Stable equivalences of Morita type for $\Phi$-Beilinson-Green algebras

Shengyong Pan

arXiv:1907.01915·math.RT·August 13, 2019

Stable equivalences of Morita type for $\Phi$-Beilinson-Green algebras

Shengyong Pan

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TL;DR

This paper introduces a method to generate new stable equivalences of Morita type between algebras using $ ext{Phi}$-Beilinson-Green algebras, expanding understanding of algebra equivalences and their invariants.

Contribution

It presents a novel approach to construct stable Morita type equivalences via $ ext{Phi}$-Beilinson-Green algebras, generalizing previous results and exploring graded algebra cases.

Findings

01

Constructed new stable equivalences of Morita type for $ ext{Phi}$-Beilinson-Green algebras.

02

Provided examples of derived equivalent algebras not stably equivalent of Morita type.

03

Established a link between graded stable equivalences and Beilinson-Green algebras.

Abstract

In this paper, we present a method to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between finite dimensional algebras $A$ and $B$ is defined by a $B$ - $A$ -bimodule $N$ . Then, for any finite admissible set $Φ$ of natural numbers and any generator $X$ of the $A$ -module category, the $Φ$ -Beilinson-Green algebras $\scr G_{A}^{Φ} (X)$ and $\scr G_{B}^{Φ} (N \otimes_{A} X)$ are stably equivalent of Morita type. In particular, if $Φ = {0}$ , we get a known result in literature. As another consequence, we construct an infinite family of derived equivalent algebras of the same dimension and of the same dominant dimension such that they are pairwise not stably equivalent of Morita type. Finally, we will prove that, if there is a graded stable equivalence of Morita type between graded algebras, then we can get a stable equivalence of Morita…

Equations80

Hom_{C_{cov}^{F, Φ}} (X, Y) := i, j \in Φ ⨁ Hom_{C} (F^{i} X, F^{j} Y),

Hom_{C_{cov}^{F, Φ}} (X, Y) := i, j \in Φ ⨁ Hom_{C} (F^{i} X, F^{j} Y),

G_{A}^{Φ} (X, Y) = Hom_{G_{A}^{Φ}} (X, Y) := i, j \in Φ ⨁ Hom_{D^{b} (A)} (X [i], Y [j]) = i, j \in Φ ⨁ Hom_{D^{b} (A)} (X, Y [j - i]),

G_{A}^{Φ} (X, Y) = Hom_{G_{A}^{Φ}} (X, Y) := i, j \in Φ ⨁ Hom_{D^{b} (A)} (X [i], Y [j]) = i, j \in Φ ⨁ Hom_{D^{b} (A)} (X, Y [j - i]),

G_{A}^{Φ} (X, Y) \times G_{A}^{Φ} (Y, Z) ⟶ G_{A}^{Φ} (X, Z)

G_{A}^{Φ} (X, Y) \times G_{A}^{Φ} (Y, Z) ⟶ G_{A}^{Φ} (X, Z)

\big{(}(f_{ij})_{u\in\Phi},(g_{kl})_{v\in\Phi}\big{)}\mapsto(h_{st})_{s,t\in\Phi},

\big{(}(f_{ij})_{u\in\Phi},(g_{kl})_{v\in\Phi}\big{)}\mapsto(h_{st})_{s,t\in\Phi},

h_{st}:=\left\{\begin{array}[]{cc}0&if\ \ 0\leq s<t,\\ \sum_{\begin{subarray}{c}k-t\in\Phi\\ l-k\in\Phi\end{subarray}}f_{lk}F^{l-k}g_{kt}&if\ 0\leq t\leq s.\end{array}\right.

h_{st}:=\left\{\begin{array}[]{cc}0&if\ \ 0\leq s<t,\\ \sum_{\begin{subarray}{c}k-t\in\Phi\\ l-k\in\Phi\end{subarray}}f_{lk}F^{l-k}g_{kt}&if\ 0\leq t\leq s.\end{array}\right.

G_{A}^{Φ} (X) = E (X)_{0, 0} 0 ⋮ 00 \dots E (X)_{1, 1} ⋮ 00 \dots \dots ⋮ \dots \dots E (X)_{0, m - 1} E (X)_{1, m - 1} ⋮ E (X)_{m - 1, m - 1} 0 E (X)_{0, m} E (X)_{1, m} ⋮ E (X)_{m - 1, m} E (X)_{m, m}_{Φ \times Φ},

G_{A}^{Φ} (X) = E (X)_{0, 0} 0 ⋮ 00 \dots E (X)_{1, 1} ⋮ 00 \dots \dots ⋮ \dots \dots E (X)_{0, m - 1} E (X)_{1, m - 1} ⋮ E (X)_{m - 1, m - 1} 0 E (X)_{0, m} E (X)_{1, m} ⋮ E (X)_{m - 1, m} E (X)_{m, m}_{Φ \times Φ},

e_{i} = 00 ⋮ 00 00 ⋮ 00 00 ⋮ 00 \dots \dots 1_{End_{D^{b} (A)} (X)} \dots \dots 00 ⋮ 00 00 ⋮ 00 .

e_{i} = 00 ⋮ 00 00 ⋮ 00 00 ⋮ 00 \dots \dots 1_{End_{D^{b} (A)} (X)} \dots \dots 00 ⋮ 00 00 ⋮ 00 .

rad (G_{A}^{Φ} (X)) = rad (End_{A} (X)) 0 ⋮ 0 Ext^{1} (X, X) rad (End_{A} (X)) ⋮ 0 Ext^{2} (X, X) Ext^{1} (X, X) ⋮ 0 \dots \dots ⋮ \dots Ext^{i} (X, X) Ext^{i} (X, X) ⋮ rad (End_{A} (X)),

rad (G_{A}^{Φ} (X)) = rad (End_{A} (X)) 0 ⋮ 0 Ext^{1} (X, X) rad (End_{A} (X)) ⋮ 0 Ext^{2} (X, X) Ext^{1} (X, X) ⋮ 0 \dots \dots ⋮ \dots Ext^{i} (X, X) Ext^{i} (X, X) ⋮ rad (End_{A} (X)),

φ_{Φ} : G_{A}^{Φ} (X, Y) \otimes_{G_{B}^{Φ} (B)} P_{C} 0 ⋮ 0 0 P_{C} ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ P_{C} \to G_{A}^{Φ} (X, Y \otimes_{B} P_{C})

φ_{Φ} : G_{A}^{Φ} (X, Y) \otimes_{G_{B}^{Φ} (B)} P_{C} 0 ⋮ 0 0 P_{C} ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ P_{C} \to G_{A}^{Φ} (X, Y \otimes_{B} P_{C})

φ ((c \otimes d) \cdot a) = φ (c \otimes (d F (a))) = c G (d F (a)) = c G (d) (GF) (a) = φ (c \otimes d) \cdot a .

φ ((c \otimes d) \cdot a) = φ (c \otimes (d F (a))) = c G (d F (a)) = c G (d) (GF) (a) = φ (c \otimes d) \cdot a .

(*) U \otimes_{Γ} V ≃ G_{A}^{Φ} (X, (T_{M} T_{N}) (X)) ≃ G_{A}^{Φ} (X, X) \oplus G_{A}^{Φ} (X, P \otimes_{A} X) = Λ \oplus G_{A}^{Φ} (X, P \otimes_{A} X),

(*) U \otimes_{Γ} V ≃ G_{A}^{Φ} (X, (T_{M} T_{N}) (X)) ≃ G_{A}^{Φ} (X, X) \oplus G_{A}^{Φ} (X, P \otimes_{A} X) = Λ \oplus G_{A}^{Φ} (X, P \otimes_{A} X),

Λ \to G_{A}^{Φ} (P \otimes_{A} X),

Λ \to G_{A}^{Φ} (P \otimes_{A} X),

α_{A \otimes_{k} A} : Λ \to G_{A}^{Φ} ((A \otimes_{k} A) \otimes_{A} X),

α_{A \otimes_{k} A} : Λ \to G_{A}^{Φ} ((A \otimes_{k} A) \otimes_{A} X),

π : Λ \to End_{A} (X) 0 ⋮ 0 0 End_{A} (X) ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ End_{A} (X)

π : Λ \to End_{A} (X) 0 ⋮ 0 0 End_{A} (X) ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ End_{A} (X)

{\rm End}_{A}(X)\rightarrow{\rm End}_{A}\big{(}(A\otimes_{k}A)\otimes_{A}X\big{)},

{\rm End}_{A}(X)\rightarrow{\rm End}_{A}\big{(}(A\otimes_{k}A)\otimes_{A}X\big{)},

\begin{array}[]{rl}\Lambda=\mathscr{G}^{\Phi}_{A}(X)\xrightarrow{\pi}\begin{pmatrix}{\rm End}_{A}(X)&0&0&\cdots&0\\ 0&{\rm End}_{A}(X)&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&{\rm End}_{A}(X)\end{pmatrix}\xrightarrow{\begin{pmatrix}\mu&0&0&\cdots&0\\ 0&\mu&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&\mu\end{pmatrix}}\\ \begin{pmatrix}{\rm End}_{A}((A\otimes_{k}X))&0&0&\cdots&0\\ 0&{\rm End}_{A}((A\otimes_{k}X))&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&{\rm End}_{A}((A\otimes_{k}X))\end{pmatrix}=\mathscr{G}^{\Phi}_{A}(A\otimes_{k}X),\end{array}

\begin{array}[]{rl}\Lambda=\mathscr{G}^{\Phi}_{A}(X)\xrightarrow{\pi}\begin{pmatrix}{\rm End}_{A}(X)&0&0&\cdots&0\\ 0&{\rm End}_{A}(X)&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&{\rm End}_{A}(X)\end{pmatrix}\xrightarrow{\begin{pmatrix}\mu&0&0&\cdots&0\\ 0&\mu&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&\mu\end{pmatrix}}\\ \begin{pmatrix}{\rm End}_{A}((A\otimes_{k}X))&0&0&\cdots&0\\ 0&{\rm End}_{A}((A\otimes_{k}X))&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&{\rm End}_{A}((A\otimes_{k}X))\end{pmatrix}=\mathscr{G}^{\Phi}_{A}(A\otimes_{k}X),\end{array}

G_{A}^{Φ} (X, A \otimes_{k} X) ≃ G_{A}^{Φ} (X, A) \otimes X 0 ⋮ 0 0 X ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ X

G_{A}^{Φ} (X, A \otimes_{k} X) ≃ G_{A}^{Φ} (X, A) \otimes X 0 ⋮ 0 0 X ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ X

Λ π End_{A} (X) 0 ⋮ 0 0 End_{A} (X) ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ End_{A} (X),

Λ π End_{A} (X) 0 ⋮ 0 0 End_{A} (X) ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ End_{A} (X),

G_{A}^{Φ} (X, A \otimes_{k} X) ≃ G_{A}^{Φ} (X, A) \otimes_{G_{k}^{Φ} (k)} X 0 ⋮ 0 0 X ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ X

G_{A}^{Φ} (X, A \otimes_{k} X) ≃ G_{A}^{Φ} (X, A) \otimes_{G_{k}^{Φ} (k)} X 0 ⋮ 0 0 X ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ X

X 0 ⋮ 0 0 X ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ X ≃ G_{A}^{Φ} (A, X)

X 0 ⋮ 0 0 X ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ X ≃ G_{A}^{Φ} (A, X)

G_{A}^{Φ} (X, A) \otimes_{G_{k}^{Φ} (k)} X 0 ⋮ 0 0 X ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ X \in (G_{A}^{Φ} (X, A) \otimes_{G_{k}^{Φ} (k)} G_{A}^{Φ} (A, X)

G_{A}^{Φ} (X, A) \otimes_{G_{k}^{Φ} (k)} X 0 ⋮ 0 0 X ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ X \in (G_{A}^{Φ} (X, A) \otimes_{G_{k}^{Φ} (k)} G_{A}^{Φ} (A, X)

(* *) V \otimes_{Λ} U ≃ Z ≃ Γ \oplus G_{B}^{Φ} (N \otimes_{A} X, Q \otimes_{B} (N \otimes_{A} X)),

(* *) V \otimes_{Λ} U ≃ Z ≃ Γ \oplus G_{B}^{Φ} (N \otimes_{A} X, Q \otimes_{B} (N \otimes_{A} X)),

G_{A}^{Φ} (X, A) = Hom_{A} (X, A) 0 ⋮ 0 0 Hom_{A} (X, A) ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ Hom_{A} (X, A) .

