New invariants of stable equivalences of algebras

Changchang Xi and Jinbi Zhang∗

††∗Corresponding author’s Email<EMAIL_ADDRESS>Mathematics Subject
Classification: Primary 16G10, 16E05, 18G65; Secondary 15A27, 18G80,
16L60.††Keywords: Auslander-Reiten conjecture; Centralizer algebra; Delooping
level; Derived equivalence; Finitistic dimension; $\phi$-dimension; Stable
equivalence.

###### Abstract

We show that the Auslander-Reiten conjecture on stable equivalences holds true
for principal centralizer matrix algebras and for Frobenius-finite algebras
over an algebraically closed field, that stable equivalences of algebras with
positive $\nu$-dominant dimensions preserve stable equivalences of their
Frobenius parts, and that the delooping levels, $\phi$-dimensions and
$\psi$-dimensions are invariants of stable equivalences of Artin algebras
without nodes.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Proofs of the statements
    1. 3.1 Proof of Theorem 1.1
    2. 3.2 Proof of Theorem 1.3
    3. 3.3 Proof of Theorem 1.4

##  1 Introduction

Stable equivalence of algebras is one of the prominent equivalences in the
representation theory of algebras and groups, and has been studied by many
authors. For instance, Auslander and Reiten showed that any algebra with
radical-square-zero is stably equivalent to a hereditary algebra [4].
Martínez-Villa showed that the global and dominant dimensions are invariants
of stable equivalences of algebras with no nodes and no semisimple summands
[26]. Considerable efforts notwithstanding, stable equivalences seem still to
be understood. For example, a long-standing, unsolved problem is the famous
Auslander-Reiten conjecture on stable equivalences, which says that stably
equivalent Artin algebras have the same number of non-isomorphic, non-
projective simple modules (see, for instance, [6, Conjecture (5), p.409], or
[29, Conjecture 2.5]). So the conjecture predicts that the number of non-
isomorphic, non-projective simple modules should be invariant under stable
equivalences. Auslander and Reiten proved that if an Artin algebra is stably
equivalent to a hereditary algebra then the two algebras have the same number
of non-isomorphic, non-projective simple modules [4]. For algebras over
algebraically closed field, the conjecture was verified for representation-
finite algebras, and reduced to self-injective algebras without nodes (see
[25, 26]). Recently, the conjecture is proved for stable equivalences of
Morita type between Frobenius-finite algebras without semisimple summands (see
[17]), and for stable equivalences between special biserial algebras (see
[1],[28]). In general, the conjecture still remains open.

The purpose of this note is to prove the following results on stable
equivalences of algebras. The first one is that the Auslander-Reiten
conjecture holds true for Frobenius-finite $k$-algebras and for principal
centralizer matrix algebras over an algebraically closed field $k$. The second
one is that stable equivalences of Artin algebras with positive $\nu$-dominant
dimensions preserve stable equivalences of their Frobenius parts, and the
third one is that stable equivalences preserve the delooping levels,
$\phi$-dimensions and $\psi$-dimensions of Artin algebras without nodes.

Let $A$ be an Artin algebra over a commutative Artin ring $k$. By
$A\mbox{{\rm-mod}}$ we denote the category of all finitely generated left
$A$-modules. Related to simple $A$-modules, Vincent Gélinas introduces
recently the delooping levels of algebras in [14]. The significance of
delooping levels is that the finitistic dimensions of algebras can be bounded
by the delooping levels (see [14, Proposition 1.3]), thus providing a way to
understand the unsolved finitistic dimension conjecture which states that
Artin algebras always have finite finitistic dimension (see [7], or [6,
Conjecture (11), p.410]). Recall that the _finitistic dimension_ of $A$,
denoted ${\rm findim}(A)$, is the supremum of the projective dimensions of
modules $X\in A\mbox{{\rm-mod}}$ that have finite projective dimension. By
definition, the _delooping level_ of an $A$-module $X\in A\mbox{{\rm-mod}}$,
denoted ${\rm del}(X)$, is the smallest number $d\geq 0$ such that the $d$-th
syzygy $\Omega^{d}(X)$ of $X$ is a direct summand of a module of the form
$P\oplus\Omega^{d+1}(M)$ for an $A$-module $M$ and a projective $A$-module
$P$, where $\Omega$ is the syzygy (or loop) operator of $A$. If such a number
$d$ does not exist, one defines ${\rm del}(X)=\infty$. The _delooping level_
of $A$, denoted ${\rm del}(A)$, is the maximum of the delooping levels of all
non-isomorphic simple $A$-modules. Clearly, ${\rm del}(X\oplus Y)=\max\\{{\rm
del}(X),{\rm del}(Y)\\}$ for $X,Y\in A\mbox{{\rm-mod}}$, and ${\rm
del}(A)={\rm del}({\rm top}(_{A}A))$, where ${\rm top}(_{A}X)$ denotes the top
of an $A$-module ${}_{A}X$. It was shown in [14] that ${\rm findim}(A)\leq{\rm
del}(A^{{}^{\rm op}})$, where $A^{{}^{\rm op}}$ stands for the opposite
algebra of $A$.

