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1.   Introduction

Throughout this paper, we denote a field by $k$, an algebra $R$ means a finite dimensional algebra over $k$, mod$\, R$ denotes the category of all finitely generated right $R$-modules, while ind$\, R$ denotes a full subcategory of mod$\, R$ containing one representative of each isomorphism class of indecomposable $R$-modules. We will use freely Auslander-Reiten translation $\tau = D\rm{Tr}$, irreducible maps and properties of the Auslander-Reiten sequences. For an $R$-module $X$, the injective hull of $X$ will be denoted by ${\rm E}(X)$, the projective (resp. injective) dimension of $X$ will be denoted by ${\rm pd}_RX$ (resp. ${\rm id}_RX$), and the global dimension of $R$ will be denoted by ${\rm gl.dim}R$.

As a proper generalization of tilted algebras, Happel has introduced quasi-tilted algebras in ^[1]. An algebra $R$ is called quasi-tilted if it satisfies the two conditions: (ⅰ) If $U$ is an indecomposable $R$-module, then ${\rm pd}_RU\leq 1$ or ${\rm id}_RU\leq 1$, and (ⅱ) ${\rm gl.dim}R\leq 2.$

Let $R$ and $S$ be algebras and $N$ be a finite dimensional left-$S$, right-$R$ bimodule over $k$, and set

Meanwhile, the addition and multiplication are defined as follows:

Then, $T$ is a finite-dimensional $k$-algebra, and $T$ is called a generalized lower triangular matrix algebra.

It follows from a result of ^[1] that every quasi-tilted artin algebra is isomorphic to a generalized lower triangular matrix algebra $\left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right)$, where $R$ and $S$ are hereditary. So to find all quasi-tilted algebras, it is enough to investigate when such generalized lower triangular matrix algebras are quasi-tilted.

The conditions for the one-point extension algebra of a quasi-tilted algebra to be again quasi-tilted were given by Huard ^[2] and Coelho ^[3,4]. Recently, ^[5] has obtained a new proof concerning quasi-tilted algebras. In ^[6], Assem characterized cluster-tilted and quasi-tilted algebras. For more results, one can see ^[7,8,9].

The construction of the generalized lower triangular matrix ring or algebra is a major issue in ring and algebraic representation theory, and it has been often used to construct counterexamples in rings and modules. Following ^[10], Gorenstein conditions over formal triangular matrix rings are studied by Enochs, and Gorenstein-projective modules over triangular matrix Artin algebras are also studied by Xiong and Zhang in ^[8]. As for more examples of generalized lower triangular matrix algebras, we refer to ^[7,11,12] and so on.

Based on the above discussion, it is meaningful to discuss quasi-tilted property of generalized lower triangular matrix algebra. In the present paper, we consider the problem of characterizing when a generalized lower triangular matrix algebra is quasi-tilted.

It is widely known that mod$\, T$ is equivalent to a category $\Omega$ of triples $(U, W, \varphi)$, where $U\in {\rm{mod}}\, R, \, W\in {\rm{mod}}\, S$ and $\varphi: W\otimes_S N\rightarrow U$ is a map in ${\rm{mod}} R.$ The right $T$-module which corresponds to $(U, W, \varphi)$ is the additive group $U\oplus W$ with right $T$-action given by:

We consider the right $T$-modules as triples $(U, W, \varphi)$. Given $(U_1, W_1, \varphi_1)$ and $(U_2, W_2, \varphi_2)$, a morphism $f: (U_1, W_1, \varphi_1)\longrightarrow (U_2, W_2, \varphi_2)$ is a pair $(f_1, f_2)$, where $f_1: U_1\rightarrow U_2$ is homomorphism in mod$\, R$ and $f_2: W_1\rightarrow W_2$ is homomorphism in mod$\, S$; furthermore, we can get that

is a commutative diagram. If $f = (f_1, f_2): (U_1, W_1, \varphi_1)\longrightarrow (U_2, W_2, \varphi_2)$ is a homomorphism map, then we have $f$ is surjective (resp. injective) if and only if $f_1: U_1\rightarrow U_2$ and $f_2: W_1\rightarrow W_2$ are surjective (resp., injective).