G_{A}^{Φ} (X, A) = Hom_{A} (X, A) 0 ⋮ 0 0 Hom_{A} (X, A) ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ Hom_{A} (X, A) .

000 ⋮ 0 Ext^{1} (X, X) 00 ⋮ 0 Ext^{2} (X, X) Ext^{1} (X, X) 0 ⋮ 0 \dots \dots \dots ⋮ \dots Ext^{m} (X, X) Ext^{m - 1} (X, X) Ext^{1} (X, X) ⋮ 0,

000 ⋮ 0 Ext^{1} (X, X) 00 ⋮ 0 Ext^{2} (X, X) Ext^{1} (X, X) 0 ⋮ 0 \dots \dots \dots ⋮ \dots Ext^{m} (X, X) Ext^{m - 1} (X, X) Ext^{1} (X, X) ⋮ 0,

e^{'} Λ e^{'} ≃ A 0 ⋮ 0 0 A ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ A ≃ A \times A \times \dots \times A

e^{'} Λ e^{'} ≃ A 0 ⋮ 0 0 A ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ A ≃ A \times A \times \dots \times A

f^{'} Γ f^{'} ≃ B 0 ⋮ 0 0 B ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ B ≃ \times B \times B \times \dots \times B

f^{'} Γ f^{'} ≃ B 0 ⋮ 0 0 B ⋮ 0 00 ⋮ 0 \dots \dots ⋮ \dots 00 ⋮ B ≃ \times B \times B \times \dots \times B

di m_{_{k}} (Λ_{m}^{{0}}) = di m_{_{k}} End_{A} (M) + di m_{_{k}} Hom_{A} (M, I_{m}) + di m_{_{k}} Hom_{A} (I_{m}, M) + di m_{_{k}} End_{A} (I_{m})

di m_{_{k}} (Λ_{m}^{{0}}) = di m_{_{k}} End_{A} (M) + di m_{_{k}} Hom_{A} (M, I_{m}) + di m_{_{k}} Hom_{A} (I_{m}, M) + di m_{_{k}} End_{A} (I_{m})

di m_{_{k}} (Λ_{m}^{{0, 1}}) = 2 di m_{_{k}} (Λ_{m}^{{0}}) + di m_{_{k}} Ext_{A}^{1} (M, M) + di m_{_{k}} Ext_{A}^{1} (I_{m}, I_{m}) .

di m_{_{k}} (Λ_{m}^{{0, 1}}) = 2 di m_{_{k}} (Λ_{m}^{{0}}) + di m_{_{k}} Ext_{A}^{1} (M, M) + di m_{_{k}} Ext_{A}^{1} (I_{m}, I_{m}) .

di m_{_{k}} End_{A} (I_{m}) = 3, di m_{_{k}} Ext_{A}^{1} (I_{m}, I_{m}) = 1, di m_{_{k}} Hom_{A} (M, I_{m}) = di m_{_{k}} Hom_{A} (I_{m}, M) = 2 n + 6.

di m_{_{k}} End_{A} (I_{m}) = 3, di m_{_{k}} Ext_{A}^{1} (I_{m}, I_{m}) = 1, di m_{_{k}} Hom_{A} (M, I_{m}) = di m_{_{k}} Hom_{A} (I_{m}, M) = 2 n + 6.

Λ_{m}^{{0, 1}} = (Λ_{m}^{{0}} 0 Ext^{1} (M \oplus I_{m}) Λ_{m}^{{0}})

Λ_{m}^{{0, 1}} = (Λ_{m}^{{0}} 0 Ext^{1} (M \oplus I_{m}) Λ_{m}^{{0}})

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TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Rings, Modules, and Algebras

Full text

Abstract

We present a method to construct new stable equivalences of Morita type. Suppose that a $B$ - $A$ -bimodule $N$ define a stable equivalence of Morita type between finite dimensional algebras $A$ and $B$ . Then, for any generator $X$ of the $A$ -module category and any finite admissible set $\Phi$ of natural numbers, the $\Phi$ -Beilinson-Green algebras $\mathscr{G}^{\Phi}_{A}(X)$ and $\mathscr{G}^{\Phi}_{B}(N\otimes_{A}X)$ are stably equivalent of Morita type. In particular, if $\Phi=\{0\}$ , we get a known result in literature. As another consequence, we construct an infinite family of derived equivalent algebras of the same dimension and of the same dominant dimension such that they are pairwise not stably equivalent of Morita type. Finally, we develop some techniques for proving that, if there is a graded stable equivalence of Morita type between graded algebras, then we can get a stable equivalence of Morita type between Beilinson-Green algebras associated with graded algebras.

Stable equivalences of Morita type for $\Phi$ -Beilinson-Green algebras

Shengyong Pan

Department of Mathematics,

Beijing Jiaotong University,

100044 Beijing, People’s Republic of China

E-mail: [email protected]

†† ∗ Corresponding author. Email: [email protected].††2000 Mathematics Subject Classification: 18E30,16G10;16S10,18G15.††Keywords: Admissible sets, stable equivalences of Morita type, self-injective algebras, derived equivalences, Beilinson-Green algebras.

1 Introduction

In the representation theory of algebras and groups, there are three fundamental equivalences: Morita, derived and stable equivalences. A cycle of interconnected questions and conjectures known as “Morita invariances” have inspired a lot of work over the last few decades. As is known, for the first two, there is a corresponding Morita theory for each (see [24, 29]), while much less is known for the last. Recently, a special class of stable equivalences, called stable equivalences of Morita type, are introduced by Broué [4] in modular representations of finite groups. They are close to derived equivalences, are induced by bimodules, and are shown to be of great interest in modern representation theory since they preserve many homological and structural invariants of algebras and modules (see, for example, [4, 7, 15, 16, 27, 32, 33]). Although we have a better understanding of stable equivalences of Morita type than on general stable equivalences, we still cannot give an answer for Auslander-Reiten conjecture (see, for Auslander-Reiten conjecture, [2, 22]). In order to understand this kind of equivalences, one has to know, examples and basic properties of stable equivalences of Morita type as many as possible.

Only a few methods so far using one-point extensions, endomorphism algebras and Auslander-Yoneda algebras were known in [19, 20, 21, 5]. Rickard’s result that the existence of derived equivalences for self-injective algebras imply the one of stable equivalences of Morita type provides another way to construct stable equivalences of Morita type. This method is no longer true for general finite-dimensional algebras (see [11] for some new advances in this direction). Moreover, unlike tensor products and trivial extensions preserved by derived equivalences [28], stable equivalences of Morita type do not preserve tensor products and trivial extensions [22]. So, a systematical method for constructing stable equivalences of Morita type seems not yet to be available.

In this paper, we shall look for a more general method to construct new stable equivalences of Morita type for general finite-dimensional algebras. The procedure has two flexibilities, one is the choice of generators, and the other is the one of finite admissible sets. Thus this construction provides a large variety of stable equivalences of Morita type.

To state our first main result, let us recall the definition of $\Phi$ -Beilinson-Green algebras. Let $A$ be a finite-dimensional algebra and $X$ an $A$ -module. Then, for an admissible set $\Phi$ of natural numbers, there is defined an algebra $\mathscr{G}^{\Phi}_{A}(X)$ , called the $\Phi$ -Beilinson-Green algebra of $X$ in [25, 26], which is equal to $\bigoplus_{i,j\in\Phi}\mbox{Ext}^{j-i}_{A}(X,X)$ as a vector space, and its multiplication is defined in a natural way (see Subsection 2.2 for details). Our main result reads as follows:

Theorem 1.1.

Let $A$ and $B$ be finite-dimensional $k$ -algebras over a field $k$ . Assume that two bimodules ${}_{A}M_{B}$ and ${}_{B}N_{A}$ define a stable equivalence of Morita type between $A$ and $B$ . Let $X$ be an $A$ -module which is a generator for $A$ -module category. Then, for any finite admissible set $\Phi$ of natural numbers, there is a stable equivalence of Morita type between $\mathscr{G}^{\Phi}_{A}(X)$ and $\mathscr{G}^{\Phi}_{B}(N\otimes_{A}X)$ .

Remark 1.2.

$(1)$ Note that if $\Phi=\{0\}$ , then the above result was known in [5, 21]. Thus Theorem 1.1 generalizes the main result in [21], and provides much more possibilities through $\Phi$ for constructing stable equivalences of Morita type. Also, our proof of Theorem 1.1 is different from that in [5, 21].

$(2)$ Note that $\Phi$ -Beilinson-Green algebra is a C-construction which is different from $\Phi$ -Auslander-Yoneda algebra, however, $\Phi$ -Auslander-Yoneda algebra is B-construction in the sense of Mori [23].

We have the following characterization of stable equivalences of Morita type for self-injective algebras from Theorem 1.1 and [5, Theorem 2].

Corollary 1.3.

Suppose that $A$ and $B$ are finite-dimensional self-injective $k$ -algebras over a field $k$ such that neither $A$ nor $B$ has semisimple direct summands. Let $X$ be an $A$ -module and let $Y$ be a $B$ -module. If there is a finite admissible set $\Phi$ of natural numbers such that $\mathscr{G}^{\Phi}_{A}(A\oplus X)$ and $\mathscr{G}^{\Phi}_{B}(B\oplus Y)$ are stably equivalent of Morita type, then, for any finite admissible set $\Psi$ of natural numbers, the algebras $\mathscr{G}^{\Psi}_{A}(A\oplus X)$ and $\mathscr{G}^{\Psi}_{B}(B\oplus Y)$ are stably equivalent of Morita type.

As another byproduct of our considerations in this paper, we can construct a family of derived equivalent algebras with certain special properties, but derived equivalent algebras are not stably equivalent of Morita type.

Corollary 1.4.

There is an infinite series of $k$ -algebras of the same dimension over an algebraically closed field such that they have the same dominant and global dimensions, and are all derived equivalent, but not pairwise stably equivalent of Morita type.

Finally, we develop some techniques for proving that, for finite group $G$ , we get stable equivalences of Morita type between Beilinson-Green algebra of $G$ -graded algebra from $G$ -graded stable equivalences of Morita type. For a $G$ -graded $k$ -algebra $A=\oplus_{g\in G}A_{g}$ , there is a Beilinson-Green algebra $\bar{A}$ defined as a $G\times G$ -matrix algebra with $(\bar{A}_{gh})_{g,h\in G}$ , where $\bar{A}_{gh}=A_{gh^{-1}}$ .

Theorem 1.5.

Suppose that there is a $G$ -graded stable equivalence of Morita type between $G$ -graded algebras $A$ and $B$ , then there is a stable equivalence of Morita type between the Beilinson-Green algebras $\bar{A}$ and $\bar{B}$ of $A$ and $B$ , respectively.