To understand the finitistic dimensions of algebras, Igusa and Todorov
introduced the $\phi$\- and $\psi$-dimensions for Artin algebras in [20]. Let
$K(A)$ be the Grothendieck group of $A$, that is, the quotient of the free
abelian group generated by the isomorphism classes $[X]$ with $X\in
A\mbox{{\rm-mod}}$, modulo the relations: (1) $[Z]=[X]+[Y]$ if
${}_{A}Z\simeq{}_{A}X\oplus{}_{A}Y$; (2) $[P]=0$ if ${}_{A}P$ is projective.
Then $K(A)$ is the free abelian group generated by the isomorphism classes of
non-projective indecomposable $A$-modules $X\in A\mbox{{\rm-mod}}$. Now, we
recall two functions $\phi$ and $\psi$ from $A\mbox{{\rm-mod}}$ to
$\mathbb{N}$, the set of natural numbers, defined in [20].

The syzygy functor $\Omega:A\mbox{{\rm-{\text@underline{mod}}}}\rightarrow
A$-mod on the stable module category $A$-mod induces a group homomorphism
$\Omega:K(A)\rightarrow K(A)$ of abelian groups, given by
$\Omega([X]):=[\Omega(X)]$. For $X\in A\mbox{{\rm-mod}}$, let $\langle
X\rangle$ be the $\mathbb{Z}$-submodule of $K(A)$ generated by the isomorphism
classes of non-projective, indecomposable direct summands of $X$. Since the
rank of the image $\Omega\langle X\rangle$ of $\langle X\rangle$ under
$\Omega$ does not exceed the finite rank of $\langle X\rangle$, it follows
from Fitting’s Lemma that there exists a smallest nonnegative integer
$\phi(X)$ such that $\Omega:\Omega^{n}\langle
X\rangle\rightarrow\Omega^{n+1}\langle X\rangle$ is an isomorphism for all
$n\geq\phi(X)$. Furthermore, let

$\psi(X):=\phi(X)+\sup\\{{\rm projdim}(Y)\mid Y\mbox{ is a direct summand of
}\Omega^{\phi(X)}(X),\;{\rm projdim}(Y)<\infty\\}.$

Then the $\phi$-dimension and $\psi$-dimension of $A$ are defined by

$\phi\dim(A):=\sup\\{\phi(X)\mid X\in A\mbox{{\rm-mod}}\\}\quad\mbox{ and
}\quad\psi\dim(A):=\sup\\{\psi(X)\mid X\in A\mbox{{\rm-mod}}\\}.$

According to [20, Lemma 0.3], $\psi(X)=\phi(X)={\rm projdim}(X)$ if ${\rm
projdim}(X)<\infty$. Thus

${\rm findim}(A)\leq\phi\dim(A)\leq\psi\dim(A)\leq{\rm gl.dim}(A),$

where ${\rm gl.dim}(A)$ means the global dimension of $A$.

Following [24], a non-projective, non-injective simple $A$-module $S$ is
called a _node_ of $A$ if the middle term $P$ of the almost split sequence
starting at $S$, $0\rightarrow S\rightarrow P\rightarrow{\rm
Tr\,}D(S)\rightarrow 0$, is projective, where $D$ is the usual duality of
Artin algebras and Tr stands for the transpose of modules.

######  Theorem 1.1.

If $A$ and $B$ are stably equivalent Artin algebras without nodes, then ${\rm
del}(A)={\rm del}(B)$, $\phi\dim(A)=\phi\dim(B)$ and
$\psi\dim(A)=\psi\dim(B)$.

As a consequence of Theorems 1.1, we have the result (see Proposition 3.4).