It is well-known that the right $T$-module $(U, W, \varphi)$ is a projective module if, and only if, $(U/\varphi (W\otimes N))_R$ and $W_S$ are projective modules and $\varphi: \, W\otimes N \longrightarrow U$ is one-one. So, we can get the following.

The $T$-module $(U, W, \varphi)$ is a projective module if, and only if, $W_S$ is a projective module, $\varphi: \, W\otimes N \longrightarrow U$ is monic, and $U = P\oplus \varphi(W\otimes N)$, where $P_R$ is projective.

For a right $T$-module $(U, W, \varphi)$, define

given by $\widetilde{\varphi}(w)(n) = \varphi(w\otimes n)$ for $n\in N, \, w\in W.$ Then, we can get $\widetilde{\varphi}$ is an $S$-homomorphism.

The right $T$-module $(U, W, \varphi)$ is an injective module if, and only if, $U_R$ is an injective module and the map $\Phi: \, W \longrightarrow {\rm Hom}_R(N, U)\oplus {\rm E}({\rm ker} \widetilde{\varphi})$ given by $\Phi(w) = (\widetilde{\varphi}(w), \tau (w))$ is an isomorphism of right $S$-modules, where $\tau : W \rightarrow {\rm E}({\rm ker} \widetilde{\varphi})$ is an extension of the inclusion ${\rm ker} \widetilde{\varphi} \hookrightarrow {\rm E}({\rm ker}\widetilde{\varphi}).$

We know that a right $T$-module $(U_R, 0, 0)$ is a projective module if, and only if, $U_R$ is projective, hence, ${\rm pd}_T (U_R, 0, 0) = {\rm pd}_R U_R$. A right $T$-module $(U_R, 0, 0)$ is an injective module if, and only if, $U_R$ is an injective module and ${\rm Hom}_R(N, U) = 0$. A right $T$-module $(0, W_S, 0)$ is a projective module if, and only if, $W_S$ is projective and $W\otimes N = 0$. A right $T$-module $(0, W_S, 0)$ is injective if, and only if, $W_S$ is injective, hence, ${\rm id}_T (0, W_S, 0) = {\rm id}_S W_S$.

2.   Quasi-tilted algebra

In this section, we consider the problem of characterizing when a generalized lower triangular matrix algebra is quasi-tilted.

Suppose $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right)$ is a generalized lower triangular matrix algebra, while Goodearl [2] gave a condition for $T$ to be hereditary.

Theorem 1 ^[13]. $T$ is a hereditary ring if, and only if,

(1) $R$ and $S$ are both hereditary.

(2) $_SN$ is a projective module.

(3) $(N/IN)_R$ is projective for all $I\leq S_S.$

We know ${\rm Hom}_R(W\otimes_SN, U)\stackrel{\sigma}{\cong} {\rm Hom}_S(W, {\rm Hom}_R(N, U))$; for $\varphi : \, W\otimes N \longrightarrow U$, let $\varphi': \, W \longrightarrow {\rm Hom}_R(N, U)$ be the map corresponding to $\varphi$, that is, $\sigma (\varphi) = \varphi'.$ So, we can get the following result.

Theorem 2. Suppose $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right), \, (U, W, \varphi)$ is a right $T$-module such that $\varphi'$ is a monomorphism. Then, the following conditions are equivalent.

(1) ${\rm id}_T (U, W, \varphi)\leq 1$.

(2) There exists an exact sequence $0\longrightarrow W\longrightarrow {\rm Hom}_R(N, I_0) \longrightarrow {\rm Hom}_R(N, I_1)\oplus I_2\longrightarrow 0$ with $I_0, \, I_1, \, I_2$ injective $R$-modules and ${\rm id}_R U\leq 1$.