We mention that the proof of Theorem 1.5 is somewhat routine. However, the assertions seem to be quite surprising.

The paper is structured as follows. In Section 2, we shall fix notations and prepare some basic facts for our proofs. In Section 3, we shall prove our main results, Theorem 1.1. In Section 4.1, we will concentrate our consideration on self-injective algebras, and establish some applications of our main results. In particular, in this section we shall prove Corollary 1.3 and supply a sufficient condition to verify when two algebras are not stably equivalent of Morita type to another algebra, which will be used in Section 4.2. In Section 4.2, we shall apply our results in the previous sections to Liu-Schulz algebras and give a proof of Corollary 1.4 which answers a question by Thorsten Holm. In Section 5, for finite group, we can construct stable equivalences of Morita type between Beilinson-Green algebra of group graded algebra from group graded stable equivalences of Morita type. We will give an example to illustrate our main result in Section 6.

Acknowledgements. Shengyong Pan is funded by China Scholarship Council. He thanks Professors Hongxing Chen and Wei Hu for useful discussions and suggestions, and also thanks his wife Ran Wen, his sons Ruipeng and Ruiqi for their encouragements. This paper is done during a visit of Shengyong Pan to the University of Edinburgh, he would like to thank Professor Susan J. Sierra for her hospitality and useful discussions.

2 Preliminaries

In this section, we fix some notations, and recall some definitions and basic results which are needed in the proofs of our main results.

2.1 Some conventions and homological facts

Throughout this paper, $k$ stands for a fixed field. All categories and functors will be $k$ -categories and $k$ -functors, respectively. Unless stated otherwise, all algebras considered are finite-dimensional $k$ -algebras, and all modules are finitely generated left modules.

Let $\mathcal{C}$ be a category. Given two morphisms $f:X\to Y$ and $g:Y\to Z$ in $\mathcal{C}$ , we denote the composition of $f$ and $g$ by $fg$ which is a morphism from $X$ to $Z$ , while we denote the composition of a functor $F:\mathcal{C}\to\mathcal{D}$ between categories $\mathcal{C}$ and $\mathcal{D}$ with a functor $G:\mathcal{D}\rightarrow\mathcal{E}$ between categories $\mathcal{D}$ and $\mathcal{E}$ by $GF$ which is a functor from $\mathcal{C}$ to $\mathcal{E}$ .

If $\cal C$ is an additive category and $X$ is an object in $\mathcal{C}$ , we denote by ${\rm add}(X)$ the full subcategory of $\mathcal{C}$ consisting of all direct summands of finite direct sums of copies of $X$ . The object $X$ is called an additive generator for $\mathcal{C}$ if add( $X$ ) = $\mathcal{C}$ .

Let $A$ be an algebra. We denote by $A\mbox{{\rm-mod}}$ the category of all $A$ -modules, by $A$ -proj (respectively, $A$ -inj) the full subcategory of $A\mbox{{\rm-mod}}$ consisting of projective (respectively, injective) modules, by $D$ the usual $k$ -duality ${\rm Hom}_{k}(-,k)$ , and by $\nu_{A}$ the Nakayama functor $D{\rm Hom}_{A}(-,\,_{A}A)$ of $A$ . Note that $\nu_{A}$ is an equivalence from $A$ -proj to $A$ -inj with the inverse ${\rm Hom}_{A}(D(A),-)$ . We denote the global and dominant dimensions of $A$ by gl.dim $(A)$ and dom.dim $(A)$ , respectively.

As usual, by ${\mathscr{D}}^{b}(A)$ we denote the bounded derived category of complexes over $A$ -mod. It is known that $A$ -mod is fully embedded in ${\mathscr{D}}^{b}(A)$ and that ${\rm Hom}_{{\mathscr{D}}^{b}(A)}(X,Y[i])\simeq{\rm Ext}^{i}_{A}(X,Y)$ for all $i\geq 0$ and $A$ -modules $X$ and $Y$ . Let $X$ be an $A$ -module. We denote by $\Omega^{i}_{A}(X)$ the $i$ -th syzygy, by ${\rm soc}(X)$ the socle, and by rad $(X)$ the Jacobson radical of $X$ .

Let $X$ be an additive generator for $A$ -mod. The endomorphism algebra of $X$ is called the Auslander algebra of $A$ . This algebra is, up to Morita equivalence, uniquely determined by $A$ . Note that Auslander algebras can be described by two homological properties: An algebra $A$ is an Auslander algebra if gl.dim( $A$ ) $\leq 2\leq{\rm dom.dim}(A)$ .

An $A$ -module $X$ is called a generator for $A$ -mod if ${\rm add}(_{A}A)\subseteq{\rm add}(X)$ ; a cogenerator for $A$ -mod if ${\rm add}(D(A_{A}))\subseteq{\rm add}(X)$ , and a generator-cogenerator if it is both a generator and a cogenerator for $A$ -mod. Clearly, an additive generator for $A$ -mod is a generator-cogenerator for $A$ -mod. But the converse is not true in general.

2.2 Admissible sets and perforated covering categories

In [25, 26], a class of algebras, called $\Phi$ -Green algebras, were introduced. Recall that a subset $\Phi$ of $\mathbb{N}$ is said to be admissible provided that $0\in\Phi$ and that for any $p,q,r\in\Phi$ with $p+q+r\in\Phi$ we have $p+q\in\Phi$ if and only if $q+r\in\Phi$ . As shown in [13], there are a lot of admissible subsets of $\mathbb{N}$ . For example, given any subset $S$ of $\mathbb{N}$ containing [math], the set $\{x^{m}\mid x\in S\}$ is admissible for all $m\geq 3.$

Assume that $\Phi$ is an admissible subset of $\mathbb{N}$ . Let $\mathcal{C}$ be a $k$ -category, and let $F$ be an additive functor from $\mathcal{C}$ to itself. The $(F,\Phi)$ -covering category ${\mathcal{C}}_{{\rm cov}}^{F,\Phi}$ of $\mathcal{C}$ is a category in which the objects are the same as that of $\mathcal{C}$ , and the morphism set between two objects $X$ and $Y$ is defined to be

[TABLE]

and the composition is defined in an obvious way. Since $\Phi$ is admissible, ${\mathcal{C}}_{{\rm cov}}^{F,\Phi}$ is an additive $k$ -category. In particular, ${\rm Hom}_{{\mathcal{C}}_{{\rm cov}}^{F,\Phi}}(X,X)$ is a $k$ -algebra (which may not be finite-dimensional), and ${\rm Hom}_{{\mathcal{C}}_{{\rm cov}}^{F,\Phi}}(X,Y)$ is an ${\rm Hom}_{{\mathcal{C}}^{F,\Phi}}(X,X)$ - ${\rm Hom}_{{\mathcal{C}}^{F,\Phi}}(Y,Y)$ -bimodule. For more details, we refer the reader to [25, 26]. In this paper, the category ${\mathcal{C}}_{{\rm cov}}^{F,\Phi}$ is simply called a perforated covering category, and the algebra ${\rm Hom}_{{\mathcal{C}}^{F,\Phi}}(X,X)$ is called perforated covering algebra of $X$ without mentioning $F$ and $\Phi$ .

In case $\mathcal{C}$ is the bounded derived category ${\mathscr{D}}^{b}(A)$ with $A$ a $k$ -algebra, and $F$ is the shift functor [1] of ${\mathscr{D}}^{b}(A)$ , we denote simply by ${\mathcal{G}}^{\Phi}_{A}$ the $(F,\Phi)$ -covering category ${\mathcal{C}}^{F,\Phi}$ , by $\mathscr{G}_{A}^{\Phi}(X,Y)$ the set ${\rm Hom}_{{\mathcal{G}}^{\Phi}_{A}}(X,Y)$ ,

[TABLE]

Denote by $\mathscr{G}_{A}^{\Phi}(X)$ the endomorphism algebra ${\rm Hom}_{{\mathcal{G}}^{\Phi}_{A}}(X,X)$ of $X$ in ${\mathcal{G}}^{\Phi}_{A}$ . In this case, each element in $\mathscr{G}_{A}^{\Phi}(X,Y)$ can be written as $(f_{ij})_{i,j\in\Phi}$ with $f_{ij}\in{\rm Hom}_{{\mathscr{D}}^{b}(A)}(X,Y[j-i])$ . The composition of morphisms in ${\mathcal{G}}^{\Phi}_{A}$ can be interpreted as follows: for each triple $(X,Y,Z)$ of objects in ${\mathscr{D}}^{b}(A)$ ,

[TABLE]

where

[TABLE]

for $s,t\in\Phi$ . Clearly, if $\Phi$ is finite, then $\mathscr{G}_{A}^{\Phi}(X,Y)$ is finite-dimensional for all $X,Y\in A\mbox{{\rm-mod}}$ .

Let $\Phi$ be a finite admissible subset of $\mathbb{N}$ and let $m=max\{i|i\in\Phi\}$ . Thus we get $m\geq 0$ . Then the algebras $\mathscr{G}_{A}^{\Phi}(X)$ for an object $X$ in ${\mathscr{D}}^{b}(A)$ , and mention some basic properties of these algebras. Firstly, let us define an $R$ -module $\mathscr{G}_{A}^{\Phi}(X)$ as follows:

[TABLE]

where ${\rm E}(X)_{i,j}={\rm Hom}_{{\mathscr{D}}^{b}(A)}(X,X[j-i])$ , $s\leq i,j\leq m$ . If $j-i\neq\Phi$ , then ${\rm E}(X)_{i,j}={\rm Hom}_{{\mathscr{D}}^{b}(A)}(X,X[j-i])=0$ . That is, $\mathscr{G}_{A}^{\Phi}(X)=({\rm E}(X)_{i,j})_{s\leq i,j\leq m}$ .

Recall that the triangular matrix algebra of this form seems first to appear in the paper [9] by Edward L. Green in 1975. A special case of this kind of algebras appeared in the paper [3] by A. A. Beilinson in 1978, where he described the derived category of coherent sheaves over $\mathbb{P}^{n}$ as the one of this triangular matrix algebra. Perhaps it is more appropriate to name this triangular matrix algebra as the $\Phi$ -Beilinson-Green algebra of $X$ .

The following lemma tells us that $\mathscr{G}_{A}^{\Phi}(X)$ is an associative ring by the multiplication defined above.

Lemma 2.1.

With the notations above, $\Phi$ is admissible if and only if $\mathscr{G}_{A}^{\Phi}(X)$ is an associative ring.

Proof. It follows from that $\Phi$ is admissible and the multiplication of $\mathscr{G}_{A}^{\Phi}(X)$ is defined by compositions. $\square$

The following fact of the $\Phi$ -Green algebra $\mathscr{G}_{A}^{\Phi}(X)$ is useful, which can be easily check. Let

[TABLE]

Therefore, $1_{\mathscr{G}_{A}^{\Phi}(X)}=\Sigma^{m}_{i=s}{e_{i}}$ is the identity element of $\mathscr{G}_{A}^{\Phi}(X)$ and as a left $\mathscr{G}_{A}^{\Phi}(X)$ -module, $\mathscr{G}_{A}^{\Phi}(X)e_{i}\cong{\rm E}^{\Phi,i}(X),$ where ${\rm E}^{\Phi,i}(X)=\oplus_{j=s}^{m}{\rm E}^{i-j,\Phi}_{{\mathscr{D}}^{b}(A)}(X)$ . Then $\mathscr{G}_{A}^{\Phi}(X)\cong\oplus_{i=s}^{m}{\rm E}^{\Phi,i}(X)$ as left $\mathscr{G}_{A}^{\Phi}(X)$ -modules.