######  Proposition 1.2.

If $A$ is a finite-dimensional self-injective algebra over a field and $X$ is
an $A$-module, then

${\rm del}({\rm End}_{A}(A\oplus X))={\rm del}({\rm End}_{A}(A\oplus D{\rm
Tr}(X))\mbox{ and }\;\phi\dim({\rm End}_{A}(A\oplus X))=\phi\dim({\rm
End}_{A}(A\oplus D{\rm Tr}(X)),$

where ${\rm End}_{A}(M)$ stands for the endomorphism algebra of an $A$-module
$M$.

Recall that a projective $A$-module $P$ is said to be _$\nu$ -stably
projective_ [17] if $\nu_{A}^{i}P$ is projective for all $i>0$, where $\nu$ is
the Nakayama functor of $A$. By $A$-stp we mean the full subcategory of
$A\mbox{{\rm-mod}}$ consisting of all $\nu$-stably projective $A$-modules. If
$X$ is an $A$-module such that ${\rm add}(X)=A{\mbox{\rm-stp}}$, then the
endomorphism algebra of the $A$-module $X$ is called a _Frobenius part_ of
$A$, which is self-injective (see [25], or [17, Lemma 2.7]) and unique up to
Morita equivalence. Note that Frobenius parts of Artin algebras were first
given by Martínez-Villa in different but equivalent terms in [25]. An Artin
algebra is said to be _Frobenius-finite_ if its Frobenius part is
representation-finite. By [26, Theorem 2.6], over an algebraically closed
field $k$, every stable equivalence of $k$-algebras with no nodes and no
semisimple direct summands induces a stable equivalence of their Frobenius
parts. This is also true for Artin algebras by checking the proof there. In
this note we show that the assumption that algebras have no nodes and no
semisimple direct summands can be replaced by positive $\nu$-dominant
dimensions (see Section 3.2 for definition).

######  Theorem 1.3.

Suppose that $A$ and $B$ are Artin algebras of $\nu$-dominant dimension at
least $1$. If $A$ and $B$ are stably equivalent, then so are their Frobenius
parts.

For a positive integer $n$, let $M_{n}(A)$ be the $n\times n$ matrix algebra
over $A$. Given a nonempty subset $C$ of $M_{n}(A)$, the _centralizer_ algebra
of $C$ in $M_{n}(A)$ is defined by

$S_{n}(C,A):=\\{a\in M_{n}(A)\mid ca=ac,\;\forall\;c\in C\\}.$

If $C=\\{c\\}$ consists of only a single matrix $c$, we write $S_{n}(c,A)$ for
$S_{n}(\\{c\\},A)$ and call $S_{n}(c,A)$ a _principal centralizer matrix
algebra_. Clearly, $S_{n}(C,A)=\cap_{c\in C}S_{n}(c,A)$. If $C$ consists of
invertible matrices, $S_{n}(C,A)$ is nothing else than the invariant algebra
under the action of conjugation. Particularly, centralizer matrix algebras
include algebras of centrosymmetric matrices which have applications in
engineering problems and quantum physics [10], and the Auslander algebras of
the truncated polynomial algebras (see [31]), which are quasi-hereditary and
useful in the study of unipotent radicals of reductive groups. For further
homological properties of principal centralizer matrix algebras, we refer to
[32].

######  Theorem 1.4.

Let $k$ be an algebraically closed field. Then the Auslander-Reiten conjecture
on stable equivalences holds true for the following finite-dimensional
$k$-algebras:

$(1)$ Frobenius-finite $k$-algebras.

$(2)$ Principal centralizer matrix algebras over $k$.

Note that if we assume in Theorem 1.4(1) that the stable equivalences are of
Morita type and both algebras have no semisimple direct summands, then Theorem
1.4(1) follows from [17, Theorem 1.1]. Since Frobenius-finite alegbras
properly contain representation-finite algebras, Theorem 1.4(1) generalizes
also a result in [25, Theorem 3.4] which states that the Auslander-Reiten
conjecture on stable equivalences holds true for representation-finite
algebras over an algebraically closed field.

This paper is structured as follows. In Section 2, we recall basic facts on
stable equivalences. In Section 3, we prove all results mentioned in
Introduction. In the course of our proofs, we point out that almost
$\nu$-stable derived equivalences preserve the delooping levels,
$\phi$-dimensions and $\psi$-dimensions of algebras over a field (see
Proposition 3.4), though derived equivalences in general may not have this
property. We also conjecture that the finiteness of delooping levels of Artin
algebras is invariant under derived equivalences.

Acknowledgements. The research work was partially supported by the National
Natural Science Foundation of China (Grant 12031014).

##  2 Preliminaries

In this section, we fix notations and recall basic results of stable
equivalences.