Proof: Since ${\rm Hom}_R(W\otimes_SN, U)\stackrel{\sigma}{\cong} {\rm Hom}_S(W, {\rm Hom}_R(N, U)),$ we can view $(U, W, \varphi)$ as the triple $(U, W, \varphi').$

Suppose

is a minimal injective resolution of $U$; thus, we can get

Set $\varphi'' = {\rm Hom}_R(N, i)\circ \varphi',$ then $\varphi''$ is a monomorphism because $\varphi'$ is a monomorphism. We have that

is an injective envelope of $T$-module $(U, W, \varphi').$

Then, we can get the following diagram:

So, ${\rm Coker}(i, \varphi'') = (I_0(U)/{i(U)}, {\rm Hom}_R(N, I_0(U))/{\varphi''(W)}, g).$ The injective envelope of ${\rm Coker}(i, \varphi'')$ is

where $I = {\rm ker}g\cong {\rm Coker}\varphi'.$ Set $F = {\rm Hom}_R(N, -)$, hence, ${\rm id}_T (U, W, \varphi)\leq 1$ if, and only if,

is exact if, and only if,

and

are exact.

Given an algebra $C$, suppose ${{\mathcal{L}}_C}$ denotes the subset of ${\rm ind} C$ defined by

Also, we denote the full subcategory of ${{\mathcal{L}}_C}$ of direct summands of finite direct sums of copies of ${{\mathcal{L}}_C}$ by ${{\rm add} {\mathcal{L}}_C}.$

We now give the following necessary conditions for generalized lower triangular matrix algebra $R$ to be quasi-tilted.

Theorem 3. Let $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right)$. If $T$ is quasi-tilted, then

(1) Both $R$ and $S$ are quasi-tilted.

(2) $N_R\in {{\rm add}{\mathcal{L}}_R}.$

Proof: Since $T$ is quasi-tilted, we have ${\rm gl.dim} T\leq 2$, and it follows from [3] that ${\rm gl.dim} R\leq\; 2, \, {\rm gl.dim} S\leq\; 2$.

(1) If $S$ is not quasi-tilted, then there is an indecomposable $R$-module $W_S$ with ${\rm pd}_S W = 2 = {\rm id}_S W$. Since ${\rm id}_T (0, W_S, 0) = {\rm id}_S W = 2$ and $T$ is quasi-tilted, we know that ${\rm pd}_T (0, W_S, 0)\leq 1$. Assume that $\pi : \, P\rightarrow W_S$ is a projective cover of $W_S$, let $Z_S$ be the kernel of $\pi$, then $(0, \pi): \, (P\otimes N, P, 1)\rightarrow (0, W_S, 0)$ is a projective cover of $(0, W_S, 0)$ with kernel $(P\otimes N, Z_S, g)$. Since ${\rm pd}_T (0, W_S, 0)\leq 1$, we know that $(P\otimes N, Z_W, g)$ is a projective $T$-module, and then $Z_W$ is a projective $S$-module, thus, ${\rm pd}_S W \leq 1$. This contradicts that ${\rm pd}_S W = 2,$ so $S$ is quasi-tilted.

If $R$ is not quasi-tilted, then there exists an indecomposable $R$-module $U_R$ with ${\rm pd}_R U = 2 = {\rm id}_R U$. Since ${\rm pd}_T (U_R, 0, 0) = {\rm pd}_R U = 2$ and $T$ is quasi-tilted, we know ${\rm id}_T (U_R, 0, 0)\leq 1$, and by Theorem 2, ${\rm id}_R U \leq 1$. This contradicts that ${\rm id}_R U = 2,$ so $R$ is quasi-tilted.

(2) By (1), $R$ is quasi-tilted. If some indecomposable summand $N_1$ of $N$ is not in ${\mathcal{L}}_R,$ then, by definition of ${\mathcal{L}}_R,$ there exists some indecomposable predecessor $U_R$ of $N_1$ such that ${\rm pd}_R U = 2$. Then, ${\rm pd}_T (U, 0, 0) = 2$, and by [4], there exists an indecomposable injective $T$-module $L$ such that ${\rm Hom}_T(L, \tau(U, 0, 0))\neq 0.$ Thus, we have a sequence

of nonzero maps between indecomposable $T$-modules starting with an injective module and ending with a projective module. Because no refinement of the path can be sectional, we have a contradiction to $T$ being quasi-tilted. So, $M_R\in {{\rm add}{\mathcal{L}}_R}.$

Corollary 4 ^[1]. Suppose $k$ is a field, and $T = R[N] = \left(\begin{array}{cc} R & 0 \\ N & k \end{array} \right)$ is the one-point extension of $R$ by $N$. If $T$ is quasi-tilted, then

(1) $R$ is quasi-tilted.