Now, let us state some elementary properties of the Hom-functor $\mathscr{G}^{\Phi}_{A}(X,-)$ .

Lemma 2.2.

Suppose that $A$ is an algebra, that $X$ is an $A$ -module, and that $\Phi$ is a finite admissible subset of $\mathbb{N}$ .

$(1)$ * Let ${\rm add}_{A}^{\Phi}(X)$ stand for the full subcategory of $\mathscr{G}_{A}^{\Phi}$ consisting of objects in ${\rm add}(_{A}X)$ . Then the Hom-functor $\mathscr{G}_{A}^{\Phi}(X,-):{\rm add}_{A}^{\Phi}(X)\longrightarrow\mathscr{G}_{A}^{\Phi}(X)$ -proj is an equivalence of categories;*

$(2)$ * Let $B$ be a $k$ -algebra, and let $P$ be a $B$ - $A$ -bimodule such that $P_{A}$ is projective. Then there is a canonical algebra homomorphism $\alpha_{P}:\mathscr{G}^{\Phi}_{A}(X)\longrightarrow\mathscr{G}^{\Phi}_{B}(P\otimes_{A}X)$ defined by $(f_{ij})_{i,j\in\Phi}\mapsto(P\otimes_{A}f_{ij})_{i,j\in\Phi}$ for $(f_{ij})_{i,j\in\Phi}\in\mathscr{G}^{\Phi}_{A}(X)$ . Thus every left (or right) $\mathscr{G}^{\Phi}_{B}(P\otimes_{A}X)$ -module can be regarded as a left (or right) $\mathscr{G}^{\Phi}_{A}(X)$ -module via $\alpha_{P}$ .*

Proof. (1) Note that the objects of ${\rm add}_{A}^{\Phi}(X)$ are the same objects as ${\rm add}_{A}(X)$ , and ${\rm Hom}_{{\rm add}_{A}^{\Phi}(X)}(X,Y)=\mathscr{G}_{A}^{\Phi}(X,Y)$ . Then $\mathscr{G}_{A}^{\Phi}(X,Y)$ is the Hom-functor ${\rm Hom}_{{\rm add}_{A}^{\Phi}(X)}(X,-):{\rm add}_{A}^{\Phi}(X)\longrightarrow\mathscr{G}_{A}^{\Phi}(X)$ -proj. It is known that

[TABLE]

for some $i\in\Phi$ . Now, it is easy to verify that the functor in (1) is an equivalence of additive categories.

(2) We can check it directly. $\square$

The following homological result plays an important role in proving Theorem 1.1.

Lemma 2.3.

Suppose that $A,B$ and $C$ are $k$ -algebras. Let ${}_{A}X$ be an $A$ -module, and let ${}_{A}Y_{B}$ and ${}_{B}P_{C}$ be bimodules with ${}_{B}P$ projective. Then, for each $i\geq 0$ , we have ${\rm Ext}^{i}_{A}(X,Y\otimes_{B}P_{C})\simeq{\rm Ext}^{i}_{A}(X,Y)\otimes_{B}P_{C}$ as $C^{op}$ -modules. Moreover, for each admissible subset $\Phi$ of $\mathbb{N}$ , we have $\mathscr{G}^{\Phi}_{A}(X,Y\otimes_{B}P_{C})\simeq\mathscr{G}^{\Phi}_{A}(X,Y)\otimes_{\mathscr{G}_{B}^{\Phi}(B)}\begin{pmatrix}P_{C}&0&0&\cdots&0\\ 0&P_{C}&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&P_{C}\end{pmatrix}$ as $\mathscr{G}^{\Phi}_{A}(X)\mbox{-}\mathscr{G}_{C}^{\Phi}(C)$ -bimodules.

Proof. The first part of Lemma 2.3 is proved in [5, Lemma 2.5].

Second, for each admissible subset $\Phi$ of $\mathbb{N}$ , we define a map

[TABLE]

by $(\overline{f_{ij}})\otimes(p_{jj})\mapsto(\varphi_{ij}(\overline{f_{ij}}\otimes p_{jj}))$ , where $p_{jj}\in P$ , and $f_{ij}\in{\rm Hom}{{}_{A}}(\Omega_{A}^{j-i}(X),Y)$ with $j-i\in\Phi$ . From the proof of first part of [5, Lemma 2.5], we know that $\varphi_{\Phi}$ is an isomorphism of $\mathscr{G}_{C}^{\Phi}(C)^{op}$ -modules. It suffices to show that $\varphi_{\Phi}$ is an isomorphism of left $\mathscr{G}_{A}^{\Phi}(X)$ -modules, this follows from the proof of the second part of [5, Lemma 2.5]. $\square$

3 Stable equivalences of Morita type for $\Phi$ -Beilinson-Green algebras

In this section, we shall prove Theorem 1.1. First, let us recall the definition of stable equivalences of Morita type in [4].

Definition 3.1.

Let $A$ and $B$ be (arbitrary) $k$ -algebras. We say that $A$ and $B$ are stably equivalent of Morita type if there is an $A$ - $B$ -bimodule ${}_{A}M_{B}$ and a $B$ - $A$ -bimodule ${}_{B}N_{A}$ such that

$(1)$ * $M$ and $N$ are projective as one-sided modules, and*

$(2)$ * $M\otimes_{B}N\simeq{A\oplus P}$ as $A$ - $A$ -bimodules for some projective $A$ - $A$ -bimodule $P$ , and $N\otimes_{A}M\simeq{B\oplus Q}$ as $B$ - $B$ -bimodules for some projective $B$ - $B$ -bimodule $Q$ .*

In this case, we say that $M$ and $N$ define a stable equivalence of Morita type between $A$ and $B$ . Moreover, we have two exact functors $T_{N}:=N\otimes_{A}-:A\mbox{{\rm-mod}}\to B\mbox{{\rm-mod}}$ and $T_{M}:=M\otimes_{B}-:B\mbox{{\rm-mod}}\to A\mbox{{\rm-mod}}$ . Similarly, the bimodules $P$ and $Q$ define two exact functors $T_{P}$ and $T_{Q}$ , respectively. Note that the images of $T_{P}$ and $T_{Q}$ consist of projective modules.

Let us remark that if $A$ and $B$ have no separable direct summands, then we may assume that $M$ and $N$ have no non-zero projective bimodules as direct summands. In fact, If $M=M^{\prime}\oplus M^{\prime\prime}$ and $N=N^{\prime}\oplus N^{\prime\prime}$ such that $M^{\prime}$ and $N^{\prime}$ have no non-zero projective bimodules as direct summands, and that $M^{\prime\prime}$ and $N^{\prime\prime}$ are projective bimodules, then it follows from [20, Lemma 4.8] that $M^{\prime}$ and $N^{\prime}$ also define a stable equivalence of Morita type between $A$ and $B$ .

From now on, we assume that $A,B,M,N,P$ and $Q$ are fixed as in Definition 3.1, and that $X$ is a generator for $A\mbox{{\rm-mod}}$ . Moreover, we fix a finite admissible subset $\Phi$ of $\mathbb{N}$ , and define $\Lambda:=\mathscr{G}^{\Phi}_{A}(X)$ and $\Gamma:=\mathscr{G}^{\Phi}_{B}(N\otimes_{A}X)$ .

Since the functors $T_{N}$ and $T_{M}$ are exact, they preserve acyclicity, and can be extended to triangle functors $T_{N}^{\prime}:{\mathscr{D}}^{b}(A)\to{\mathscr{D}}^{b}(B)$ and $T_{M}^{\prime}:{\mathscr{D}}^{b}(B)\to{\mathscr{D}}^{b}(A)$ , respectively. Furthermore, $T^{\prime}_{N}$ and $T^{\prime}_{M}$ induce canonically two functors $F:{\mathcal{G}}_{A}^{\Phi}\to{\mathcal{G}}_{B}^{\Phi}$ and $G:{\mathcal{G}}_{B}^{\Phi}\to{\mathcal{G}}_{A}^{\Phi}$ , respectively. More precisely, if $X^{\bullet}\in{\mathscr{D}}^{b}(A)$ , then $F(X^{\bullet}):=(N\otimes_{A}X^{i},N\otimes_{A}d_{X}^{i})$ , and if $f:=(f_{i,j})_{i,j\in\Phi}\in{\rm Hom}_{{\mathcal{G}}^{\Phi}_{A}}(X^{\bullet},Y^{\bullet})$ with $Y^{\bullet}\in{\mathscr{D}}^{b}(A)$ , then $F(f):=(N\otimes_{A}f_{i,j})_{i,j\in\Phi}\in{\rm Hom}_{{\mathcal{E}}^{\Phi}_{B}}(F(X^{\bullet}),F(Y^{\bullet}))$ . Similarly, we define the functor $G$ .

The functor $F$ gives rise to a canonical algebra homomorphism $\alpha_{N}:\mathscr{G}_{A}^{\Phi}(X^{\bullet})\to\mathscr{G}_{B}^{\Phi}(F(X^{\bullet}))$ for each object $X^{\bullet}\in{\mathscr{D}}^{b}(A)$ . In particular, for any $Z^{\bullet}\in{\mathscr{D}}^{b}(B)$ , we can regard $\mathscr{G}_{B}^{\Phi}(Z^{\bullet},F(X^{\bullet}))$ as an $\mathscr{G}_{B}^{\Phi}(Z^{\bullet})$ - $\mathscr{G}_{A}^{\Phi}(X^{\bullet})$ -bimodule via $\alpha_{N}$ . Note that the homomorphism $\alpha_{N}$ coincides with the one defined in Lemma 2.2, when $X^{\bullet}$ is an $A$ -module.

Proof of Theorem 1.1. We define $U:=\mathscr{G}^{\Phi}_{A}(X,T_{M}(N\otimes_{A}X))$ and $V:=\mathscr{G}^{\Phi}_{B}(N\otimes_{A}X,T_{N}(X))$ . In the following we shall prove that $U$ and $V$ define a stable equivalence of Morita type between $\Lambda$ and $\Gamma$ .

First, we endow $U$ with a right $\Gamma$ -module structure by $u\cdot\gamma:=uG(\gamma)$ for $u\in U$ and $\gamma\in\Gamma$ , and endow $V$ with a right $\Lambda$ -module structure by $v\cdot\lambda:=vF(\lambda)$ for $v\in V$ and $\lambda\in\Lambda$ . Then, $U$ becomes a $\Lambda$ - $\Gamma$ -bimodule, and $V$ becomes a $\Gamma$ - $\Lambda$ -bimodule.

By definition, we know $V=\Gamma$ , and it is a projective left $\Gamma$ -module. Note that ${}_{A}X$ is a generator and the images of $T_{P}$ consists of projective modules. We conclude that $T_{M}(N\otimes_{A}X)=M\otimes_{B}(N\otimes_{A}X)\simeq X\oplus P\otimes_{A}X\in{\rm add}(X)$ . Thus $U$ is projective as a left $\Lambda$ -module by Lemma 2.2.

(1) $U\otimes_{\Gamma}V$ , as a $\Lambda$ - $\Lambda$ -bimodule, satisfies the condition (2) in Definition 3.1.