Let $A$ be an Artin algebra over a commutative Artin ring $k$. By
$A\mbox{{\rm-mod}}$ we denote the category of all finitely generated left
$A$-modules. Let $A^{{}^{\rm op}}$ be the opposite algebra of $A$, we
understand a right $A$-module as a left $A^{{}^{\rm op}}$-module. We denote by
$D$ the usual duality of Artin algebra from $A\mbox{{\rm-mod}}$ to $A^{{}^{\rm
op}}\mbox{{\rm-mod}}$. For $M\in A\mbox{{\rm-mod}}$, let $\Omega_{A}(M)$ be
the syzygy of ${}_{A}M$; Tr$(M)$ the transpose of $M$, which is an $A^{{}^{\rm
op}}$-module; and ${\rm add}(M)$ the full additive subcategory of
$A\mbox{{\rm-mod}}$ consisting of all direct summands of finite sums of copies
of $M$. We write ${\rm End}_{A}(M)$ for the endomorphism algebra of ${}_{A}M$.

Let $A\mbox{{\rm-mod}}_{\mathscr{P}}$ (respectively, $A\mbox{{\rm-
mod}}_{\mathscr{I}}$) be the full subcategory of $A\mbox{{\rm-mod}}$
consisting of those modules that do not have nonzero projective (respectively,
injective) direct summands. Let $\mathscr{P}(A)_{\mathscr{I}}$ (respectively,
$\mathscr{I}(A)_{\mathscr{P}}$) denote the set of all isomorphism classes of
indecomposable projective (respectively, injective) $A$-modules without any
nonzero injective (respectively, projective) direct summands.

The stable category $A$-mod of $A$ has the same objects as $A\mbox{{\rm-
mod}}$, its morphism set $\underline{{\rm Hom}}_{A}(X,Y)$ of two modules $X$
and $Y$ is the quotient $k$-module of ${\rm Hom}_{A}(X,Y)$ modulo all
homomorphisms that factorize through a projective $A$-module. For $f\in{\rm
Hom}_{A}(X,Y)$, we write $\underline{f}$ for the image of $f$ in
${\rm\underline{Hom}}_{A}(X,Y)$. Note that $X\simeq Y$ in $A$-mod if and only
if there are two projective modules $P,Q\in A\mbox{{\rm-mod}}$ such that
$X\oplus P\simeq Y\oplus Q$ as $A$-modules. In this case, ${\rm
End}_{A}(A\oplus X)$ and ${\rm End}_{A}(A\oplus Y)$ are Morita equivalent.

Let $(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$ be the category
of all finitely presented functors from $(A\mbox{{\rm-mod}})^{\rm op}$ to the
category $\mathscr{A}$ of all abelian groups. Recall that a functor
$H:(A\mbox{{\rm-mod}})^{\rm op}\rightarrow\mathscr{A}$ is said to be _finitely
presented_ if there exists an exact sequence of functors
${\rm\underline{Hom}}_{A}(-,X)\rightarrow{\rm\underline{Hom}}_{A}(-,Y)\rightarrow
H\rightarrow 0$ with $X$ and $Y$ in $A\mbox{{\rm-{\text@underline{mod}}}}$. It
was known from [2] that $(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-
mod}}$ is an abelian category, its projective objects are precisely the
functors ${\rm\underline{Hom}}_{A}(-,X)$ for $X\in A\mbox{{\rm-
mod}}_{\mathscr{P}}$, and its injective objects of
$(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$ are precisely the
functors ${\rm Ext}^{1}_{A}(-,X)$ for $X\in A\mbox{{\rm-mod}}_{\mathscr{P}}$.

Artin algebras $A$ and $B$ over a commutative Artin ring $k$ are said to be
_stably equivalent_ if the two stable categories $A$-mod and $B$-mod are
equivalent as $k$-categories.

Assume that $F:A\mbox{{\rm-{\text@underline{mod}}}}\rightarrow
B\mbox{{\rm-{\text@underline{mod}}}}$ is an equivalence of $k$-categories with
a quasi-inverse functor $G:B\mbox{{\rm-{\text@underline{mod}}}}\rightarrow
A\mbox{{\rm-{\text@underline{mod}}}}$. So $F$ and $G$ are additive functors
and induce two equivalences $\alpha$ and $\beta$ of abelian categories (see
[3, Section 8])