(2) $N\in {\rm add}{\mathcal{L}}_R.$

Next, we will investigate what conditions need to be added to ensure that $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right)$ is quasi-tilted.

To investigate when $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right)$ is quasi-tilted, we first investigate when ${\rm gl.dim} T\leq 2$. By ^[1], we have:

Lemma 5. Let $R$ and $S$ be finite dimensional algebras over a field $k$ and $N$ a $S$-$R$ bimodule finite dimensional over $k$. Let $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right)$. Then, we can get

From this, we get that ${\rm gl.dim}T\leq 2$ if, and only if, ${\rm gl.dim}R\leq 2$ and ${\rm gl.dim}S\leq 2$ and ${\rm pd}_{S\otimes_kR^{op}} N\leq 1.$ From Lemma 5, we have

Theorem 6. Let $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right), \, (U, W, \varphi)$ be a right $T$-module. Then,

If $W_S$ is a projective module, then

Proof: Consider the following exact sequence:

in mod $T$. Then, it follows from a well-known result that

Since ${\rm pd}_T(U, 0, 0) = {\rm pd}_R U,$ then

If $W_S$ is a projective module, then we can get the following exact sequence:

in mod $T$. Because $(W\otimes N, W, 1)$ is projective, we can get that

In the equation above, suppose $W_S$ is not projective, and

is a projective cover of $W_S$ with the first syzygy $Z_S$, then we have the projective cover

with the first syzygy $(Q\otimes N, Z_S, \eta\otimes 1)$, where $\eta : \, Z_S \hookrightarrow Q_S$ is the inclusion map. Hence,

Corollary 7 ^[2]. Suppose $k$ is a field and $T = R[N] = \left(\begin{array}{cc} R & 0 \\ N & k \end{array} \right)$ is the one-point extension of $R$ by $N$. If $(U, k^s, \varphi)$ is a right $T$-module, then

Corollary 8. Let $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right)$. If $S$ is semi-simple and $N_R$ is projective, then ${\rm pd}_T (U, W, \varphi)\leq {\rm sup}\{{\rm pd}_R U, \, 1 \}$ for any $T$-module $(U, W, \varphi)$.

Corollary 9 ^[13]. Let $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right)$. If $R$ is hereditary, $N_R$ is projective, and $S$ is semi-simple, then $T$ is hereditary.

From Theorem 6, we also have

Corollary 10. Let $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right)$. If ${\rm gl.dim}R\leq 2$ and $S$ is semi-simple and ${\rm pd}_R N \leq 1$, then ${\rm gl.dim}T\leq 2.$ Proof: By Theorem 6, for any $T$-module $(U, W, \varphi)$, we can get

Theorem 11. Let $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right)$ with ${\rm gl.dim}R\leq 2.$ For any $T$-module $(U, W, \varphi),$ if ${\rm ker}\varphi$ is not projective, then ${\rm pd}_T (U, W, \varphi)\geq 2.$

Proof: Let $\varepsilon_2 : \, H \rightarrow W$ be a projective cover, and $\delta_1 : \, P \rightarrow U/\phi (W\otimes N)$ is a projective cover. Set $\theta = \varphi \circ (\varepsilon_2\otimes 1_N): \, H\otimes N \rightarrow U.$ Let $\eta : \, U \rightarrow U/\varphi (W\otimes N)$ be a natural homomorphism, $\varepsilon_1: \, (H\otimes N)\oplus P \rightarrow U$ such that $\varepsilon_1|_{H\otimes N} = \theta, \, \varepsilon_1|_P = \gamma_1,$ where $\gamma_1 : \, P \rightarrow U$ such that $\delta_1 = \eta\circ\gamma_1.$ We have a projective cover

According to the above discussion, we can get the following commutative diagram:

According to the snake lemma, it is easy to get the two exact sequences

and

We know that ${\rm pd}_T (U, W, \varphi)\leq 1$ if, and only if, $(K, {\rm ker} \varepsilon_2, \varphi)$ is projective. Since ${\rm gl.dim} R\leq 2$, we have ${\rm pd}_R {\rm coker}\varphi \leq 2$. Hence, ${\rm ker}\varphi$ is projective. Therefore, if ${\rm ker}\varphi$ is not projective, then ${\rm pd}_T (U, W, \varphi)\geq\; 2.$

From the above proof, if $R$ is a hereditary ring, then we can get that ${\rm pd}_T (U, W, \varphi)\leq 1$ if, and only if, ${\rm ker}\varphi$ is projective.