Indeed, we write $W:=\mathscr{G}^{\Phi}_{A}(X,(T_{M}T_{N})(X))$ , and define a right $\Lambda$ -module structure on $W$ by $w\cdot\lambda^{\prime}:=w(GF)(\lambda^{\prime})$ for $w\in W$ and $\lambda^{\prime}\in\Lambda$ . Then $W$ becomes a $\Lambda$ - $\Lambda$ -bimodule. Note that there is a natural $\Lambda$ -module isomorphism $\varphi:U\otimes_{\Gamma}V\rightarrow W$ defined by $x\otimes y\mapsto xG(y)$ for $x\in U$ and $y\in V$ . We claim that $\varphi$ is an isomorphism of $\Lambda$ - $\Lambda$ -bimodules. In fact, it suffices to show that $\varphi$ respects the structure of right $\Lambda$ -modules. However, this follows immediately from a verification: for $c\in U,d\in V$ and $a\in\Lambda$ , we have

[TABLE]

Combining this bimodule isomorphism $\varphi$ with Lemma 2.2, we get the following isomorphisms of $\Lambda$ - $\Lambda$ -bimodules:

[TABLE]

where the second isomorphism follows from $M\otimes_{B}N\simeq{A\oplus P}$ as $A$ - $A$ -bimodules, and where the right $\Lambda$ -module structure on $\mathscr{G}^{\Phi}_{A}(X,P\otimes_{A}X)$ is induced by the canonical algebra homomorphism

[TABLE]

which sends $(f_{i,j})_{i,j\in\Phi}$ in $\Lambda$ to $(P\otimes_{A}f_{i,j})_{i,j\in\Phi}$ (see Lemma 2.2 (2)).

Now, we show that $\mathscr{G}^{\Phi}_{A}(X,P\otimes_{A}\,X)$ is a projective $\Lambda$ - $\Lambda$ -bimodule. In fact, since $P\in{\rm add}(_{A}A\otimes_{k}A_{A})$ , we conclude that $\mathscr{G}^{\Phi}_{A}(X,P\otimes_{A}\,X)\in{\rm add}(\mathscr{G}^{\Phi}_{A}(X,(A\otimes_{k}A)\otimes_{A}\,X))$ . Thus, it is sufficient to prove that $\mathscr{G}^{\Phi}_{A}(X,(A\otimes_{k}A)\otimes_{A}\,X)$ is a projective $\Lambda$ - $\Lambda$ -bimodule. For this purpose, we first note that the right $\Lambda$ -module structure on $\mathscr{G}^{\Phi}_{A}(X,(A\otimes_{k}A)\otimes_{A}\,X)$ is induced by the canonical algebra homomorphism

[TABLE]

which sends $g:=(g_{i,j})_{i,j\in\Phi}$ in $\Lambda$ to $((A\otimes_{k}A)\otimes_{A}g_{i,j})_{i,j\in\Phi}$ . Clearly, ${}_{A}A\otimes_{k}A\otimes_{A}\,X\in{\rm add}(_{A}A)$ . It follows that ${\rm Ext}_{A}^{l}((A\otimes_{k}A)\otimes_{A}X,(A\otimes_{k}A)\otimes_{A}X)=0$ for any $l>0$ , and therefore $(A\otimes_{k}A)\otimes_{A}g_{i,j}=0$ for any $0\neq i-j\in\Phi$ . Thus we have $\alpha_{A\otimes_{k}A}(g)=(A\otimes_{k}A)\otimes_{A}g_{i,i}$ . If

[TABLE]

is the canonical projection and $\mu^{\prime}$ is the canonical algebra homomorphism

[TABLE]

then $\alpha_{A\otimes_{k}A}=\pi\begin{pmatrix}\mu^{\prime}&0&0&\cdots&0\\ 0&\mu^{\prime}&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&\mu^{\prime}\end{pmatrix}$ . Thus the right $\Lambda$ -module structure on $\mathscr{G}^{\Phi}_{A}(X,(A\otimes_{k}A)\otimes_{A}\,X)$ is induced by $\begin{pmatrix}{\rm End}_{A}(X)&0&0&\cdots&0\\ 0&{\rm End}_{A}(X)&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&{\rm End}_{A}(X)\end{pmatrix}$ . Similarly, from the homomorphisms

[TABLE]

where $\mu:{\rm End}_{A}(X)\to{\rm End}_{A}(A\otimes_{k}X)$ is induced by the tensor functor $A\otimes_{k}-$ , we see that the right $\Lambda$ -module structure on $\mathscr{G}^{\Phi}_{A}(X,A\otimes_{k}X)$ is also induced by $\begin{pmatrix}{\rm End}_{A}(X)&0&0&\cdots&0\\ 0&{\rm End}_{A}(X)&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&{\rm End}_{A}(X)\end{pmatrix}$ . Thus $\mathscr{G}^{\Phi}_{A}(X,(A\otimes_{k}A)\otimes_{A}\,X)\,\simeq\,\mathscr{G}^{\Phi}_{A}(X,A\otimes_{k}X)$ as $\Lambda$ - $\Lambda$ -bimodules. Moreover, it follows from Lemma 2.3 that

[TABLE]

as $\Lambda$ - $\begin{pmatrix}{\rm End}_{A}(X)&0&0&\cdots&0\\ 0&{\rm End}_{A}(X)&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&{\rm End}_{A}(X)\end{pmatrix}$ -bimodules. Since the $\begin{pmatrix}X&0&0&\cdots&0\\ 0&X&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&X\end{pmatrix}$ can be regarded as a right $\Lambda$ -module via the homomorphism

[TABLE]

we see that

[TABLE]

as $\Lambda$ - $\Lambda$ -bimodules and that

[TABLE]

as right $\Lambda$ -modules. Thus

[TABLE]

as $\Lambda$ - $\Lambda$ -bimodules. Since ${}_{A}A\in{\rm add}(X)$ , we know that $\mathscr{G}^{\Phi}_{A}(X,A)$ is a projective left $\Lambda$ -module and $\mathscr{G}^{\Phi}_{A}(A,X)$ is a projective right $\Lambda$ -module. Hence $\mathscr{G}^{\Phi}_{A}(X,A)\otimes_{\mathscr{G}_{k}^{\Phi}(k)}\mathscr{G}^{\Phi}_{A}(A,X)$ is a projective $\Lambda$ - $\Lambda$ -bimodule. This implies that $\mathscr{G}^{\Phi}_{A}(X,P\otimes_{A}\,X)$ is a projective $\Lambda$ - $\Lambda$ -bimodule.

(2) $V\otimes_{\Lambda}U$ , as a $\Gamma$ - $\Gamma$ -bimodule, fulfills the condition (2) in Definition 3.1.

Let $Z:=\mathscr{G}^{\Phi}_{B}(N\otimes_{A}X,T_{N}T_{M}(N\otimes_{A}X))$ . Similarly, we endow $Z$ with a right $\Gamma$ -module structure defined by $z\cdot b:=z(FG)(b)$ for $z\in Z$ and $b\in\Gamma$ . Then $Z$ becomes a $\Gamma$ - $\Gamma$ -bimodule. Observe that, for each $A$ -module $Y$ , there is a homomorphism $\Psi_{Y}:V\otimes_{\Lambda}\mathscr{G}^{\Phi}_{A}(X,Y)\to\mathscr{G}^{\Phi}_{B}(N\otimes_{A}X,T_{N}(Y))$ of $\Gamma$ -modules, which is defined by $g\otimes h\mapsto gF(h)$ for $g\in V$ and $h\in\mathscr{G}^{\Phi}_{A}(X,Y)$ . This homomorphism is natural in $Y$ . In other words, $\Psi:V\otimes_{\Lambda}\mathscr{G}^{\Phi}_{A}(X,-)\to\mathscr{G}^{\Phi}_{B}(N\otimes_{A}X,T_{N}(-))$ is a natural transformation of functors from $A\mbox{{\rm-mod}}$ to $\Gamma\mbox{{\rm-mod}}$ . Clearly, $\Psi_{X}$ is an isomorphism of $\Gamma$ -modules. It follows from $T_{M}(N\otimes_{A}X)\in{\rm add}(X)$ that $\Psi{{}_{T_{M}(N\otimes_{A}X)}}:V\otimes_{\Lambda}U\to Z$ is a $\Gamma$ -isomorphism. Similarly, we can check that $\Psi{{}_{T_{M}(N\otimes_{A}X)}}$ preserves the structure of right $\Gamma$ -modules. Thus $\Psi{{}_{T_{M}(N\otimes_{A}X)}}:V\otimes_{\Lambda}U\to Z$ is an isomorphism of $\Gamma$ - $\Gamma$ -bimodules, and there are the following isomorphisms of $\Gamma$ - $\Gamma$ -bimodules:

[TABLE]

where the second isomorphism is deduced from $N\otimes_{A}M\simeq{B\oplus Q}$ as $B$ - $B$ -bimodules. By a similar argument to that in the proof of (1), we can show that $\mathscr{G}^{\Phi}_{B}(N\otimes_{A}X,Q\otimes_{B}(N\otimes_{A}X))$ is a projective $\Gamma$ - $\Gamma$ -bimodule.

It remains to show that $U_{\Gamma}$ and $V_{\Lambda}$ are projective. This is equivalent to showing that the tensor functors $T_{U}:=U\otimes_{\Gamma}-:\Gamma\mbox{{\rm-mod}}\to\Lambda\mbox{{\rm-mod}}$ and $T_{V}:=V\otimes_{\Lambda}-:\Lambda\mbox{{\rm-mod}}\to\Gamma\mbox{{\rm-mod}}$ are exact. Since tensor functors are always right exact, the exactness of $T_{U}$ is equivalent to the property that $T_{U}$ preserve injective homomorphisms of modules. Now, suppose that $f:C\to D$ is an injective homomorphism between $\Gamma$ -modules $C$ and $D$ . Since $\mathscr{G}^{\Phi}_{B}(N\otimes_{A}X,Q\otimes_{B}(N\otimes_{A}X))$ is a right projective $\Gamma$ -module, we know from $(**)$ that the composition functor $T_{V}T_{U}$ is exact. In particular, the homomorphism $(T_{V}T_{U})(f):(T_{V}T_{U})(C)\to(T_{V}T_{U})(D)$ is injective. Let $\mu:{\rm Ker}\,\,(T_{U}(f))\to T_{U}(C)$ be the canonical inclusion. Clearly, we have $\mu T_{U}(f)=0$ , which shows $T_{V}(\mu T_{U}(f))=T_{V}(\mu)(T_{V}T_{U})(f)=0$ . It follows that $T_{V}(\mu)=0$ and $(T_{U}T_{V})(\mu)=0$ . By ( $*$ ), we get $\mu=0$ , which implies that the homomorphism $T_{U}(f)$ is injective. Hence $T_{U}$ preserves injective homomorphisms. Similarly, we can show that $T_{V}$ preserves injective homomorphisms. Consequently, $U_{\Gamma}$ and $V_{\Lambda}$ are projective.

Threrfore, the bimodules $U$ and $V$ define a stable equivalence of Morita type between $\Lambda$ and $\Gamma$ . This finishes the proof of Theorem 1.1. $\square$

Remarks. (1) If we take $\Phi=\{0\}$ in Theorem 1.1, then we get [21, Theorem 1.1]. We mention that the proof of Theorem 1.1 is somewhat routine.

(2) Since stable equivalences of Morita type preserve the global, dominant and finitistic dimensions of algebras, Theorem 1.1 asserts actually also that these dimensions are equal for algebras $\mathscr{G}_{A}^{\Phi}(X)$ and $\mathscr{G}_{B}^{\Phi}(N\otimes_{A}X)$ .

4 A family of finite dimensional algebras such that they are derived equivalent but not stably equivalences of Morita type

4.1 Stable equivalences of Morita type for self-injective algebras

Derived equivalences between self-injective algebras implies stable equivalences of Morita type by a result of Rickard [28], this makes stable equivalences of Morita type closely related to the Broué abelian defect group conjecture which essentially predicates a derived equivalence between two block algebras [4], and thus also a stable equivalence of Morita type between them.

