). Since there are finite number of elements in \(L_1\), one gets there is an \(n\geq 1\) such that \(L_n=L_{n+1}\) and hence \(L_{n+1}=L_{n+2}\) by the definition of \(L_i\). So one gets that \(L_n=L_{n+2}\). Then for any \(M\in \mathcal{C}\), one gets an \(N\in \mathcal{C}\) such that \(\Omega^n_T(M)\) is a direct summand of \(\Omega^{n+2}_T(N)\), that is, the \(2\)-delooping level of \(M\) relative to \(T\) is no more than \(n\). The assertion holds. ◻
To show the main theorem in this section, we also need the following lemma.
Lemma 38. Let \(A\) be an algebra, \(T\in {\rm mod}A\) a \(\tau\)-tilting module and \(\mathcal{C}=\mathrm{Fac}T\). If \(L_1\) is a finite set, then the delooping level of \(\mathcal{C}\) relative to \(T\) is finite.
Proof. It suffices to show that for a module \(M\) in \(\mathcal{C}\) the delooping level of \(M\) relative to \(T\) is no more than the \(2\)-delooping level of \(M\) relative to \(T\). By Lemma 5.1, we can assume that \(2\)-dell\(_T M=n\), that is, \(\Omega^n_T(M)\) is a direct summand of \(\Omega^{n+2}_T(N)\) for some \(N\in\mathcal{C}\). Then one gets that \(\Omega^n_T(M)\) is a direct summand of \(\Omega_T^{n+1}(\Omega^1_T N)\). So one gets the delooping level of \(S_i\) is no more than \(n\). By Definition 28, the assertion holds. ◻
Now we are in a position to state the main result in this section.
Theorem 39. Let \(A\) be an algebra, \(T\in {\rm mod}A\) a \(\tau\)-tilting module and \(B={\rm End}_A T\). If \({\rm Fac}T\cap {\rm Sub}T\) is of finite representation type, then the finitistic dimension of \(B^{op}\) is finite.
Proof. It is not easy to show that \({\rm Fac}T\cap {\rm Sub}T={\rm add}L_1\). Since \({\rm Fac}T\cap {\rm Sub}T\) is of finite representation type, one gets that \(L_1\) is a finite set. Then the assertion follows from Theorem 29, Lemma 37 and Lemma 38. ◻
We have the following straight corollaries.
Corollary 40. Let \(A\) be an algebra, \(T\in {\rm mod}A\) a \(\tau\)-tilting module and \(B={\rm End}_A T\). If \({\rm Fac}T\) or \({\rm Sub}T\) is of finite representation type, then the finitistic dimension of \(B^{op}\) is finite.
Corollary 41. Let \(A\) be an algebra of finite representation type, \(T\in {\rm mod}A\) a \(\tau\)-tilting module and \(B={\rm End}_A T\). Then the finitistic dimension of \(B^{op}\) is finite.
In the rest of this section we give another hint on the finitistic dimension conjecture via \(\tau\)-tilting module.
Proposition 42. Let \(A\) be an algebra, \(T\in {\rm mod}A\) a \(\tau\)-tilting module and \(B={\rm End}_A T\). Let \(\overline{A}=A/{\rm ann}T\). Then the finitistic dimension of \(B^{op}\) is finite if and only if so is \(\overline{A}\).
Proof. By Proposition 13, \(T\) is a tilting module in \({\rm mod}\overline{A}\). Since \({\rm Hom}_A(T, T)={\rm Hom}_{\overline{A}}(T, T)=B\), one gets that \(T\) is also a tilting module in \({\rm mod}B^{op}\). Therefore one gets that \(A\) and \(B\) are derived equivalent by [27]. Then by [12], the assertion holds. ◻
Recall that an algebra \(A\) is called minimal representation infinite if it is of infinite representation type and every factor algebra of \(A\) is of finite representation type. We have the following corollary.
Corollary 43. Let \(A\) be a minimal representation infinite algebra, \(T\in {\rm mod}A\) a \(\tau\)-tilting module and \(B={\rm End}_A T\). Then the finitistic dimension of \(B^{op}\) is finite.
Proof. By Proposition 13, \(T\) is a tilting module in \({\rm mod}\overline{A}\) and is also a tilting module in \({\rm mod}B^{op}\). Since \(\overline{A}\) is of finite representation type, the finitistic dimension of \(\overline{A}\) is finite. By Proposition 42, the assertion holds. ◻
Both of the authors are supported by the National Natural Science Foundation of China (Nos. 12171207 and 12371038). The authors also want to thank professors Xiao-Wu Chen, Zhi-Wei Li, Shengyong Pan and Zhibing Zhao for useful discussion and suggestions. They also thank Prof. Lei Chen for the help on latex.
Mingfei Xu and Xiaojin Zhang
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, Jiangsu, PR China.