A sufficient condition for $T$ to be quasi-tilted is given by the following theorem.

Theorem 12. Let $T = \left(\begin{array}{cc} R & 0 \\ N & S \end{array} \right)$. If $R$ is quasi-tilted, $S$ is semi-simple and $N_R$ is projective, then $T$ is quasi-tilted.

Proof: According to Corollary 10, ${\rm gl.dim}T\leq 2.$ Let $(U, W, \varphi)$ be an indecomposable $T$-module, then $W$ is a semi-simple $S$-module and $U$ is an indecomposable $R$-module. Because $R$ is quasi-tilted, then ${\rm pd}_R U\leq 1$ or ${\rm id}_R U\leq 1.$

If ${\rm pd}_R U\leq 1,$ then, by Theorem 6, ${\rm pd}_T (U, W, \varphi)\leq 1.$

Assume that ${\rm id}_R U\leq 1$ and $0\rightarrow U\stackrel{i}{\rightarrow } I_0\stackrel{j}{\rightarrow } I_1\rightarrow 0$ is the minimal injective resolution of $U$, then

is exact since $N_R$ is projective.

Since ${\rm Hom}_R(Y\otimes_S N, U)\stackrel{\sigma}{\cong} {\rm Hom}_S(W, {\rm Hom}_R(N, U))$, for $\varphi: \, W\otimes N \longrightarrow U$, let $\varphi': \, W \longrightarrow {\rm Hom}_R(N, U)$ be the map corresponding to $\varphi$, that is, $\sigma (\varphi) = \varphi'.$ We can then view $(U, W, \varphi)$ as the triple $(U, W, \varphi')$ where $\varphi': \, W \longrightarrow {\rm Hom}_R(N, U)$ is a $S$-module homomorphism. Hence, $(U, W, \varphi')$ is an indecomposable $T$-module. If $\varphi'$ is not a monomorphism, then $\varphi' = 0$ and $(U, W, \varphi') = (0, W, 0),$ hence, ${\rm id}_T (U, W, \varphi') < 1.$ Thus, we can assume that $\varphi'$ is a monomorphism. Set $\varphi'' = i_*\circ \varphi',$ then $\varphi''$ is a monomorphism. We know that

is an injective envelope of $T$-module $(U, W, \varphi').$

Furthermore, we have the following commutative diagram:

By the Five Lemma in homological algebra, $g$ is epic.

We know that ${\rm Coker}(i, \varphi'') = (I_1, {\rm Hom}_R(N, I_0)/{\varphi''(W)}, g)$, and the injective envelope of ${\rm Coker}(i, \varphi'')$ is $(I_1, {\rm Hom}_R(N, I_1), 1)\oplus (0, {\rm ker}g, 0).$

We now show that ${\rm Hom}_R(N, I_0)/{\varphi''(W)}\cong {\rm Hom}_R(N, I_1)\oplus {\rm ker}g.$

Since $j_*$ and $g$ are epics, we have

Therefore,

So ${\rm id}(U, W, \varphi')\leq 1$ and $T$ is quasi-tilted.

From Theorems 3 and 12, we have:

Corollary 13 ^[2]. Let $R$ be an algebra with ${\rm gl.dim}R\leq 2, \, N$ a projective $R$-module, and let $T = R[N]$ be the one-point extension of $R$ by $N$. Then, $T$ is quasi-tilted if, and only if, $R$ is quasi-tilted.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We would like to thank the referees for the very helpful comments and suggestions which improved the original version of this paper. This work is supported by the Natural Science Foundation for Yang Scholars of Anhui Province (No.1508085QA04). The authors are grateful to professor Xianneng Du for careful guidance and assistance in this research.

Conflict of interest

The authors declare there is no conflicts of interest.