In this section, we will apply Theorem 1.1 and [5, Theorem 1.2] to self-injective algebras. It turns out that the existence of a stable equivalence of Morita type between $\Phi$ -Beilinson-Green algebras of generators for one finite admissible set $\Phi$ implies the one for all finite admissible sets.

Throughout this section, we fix a finite admissible subset $\Phi$ of $\mathbb{N}$ , and assume that $A$ and $B$ are indecomposable, non-simple, self-injective algebras. Let $X$ be a generator for $A$ -mod with a decomposition $X:=A\oplus\displaystyle\bigoplus_{1\leq i\leq n}X_{i}$ , where $X_{i}$ is indecomposable and non-projective such that $X_{i}\ncong X_{t}$ for $1\leq i\neq t\leq n$ , and let $Y$ be a generator for $B$ -mod with a decomposition $Y:=B\oplus\displaystyle\bigoplus_{1\leq j\leq m}Y_{i}$ , where $Y_{j}$ is indecomposable and non-projective such that $Y_{j}\ncong Y_{s}$ for $1\leq j\neq s\leq m$ .

Lemma 4.1.

$(1)$ * The full subcategory of $\mathscr{G}_{A}^{\Phi}(X)\mbox{{\rm-mod}}$ consisting of projective-injective $\mathscr{G}_{A}^{\Phi}(X)$ -modules is equal to ${\rm add}(\mathscr{G}_{A}^{\Phi}(X,A))$ . Particularly, if $\mathscr{G}_{A}^{\Phi}(X)\neq\begin{pmatrix}{\rm End}_{A}(X)&0&0&\cdots&0\\ 0&{\rm End}_{A}(X)&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&{\rm End}_{A}(X)\end{pmatrix}$ , then ${\rm dom.dim}(\mathscr{G}_{A}^{\Phi}(X))=0$ .*

$(2)$ * $\mathscr{G}_{A}^{\Phi}(X)$ has no semisimple direct summands.*

Proof. (1) For convenience, we set $\Lambda=\mathscr{G}_{A}^{\Phi}(X)$ . Since $A$ is self-injective, it follows from that $\nu{{}_{\Lambda}}(\mathscr{G}_{A}^{\Phi}(X,A))\simeq\mathscr{G}_{A}(X,\nu_{A}A)\simeq\mathscr{G}_{A}^{\Phi}(X,DA)\in{\rm add}\big{(}\mathscr{G}_{A}^{\Phi}(X,A)\big{)}$ . Consequently, $\mathscr{G}_{A}^{\Phi}(X,A)$ is a projective-injective $\Lambda$ -module. We claim that, up to isomorphism, each indecomposable projective-injective $\Lambda$ -module is a direct summand of $\mathscr{G}_{A}^{\Phi}(X,A)$ . Let $\Lambda^{\prime}=\begin{pmatrix}C&M\\ 0&D\\ \end{pmatrix}$ be a triangular matrix algebra. Then each $\Lambda^{\prime}$ -module can be represented by a triple $(_{C}X,_{D}Y,f)$ with $X\in C\mbox{{\rm-mod}},Y\in C\mbox{{\rm-mod}}$ and $f:M_{D}\otimes Y\to X$ a $C$ -module homomorphism. Let $(_{C}X,_{D}Y,f)$ be an indecomposable $\Lambda^{\prime}$ -module. By [2, Proposition 2.5, p. 76], there are two possibilities:

(i) ${}_{D}Y=0$ and ${}_{C}X$ is an indecomposable projective-injective $C$ -module with $M\otimes_{C}X=0$ ,

(ii) ${}_{C}X=0$ and ${}_{D}Y$ is an indecomposable projective-injective $D$ -module with ${\rm Hom}_{D}(M,Y)=0$ .

Thus ${\rm add}(\mathscr{G}_{A}^{\Phi}(X,A))$ is just the full subcategory of $\mathscr{G}_{A}^{\Phi}(X)\mbox{{\rm-mod}}$ consisting of projective-injective modules.

Finally, we consider the dominant dimension of ${\rm dom.dim}(\mathscr{G}_{A}^{\Phi}(X))$ . Suppose $\mathscr{G}_{A}^{\Phi}(X)\neq\begin{pmatrix}{\rm End}_{A}(X)&0&0&\cdots&0\\ 0&{\rm End}_{A}(X)&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&{\rm End}_{A}(X)\end{pmatrix}$ . Since $A$ is self-injective, we have

[TABLE]

It follows that $\mathscr{G}_{A}^{\Phi}(X,A)$ is annihilated by

[TABLE]

but it is not annihilated by $\Lambda$ . Hence $\Lambda$ cannot be cogenerated by $\mathscr{G}_{A}^{\Phi}(X,A)$ . This implies that ${\rm dom.dim}(\mathscr{G}_{A}^{\Phi}(X))=0$ .

(2) Contrarily, we suppose that the algebra $\mathscr{G}_{A}^{\Phi}(X)$ has a semisimple direct summand. Then $\mathscr{G}_{A}^{\Phi}(X)$ has a simple projective-injective module $S$ . According to $(1)$ , we know that $S$ must be a simple projective-injective ${\rm End}_{A}(X)$ -module. Then it follows from the first part of the proof of [5, Corollary 4.7] that $A$ has a semisimple direct summand. Clearly, this is contrary to our initial assumption that $A$ is indecomposable and non-simple. Thus $\mathscr{G}_{A}^{\Phi}(X)$ has no semisimple direct summands. $\square$

Theorem 4.2.

If the algebras $\mathscr{G}_{A}^{\Phi}(X)$ and $\mathscr{G}_{B}^{\Phi}(Y)$ are stably equivalent of Morita type, then $n=m$ and there are bimodules ${}_{A}M_{B}$ and ${}_{B}N_{A}$ which define a stable equivalence of Morita type between $A$ and $B$ such that, up to the ordering of indices, ${}_{A}M\otimes_{B}Y_{i}\simeq{X_{i}\oplus P_{i}}$ as $A$ -modules, where ${}_{A}P_{i}$ is projective for all $i$ with $1\leq i\leq n$ . Moreover, for any finite admissible subset $\Psi$ of $\mathbb{N}$ , there is a stable equivalence of Morita type between $\mathscr{G}_{A}^{\Psi}(X)$ and $\mathscr{G}_{B}^{\Psi}(Y)$ .

Proof. For convenience, we set $\Lambda_{0}={\rm End}(X)$ , $\Lambda=\mathscr{G}_{A}^{\Phi}(X)$ and $\Gamma_{0}={\rm End}(Y),\Gamma=\mathscr{G}_{B}^{\Phi}(Y)$ . By Lemma 4.1, the algebras $\Lambda$ and $\Gamma$ have no semisimple direct summands. Let $e$ be the idempotent in $\Lambda_{0}$ corresponding to the direct summand $A$ of $X$ , and let $f$ be the idempotent in $\Gamma_{0}$ corresponding to the direct summand $B$ of $Y$ . Let $e^{\prime}=\begin{pmatrix}e&0&0&\cdots&0\\ 0&e&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&e\end{pmatrix}$ and $f^{\prime}=\begin{pmatrix}f&0&0&\cdots&0\\ 0&f&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&f\end{pmatrix}$ be the idempotents of $\Lambda$ and $\Gamma$ , respectively. Note that $\Lambda e^{\prime}\simeq\mathscr{G}_{A}^{\Phi}(X,A)$ as $\Lambda$ -modules and $\Gamma f^{\prime}\simeq\mathscr{G}_{B}^{\Phi}(Y,B)$ as $\Gamma$ -modules. Clearly, $e^{\prime}\Lambda e^{\prime}\simeq\begin{pmatrix}A&0&0&\cdots&0\\ 0&A&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&A\end{pmatrix}$ and $f^{\prime}\,\Gamma f^{\prime}\simeq\begin{pmatrix}B&0&0&\cdots&0\\ 0&B&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&B\end{pmatrix}$ as algebras. Moreover, we see that $e^{\prime}\Lambda\simeq\begin{pmatrix}X&0&0&\cdots&0\\ 0&X&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&X\end{pmatrix}$ as $e^{\prime}\Lambda e^{\prime}$ -modules, and $f^{\prime}\,\Gamma\simeq\begin{pmatrix}Y&0&0&\cdots&0\\ 0&Y&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&Y\end{pmatrix}$ as $f^{\prime}\Gamma f^{\prime}$ -modules. Suppose that a stable equivalences of Morita type between $\Lambda$ and $\Gamma$ is given. we know that the idempotent $e^{\prime}$ in $\Lambda$ and the idempotent $f^{\prime}$ in $\Gamma$ satisfy the conditions in [5, Theorem 1.2]. It follows from that there are bimodules ${}_{e^{\prime}\Lambda e^{\prime}}M^{\prime}_{f^{\prime}\Gamma f^{\prime}}$ and ${}_{f^{\prime}\Gamma f^{\prime}}N^{\prime}_{e^{\prime}\Lambda e^{\prime}}$ which define a stable equivalence of Morita type between $e^{\prime}\Lambda e^{\prime}$ and $f^{\prime}\Gamma f^{\prime}$ such that ${\rm add}(M^{\prime}\otimes_{f^{\prime}\Gamma f^{\prime}}f^{\prime}\Gamma)={\rm add}(e^{\prime}\Lambda)$ .

We know that

[TABLE]

and

[TABLE]

as algebras. By [18, Theorem 2.2] or the proof of [5, Lemma 4.1], there are stable equivalences of Morita type between summands of $e^{\prime}\Lambda e^{\prime}$ and $f^{\prime}\Gamma f^{\prime}$ . It follows from that there are bimodules ${}_{A}M_{B}$ and ${}_{B}N_{A}$ which define a stable equivalence of Morita type between $A$ and $B$ such that ${\rm add}(M\otimes_{B}Y)={\rm add}(X)$ . By the given decompositions of $X$ and $Y$ , we conclude that $n=m$ and, up to the ordering of direct summands, we may assume that ${}_{A}M\otimes_{B}Y_{i}\simeq{X_{i}\oplus P_{i}}$ as $A$ -modules, where ${}_{A}P_{i}$ is projective for all $i$ with $1\leq i\leq n$ . Now, the last statement in this corollary follows immediately from Theorem 1.1. Thus the proof is completed. $\square$

Usually, it is difficult to decide whether an algebra is not stably equivalent of Morita type to another algebra. The next corollary, however, gives a sufficient condition to assert when two algebras are not stably equivalent of Morita type.

Corollary 4.3.

Let $n$ be a non-negative integer. Let $W$ be an indecomposable non-projective $A$ -module. Suppose that $\Omega_{A}^{s}(W)\not\simeq W$ for any non-zero integer $s$ . Set $W_{n}=\bigoplus_{0\leq i\leq n}\Omega_{A}^{i}(W)$ . Then, for any finite admissible subset $\Psi$ of $\mathbb{N}$ , the algebras $\mathscr{G}_{A}^{\Psi}(A\oplus W_{n}\oplus\Omega_{A}^{l}(W))$ and $\mathscr{G}_{A}^{\Psi}(A\oplus W_{n}\oplus\Omega_{A}^{m}(W))$ are not stably equivalent of Morita type whenever $m$ and $l$ belong to $\mathbb{N}$ with $n<m<l$ .

Proof. Suppose that there is a finite admissible subset $\Psi$ of $\mathbb{N}$ such that $\mathscr{G}_{A}^{\Psi}(A\oplus W_{n}\oplus\Omega_{A}^{m}(W))$ and $\mathscr{G}_{A}^{\Psi}(A\oplus W_{n}\oplus\Omega_{A}^{l}(W))$ are stably equivalent of Morita type for some fixed $l,m\in\mathbb{N}$ with $n<m<l$ . Set $\Phi_{1}=\{0,1,\cdots,n\}\cup\{l\}$ and $\Phi_{2}=\{0,1,\cdots,n\}\cup\{m\}$ . Then, by Theorem 4.2, we know that there exist bimodules ${}_{A}M_{A}$ and ${}_{A}N_{A}$ which define a stable equivalence of Morita type between $A$ and itself, and that there is a bijection $\sigma:\Phi_{1}\to\Phi_{2}$ such that $M\otimes_{A}\Omega_{A}^{j}(W)\simeq\Omega{{}_{A}^{\sigma(j)}}(W)\oplus P_{j}$ as $A$ -modules, where $P_{j}$ is projective for each $j\in\Phi_{1}$ . In particular, we have $M\otimes_{A}W\simeq\Omega{{}_{A}^{\sigma(0)}}(W)\oplus P_{0}$ . Since $M$ is projective as a one-sided module, we know that $M\otimes_{A}\Omega_{A}^{l}(W)\simeq\Omega{{}_{A}^{\sigma(0)+l}}(W)\oplus P_{l}^{\prime}$ with $P_{l}^{\prime}\in{\rm add}(_{A}A)$ . Note that $M\otimes_{A}\Omega_{A}^{l}(W)\simeq\Omega{{}_{A}^{\sigma(l)}}(W)\oplus P_{l}$ . It follows that $\Omega{{}_{A}^{\sigma(0)+l}}(W)\simeq\Omega{{}_{A}^{\sigma(l)}}(W)$ . Consequently, we have $\sigma(l)=\sigma(0)+l\geq l$ since $W$ is not $\Omega$ -periodic. Hence $l\leq\sigma(l)\leq m<l$ , a contradiction. This shows that $\mathscr{G}_{A}^{\Psi}(A\oplus W_{n}\oplus\Omega_{A}^{m}(W))$ and $\mathscr{G}_{A}^{\Psi}(A\oplus W_{n}\oplus\Omega_{A}^{l}(W))$ cannot be stably equivalent of Morita type whenever $l$ and $m$ $\in\mathbb{N}$ with $n<m<l$ . $\square$

This corollary will be used in the next subsection.

4.2 A family of derived-equivalent algebras: another answer to Thorsten Holm’s question

In a talk at a workshop in Goslar, Germany, Thorsten Holm considered the following qustion on derived equivalences and stable equivalences of Morita type:

Question. Is there any infinite series of finite-dimensional $k$ -algebras such that they have the same dimension and are all derived-equivalent, but not stably equivalent of Morita type ?

It has an affirmative answer in [5]. In this section, we shall apply our results in the previous sections to give an affirmative answer for question again.

Liu and Schulz in [17] constructed a local symmetric $k$ -algebra $A$ of dimension 8 and an indecomposable $A$ -module $M$ such that all the syzygy modules $\Omega_{A}^{n}(M)$ with $n\in\mathbb{Z}$ are 4-dimensional and pairwise non-isomorphic. This algebra $A$ depends on a non-zero parameter $q\in k$ , which is not a root of unity, and has an infinite DTr-orbit in which each module has the same dimension. Ringel in [31] carried out a thorough investigation of Auslander-Reiten components of this algebra. Based on this symmetric algebra and a recent result in [25] together with the results in the previous sections, we shall construct an infinite family of algebras, which provides a positive solution to the above question.

From now on, we fix a non-zero element $q$ in the field $k$ , and assume that $q$ is not a root of unity. The $k$ -algebra $A$ defined by Liu-Schulz is an associative algebra (with identity) over $k$ with the generators: $x_{0},x_{1},x_{2}$ , and the relations: $x_{i}^{2}=0,\quad\mbox{and}\quad x_{i+1}x_{i}+qx_{i}x_{i+1}=0\quad\mbox{for}\quad i=0,1,2.$

Here, and in what follows, the subscript is modulo $3$ .

Let $n$ be a fixed natural number, and let $\Phi=\{0\}$ or $\{0,1\}$ . For $j\in\mathbb{Z}\,$ , set $u_{j}:=x_{2}+q^{j}x_{1}$ , $I_{j}:=Au_{j}$ , $J_{j}:=u_{j}A$ , $I:=\bigoplus\limits_{i=0}^{n}I_{i}$ and $\Lambda_{j}^{\Phi}:=\mathscr{G}_{A}^{\Phi}(A\oplus I\oplus I_{j})$ .

With these notations in mind, the main result in this section can be stated as follows:

Theorem 4.4.

For any $m\geq n+4$ and $\Phi=\{0,1\}$ , we have

$(1)$ * ${\rm dim_{{}_{k}}}{(\Lambda_{m}^{\Phi})}={\rm dim_{{}_{k}}}{(\Lambda_{m+1}^{\Phi})}$ .*

$(2)$ * ${\rm gl.dim}(\Lambda_{m}^{\Phi})=\infty$ .*

$(3)$ * ${\rm dom.dim}(\Lambda_{m}^{\Phi})=0.$ *

$(4)$ * $\Lambda_{m}^{\Phi}$ and $\Lambda_{m+1}^{\Phi}$ are derived-equivalent.*

$(5)$ * If $l>m$ , then $\Lambda_{l}^{\Phi}$ and $\Lambda_{m}^{\Phi}$ are not stably equivalent of Morita type.*

Remark 4.5.

Note that if $\Phi=\{0\}$ , then the above result was known in [5].

The following corollary is a consequence of Theorem 4.4, which solves the above mentioned question positively.

Corollary 4.6.

There exists an infinite series of finite-dimensional $k$ -algebras $A_{i},\,i\in\mathbb{N}$ , such that

$(1)$ * ${\rm dim_{{}_{k}}}(A_{i})={\rm dim_{{}_{k}}}(A_{i+1})$ for all $i\in{\mathbb{N}}$ ,*

$(2)$ * all $A_{i}$ have the same global and dominant dimensions,*

$(3)$ * all $A_{i}$ are derived-equivalent, and*

$(4)$ * $A_{i}$ and $A_{j}$ are not stably equivalent of Morita type for $i\neq j$ .*

The proof of Theorem 4.4 will cover the rest of this section.

The following result can be directly deduced from the work of in [25, 26].

Lemma 4.7.

Let $B$ be a $k$ -algebra. Let $Y$ and $M$ be $B$ -modules with $M$ a generator for $B\mbox{{\rm-mod}}$ . If ${\rm Ext}_{B}^{1}(M,\Omega_{B}(Y))={\rm Ext}_{B}^{1}(Y,M)=0$ , then the endomorphism algebras ${\rm End}_{B}(M\oplus Y)$ and ${\rm End}_{B}(M\oplus\Omega_{B}(Y))$ are derived equivalent. If, in addition, ${\rm Ext}_{B}^{2}(M,\Omega_{B}(Y))={\rm Ext}_{B}^{2}(Y,M)=0$ , then the $\{0,1\}$ -Auslander-Yoneda algebras $\mathscr{G}_{B}^{\{0,1\}}(M\oplus Y)$ and $\mathscr{G}_{B}^{\{0,1\}}(M\oplus\Omega_{B}(Y))$ are derived equivalent.

Having made the previous preparations, now we can prove Theorem 4.4.

Proof of Theorem 4.4. Let $m\geq n+4$ . Set $M:=\,A\oplus I$ with $I=\bigoplus\limits_{i=0}^{n}I_{i}$ , and $V_{m}:=M\oplus I_{m}$ .

(1) By [5, Lemma 6.5(5)], we know that ${\rm Ext}_{A}^{1}(M,I_{m})={\rm Ext}_{A}^{1}(I_{m},M)=0$ . Clearly, we have

[TABLE]

and

[TABLE]

By [5, Lemma 6.5], we get

[TABLE]

It follows that ${\rm dim_{{}_{k}}}{(\Lambda_{m}^{\Phi})}={\rm dim_{{}_{k}}}{(\Lambda_{m+1}^{\Phi})}$ .

(2) It follows from [5, Theorem 6.1] that ${\rm gl.dim}(\Lambda_{m}^{\{0\}})=\infty$ . We know that

[TABLE]

This yields ${\rm gl.dim}(\Lambda_{m}^{\{0,1\}})=\infty$ .

(3) By Lemma 4.1, we have ${\rm dom.dim}(\Lambda_{m}^{\{0,1\}})=0$ .

(4) Consider the exact sequence

[TABLE]

in $A\mbox{{\rm-mod}}$ . Since $m\geq n+4$ , it follows from [5, Lemmas 6.5(5) and 6.4(4)] that ${\rm Ext}_{A}^{1}(M,I_{m+1})={\rm Ext}_{A}^{1}(I_{m+1},M)={\rm Ext}_{A}^{1}(I_{m},M)={\rm Ext}_{A}^{1}(M,I_{m})=0$ . Note that $A$ is self-injective. By Lemma 4.7, we conclude that the algebras $\Lambda_{m}^{\Phi}$ and $\Lambda_{m+1}^{\Phi}$ are derived-equivalent for $\Phi=\{0,1\}$ .

(5) It follows from [5, Lemma 6.4] that $\Omega_{A}(I_{j})=I_{j+1}$ for each $j\in\mathbb{Z}$ and that the $A$ -modules $I_{j}$ are pairwise non-isomorphic for all $j\in\mathbb{Z}$ . Now, we define $W:=I_{0}$ and $W_{n}:=\oplus_{0\leq j\leq n}I_{j}$ . Then, by Corollary 4.3, the algebras $\Lambda_{l}^{\Phi}$ and $\Lambda_{m}^{\Phi}$ are not stably equivalent of Morita type if $l>m$ . Thus the proof is completed. $\square$

5 Stable equivalences of Morita type for Beilinson-Greeen algebras of associated graded algebras

In this section, we can construct stable equivalences of Morita type between Beilinson-Green algebra of group graded algebra from group graded stable equivalences of Morita type.

Let $A=A_{0}\oplus A_{1}\oplus A_{2}\cdots\oplus A_{n}$ be a graded $k$ -algebra with multiplication induced from multiplication of $A$ , given by

[TABLE]

We say a graded $A$ -module $M$ has degree $\leq n$ if $M_{i}=0$ for $i\geq n+1$ . Let ${\rm G}_{n}(A)$ be the category of graded $A$ -modules with degree $\leq n$ and degree zero maps. Recall that Beilinson-Green algebra [3, 9] is defined as follows:

[TABLE]

Let $M\in{\rm G}_{n}(A\otimes B^{op})$ and $N\in{\rm G}_{n}(B\otimes A^{op})$ . Then we have the main theorem of this section.

Theorem 5.1.

Suppose that $M$ and $N$ induce a graded stable equivalence of Morita type between $A$ and $B$ , then there is a stable equivalence of Morita type between the Beilinson-Green algebras $\bar{A}$ and $\bar{B}$ .

Proof. Suppose that $M$ and $N$ induce a graded stable equivalence of Morita type between $A$ and $B$ , then

$(1)$ $M$ is a left graded projective $A$ -module and right graded projective $B$ -module, and $N$ is a left graded projective $B$ -module and right graded projective $A$ -module.

$(2)$ $M\otimes_{B}N\simeq{A\oplus P}$ as $A$ - $A$ -graded bimodules for some projective graded $A$ - $A$ -bimodule $P$ , and $N\otimes_{A}M\simeq{B\oplus Q}$ as $B$ - $B$ -graded bimodules for some projective graded $B$ - $B$ -bimodule $Q$ .

Set $\bar{M}=\begin{pmatrix}M_{0}&M_{1}&\cdots&M_{n-1}&M_{n}\\ 0&M_{0}&\cdots&M_{n-2}&M_{n-1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&M_{0}&M_{1}\\ 0&0&\cdots&0&M_{0}\\ \end{pmatrix}\quad\text{and}\quad\bar{N}=\begin{pmatrix}N_{0}&N_{1}&\cdots&N_{n-1}&N_{n}\\ 0&N_{0}&\cdots&N_{n-2}&N_{n-1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&N_{0}&N_{1}\\ 0&0&\cdots&0&N_{0}\\ \end{pmatrix}.$ Since

[TABLE]

we thus get $\bar{A}\bar{M}\bar{B}\subset\bar{M}$ , so $\bar{M}$ is a $\bar{A}$ - $\bar{B}$ -bimodule. Similarly, $\bar{N}$ is a $\bar{B}$ - $\bar{A}$ -bimodule.

$(1)$ We claim that $\bar{M}$ is a left projective $\bar{A}$ -module and right projective $\bar{B}$ -module. It is well-known that $\{A_{n-i}(-i)|0\leq i\leq n\}$ are graded projective $A$ -modules. It follows from ${}_{A}M\in{\rm add}\oplus_{0\leq i\leq n}A_{n-i}(-i)$ that ${}_{\bar{A}}\bar{M}=\oplus_{0\leq i\leq n}M_{n-i}(-i)\in{\rm add}\bar{A}$ . Consequently, $\bar{M}$ is a left projective $\bar{A}$ -module. Similarly, $\bar{M}$ is a right projective $\bar{B}$ -module and $\bar{N}$ is a left projective $\bar{B}$ -module and right projective $\bar{A}$ -module.

$(2)$ $\bar{P}=\begin{pmatrix}P_{0}&P_{1}&\cdots&P_{n-1}&P_{n}\\ 0&P_{0}&\cdots&P_{n-2}&P_{n-1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&P_{0}&P_{1}\\ 0&0&\cdots&0&P_{0}\\ \end{pmatrix}$ and $\bar{Q}=\begin{pmatrix}Q_{0}&Q_{1}&\cdots&Q_{n-1}&Q_{n}\\ 0&Q_{0}&\cdots&Q_{n-2}&Q_{n-1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&Q_{0}&Q_{1}\\ 0&0&\cdots&0&Q_{0}\\ \end{pmatrix}$ are projective bimodules. If $P$ is a graded projective $A$ - $A$ -bimodule, then $P\in{\rm add}(A\otimes A^{op})$ in $A\otimes A^{op}\mbox{{\rm-{grmod}}}$ . Since

[TABLE]

it follows that $\bar{P}=\begin{pmatrix}P_{0}&P_{1}&\cdots&P_{n-1}&P_{n}\\ 0&P_{0}&\cdots&P_{n-2}&P_{n-1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&P_{0}&P_{1}\\ 0&0&\cdots&0&P_{0}\\ \end{pmatrix}\in{\rm add}(\bar{A}\otimes\bar{A}^{op})$ . Then $\bar{P}$ is a projective $\bar{A}$ - $\bar{A}$ -bimodule. Similarly, $\bar{Q}$ is a projective $\bar{B}$ - $\bar{B}$ -bimodule.

$(3)$

[TABLE]

as $\bar{A}$ - $\bar{A}$ -bimodules. Similarly, $\bar{N}\otimes\bar{M}\simeq\bar{B}\oplus\bar{Q}$ as $\bar{B}$ - $\bar{B}$ -bimodules This completes the proof of Theorem 5.1. $\square$

By the above methods, we can get stable equivalences of Morita type between the Beilinson-Green algebras from $G$ -graded stable equivalences of Morita type between $G$ -graded algebras $A$ and $B$ for a finite group $G$ . Associated a $G$ -graded $k$ -algebra $A$ , there is a Beilinson-Green algebra $\bar{A}$ defined as a $G\times G$ -matrix algebra with $(\bar{A}_{gh})_{g,h\in G}$ , where $\bar{A}_{gh}=A_{gh^{-1}}$ . If $M=\oplus_{g\in G}M_{g}$ is a $G$ -graded $A$ - $B$ -bimodule, then $\bar{M}=(M_{gh^{-1}})_{g,h\in G}$ is a $\bar{A}$ - $\bar{B}$ -bimodule. Hence we have the following theorem.

Theorem 5.2.

Suppose that there is a $G$ -graded stable equivalence of Morita type between $G$ -graded algebras $A$ and $B$ , then there is a stable equivalence of Morita type between the Beilinson-Green algebras $\bar{A}$ and $\bar{B}$ .

Proof. Suppose that $M$ and $N$ induce a $G$ -graded stable equivalence of Morita type between $A$ and $B$ , then

$(1)$ $M$ is a left graded projective $A$ -module and right graded projective $B$ -module, and $N$ is a left graded projective $B$ -module and right graded projective $A$ -module.

$(2)$ $M\otimes_{B}N\simeq{A\oplus P}$ as graded $A$ - $A$ bimodules for some graded projective $A$ - $A$ -bimodule $P$ , and $N\otimes_{A}M\simeq{B\oplus Q}$ as graded $B$ - $B$ bimodules for some graded projective $B$ - $B$ -bimodule $Q$ . Then we can prove that $bar{M}$ is projective $\bar{A}$ -module and right projective $\bar{B}$ -module and $\bar{N}$ is a left projective $\bar{B}$ -module and right projective $\bar{A}$ -module. We have the following

[TABLE]

Similarly, $\bar{N}\otimes_{\bar{B}}\bar{M}\simeq\bar{B}\oplus\bar{Q}$ .

Secondly, we can prove that $\bar{P}$ is a projective $\bar{A}$ - $\bar{A}$ -bimodule and $\bar{Q}$ is a projective $\bar{B}$ - $\bar{B}$ -bimodule. Then $\bar{M}$ and $\bar{N}$ define a stable equivalence of Morita type between $\bar{A}$ and $\bar{B}$ . $\square$

6 An example

In this section, we will give an example to illustrate our main result.

Let $A$ be a self-injective algebra. Then it is well-known that $\Omega_{A\otimes A^{op}}(A)$ defines a stable equivalence of Morita type between $A$ and itself (see [20]).

The following corollary is a consequence of Theorem 1.1.

Corollary 6.1.

Let $A$ be finite-dimensional self-injective $k$ -algebras over a field $k$ and let $X$ be an $A$ -module. Then, for any finite admissible set $\Phi$ of natural numbers, there is a stable equivalence of Morita type between $\mathscr{G}^{\Phi}_{A}(A\oplus X)$ and $\mathscr{G}^{\Phi}_{A}(A\oplus\Omega_{A}(X))$ .

Proof.

Note that $\Omega_{A\otimes A^{op}}(A)\otimes X\simeq X\oplus U$ , where $U$ is a projective $A$ -module. Then the result follows from Theorem 1.1. ∎

Remark 6.2.

By Lemma 4.7, $\mathscr{G}^{\Phi}_{A}(A\oplus X)$ and $\mathscr{G}^{\Phi}_{A}(A\oplus\Omega_{A}(X))$ are also derived equivalent.

Many important classes of algebras are of the form ${\rm End}_{A}(A\oplus X)$ with $A$ a self-injective algebra. From the above corollary (see also [13, Corollary 3.14]), we may get a series of algebras which are stably equivalent of Morita type to Schur algebras. For unexplained terminology in the next corollary, we refer the reader to [10].

Corollary 6.3.

Suppose that $k$ is an algebraically closed field. Let $S_{n}$ be the symmetric group of degree $n$ . We denote by $Y$ the direct sum of all non-projective Young modules over the group algebra $k[S_{n}]$ of $S_{n}$ . Then, for every finite admissible subset $\Phi$ of $\mathbb{N}$ , the algebras $\mathscr{G}^{\Phi}_{k[S_{n}]}(k[S_{n}]\oplus Y)$ and $\mathscr{G}^{\Phi}_{k[S_{n}]}(k[S_{n}]\oplus\Omega^{i}(Y))$ are stably equivalent of Morita type for all $i\in\mathbb{Z}$ .

Example 6.4.

Let $A=k[x]/(x^{n})$ . Then $A$ is a representation-finite self-injective algebra. Denote the indecomposable $A$ -module by

[TABLE]

for $r=1,2,\cdots,n$ . Corollary 6.1 thus shows that $\mathscr{G}^{\Phi}_{A}(A\oplus X_{r})$ and $\mathscr{G}^{\Phi}_{A}(A\oplus X_{n-r})$ are stable equivalent of Morita type with $\Omega(X_{r})=X_{n-r}$ .

In the following, let $\Phi=\{0,1\}$ and $r=1$ . Then we describe $\Phi$ -Beilinson-Green algebras $\mathscr{G}^{\Phi}_{A}(A\oplus X_{r})$ and $\mathscr{G}^{\Phi}_{A}(A\oplus X_{n-r})$ in terms of quivers with relations as follows

[TABLE]

and

[TABLE]

We calculate that ${\rm dim_{{}_{k}}}(\mathscr{G}^{\Phi}_{A}(A\oplus X_{1}))=2n+7,{\rm dim_{{}_{k}}}(\mathscr{G}^{\Phi}_{A}(A\oplus X_{n-1}))=4n-2$ . Motivated by Zheng’s method [34, Property 5.1], it is an interesting question to show that whether the $\Phi$ -Beilinson-Green algebras $\mathscr{G}^{\Phi}_{A}(A\oplus X_{r})$ and $\mathscr{G}^{\Phi}_{A}(A\oplus X_{n-r})$ are cellular algebras. Note that cellular algebras were introduced by Graham and Lehrer [8] and the theory of cellular algebra related to the modular representations of Hecke algebras and various topics.

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Abstract

1 Introduction

Theorem 1.1**.**

Remark 1.2**.**

Corollary 1.3**.**

Corollary 1.4**.**

Theorem 1.5**.**

2 Preliminaries

2.1 Some conventions and homological facts

2.2 Admissible sets and perforated covering categories

Lemma 2.1**.**

Lemma 2.2**.**

Lemma 2.3**.**

3 Stable equivalences of Morita type for Φ\PhiΦ-Beilinson-Green algebras

Definition 3.1**.**

4 A family of finite dimensional algebras such that they are derived equivalent but not stably equivalences of Morita type

4.1 Stable equivalences of Morita type for self-injective algebras

Lemma 4.1**.**

Theorem 4.2**.**

Corollary 4.3**.**

4.2 A family of derived-equivalent algebras: another answer to Thorsten Holm’s question

Theorem 4.4**.**

Remark 4.5**.**

Corollary 4.6**.**

Lemma 4.7**.**

5 Stable equivalences of Morita type for Beilinson-Greeen algebras of associated graded algebras

Theorem 5.1**.**

Theorem 5.2**.**

6 An example

Corollary 6.1**.**

Proof.

Remark 6.2**.**

Corollary 6.3**.**

Example 6.4**.**

Theorem 1.1.

Remark 1.2.

Corollary 1.3.

Corollary 1.4.

Theorem 1.5.

Lemma 2.1.

Lemma 2.2.

Lemma 2.3.

3 Stable equivalences of Morita type for $\Phi$ -Beilinson-Green algebras

Definition 3.1.

Lemma 4.1.

Theorem 4.2.

Corollary 4.3.

Theorem 4.4.

Remark 4.5.

Corollary 4.6.

Lemma 4.7.

Theorem 5.1.

Theorem 5.2.

Corollary 6.1.

Remark 6.2.

Corollary 6.3.

Example 6.4.