$\alpha:(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-
mod}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(B\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-
mod}}\quad\mbox{and}\quad\beta:(B\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-
mod}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-
mod}},$

and two one-to-one correspondences

$F:A\mbox{{\rm-mod}}_{\mathscr{P}}\longleftrightarrow B\mbox{{\rm-
mod}}_{\mathscr{P}}:G\quad\mbox{and}\quad F^{\prime}:A\mbox{{\rm-
mod}}_{\mathscr{I}}\longleftrightarrow B\mbox{{\rm-
mod}}_{\mathscr{I}}:G^{\prime}$

such that

$\alpha({\rm\underline{Hom}}_{A}(-,X))\simeq{\rm\underline{Hom}}_{B}(-,F(X))\quad\mbox{and}\quad\alpha({\rm
Ext}_{A}^{1}(-,Y))\simeq{\rm Ext}_{B}^{1}(-,F^{\prime}(Y)),$
$\beta({\rm\underline{Hom}}_{B}(-,U))\simeq{\rm\underline{Hom}}_{A}(-,G(U))\quad\mbox{and}\quad\beta({\rm
Ext}_{B}^{1}(-,V))\simeq{\rm Ext}_{A}^{1}(-,G^{\prime}(V)),$

for $X\in A\mbox{{\rm-mod}}_{\mathscr{P}},\,Y\in A\mbox{{\rm-
mod}}_{\mathscr{I}}$, $U\in B\mbox{{\rm-mod}}_{\mathscr{P}}$ and $V\in
B\mbox{{\rm-mod}}_{\mathscr{I}}$. For convenience, we set $F(P)=0$ for a
projective module $P$, and $F^{\prime}(I)=0$ for an injective module $I$.

The following lemma is useful for our later discussions.

######  Lemma 2.1.

([3, Section 7, p.347]) If $X,Y\in A\mbox{{\rm-mod}}_{\mathscr{I}}$, then
$X\simeq Y$ in $A\mbox{{\rm-mod}}$ if and only if ${\rm
Ext}_{A}^{1}(-,X)\simeq{\rm Ext}^{1}_{A}(-,Y)$ in
$(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$.

Recall that a node $S$ of $A$ is called an $F$-_exceptional node_ if
$F(S)\not\simeq F^{\prime}(S)$. By $\mathfrak{n}_{F}(A)$ we denote the set of
isomorphism classes of $F$-exceptional nodes of $A$. By [5, Lemma 3.4], if $X$
is a non-injective, non-projective, indecomposable $A$-module, then
$F(X)\simeq F^{\prime}(X)$. Thus $\mathfrak{n}_{F}(A)$ coincides with the set
of isomorphism classes of non-projective, non-injective, indecomposable
$A$-modules $X$ such that $F(X)\not\simeq F^{\prime}(X)$.

In the following, let

$\bigtriangleup_{A}:=\mathfrak{n}_{F}(A)\dot{\cup}\mathscr{P}(A)_{\mathscr{I}}\quad\mbox{and}\quad\bigtriangledown_{A}:=\mathfrak{n}_{F}(A)\dot{\cup}\mathscr{I}(A)_{\mathscr{P}},$

where $\dot{\cup}$ stands for the disjoint union of sets;
$\mathscr{P}(A)_{\mathscr{I}}$ (respectively, $\mathscr{I}(A)_{\mathscr{P}}$)
stands for the set of all isomorphism classes of indecomposable projective
(respectively, injective) $A$-modules without any nonzero injective
(respectively, projective) summands. By $\bigtriangleup_{A}^{c}$ we mean the
class of indecomposable, non-injective $A$-modules which do not belong to
$\bigtriangleup_{A}$. Thus each module $M\in A\mbox{{\rm-mod}}_{\mathscr{I}}$
admits a unique decomposition (up to isomorphism)

$M\simeq M_{1}\oplus M_{2}$

with $M_{1}\in{\rm add}(\bigtriangleup_{A})$ and $M_{2}\in{\rm
add}(\bigtriangleup_{A}^{c})$.

######  Lemma 2.2.

([9, Lemma 4.10(1)]) The functor $F$ induces the bijections

$F:\bigtriangledown_{A}\longleftrightarrow\bigtriangledown_{B}:G,\quad
F^{\prime}:\bigtriangleup_{A}\longleftrightarrow\bigtriangleup_{B}:G^{\prime}\quad\mbox{and}\quad
F^{\prime}:\bigtriangleup_{A}^{c}\longleftrightarrow\bigtriangleup_{B}^{c}:G^{\prime}.$

Recall that an exact sequence $0\rightarrow X\stackrel{{\scriptstyle
f}}{{\rightarrow}}Y\stackrel{{\scriptstyle g}}{{\rightarrow}}Z\rightarrow 0$
in $A\mbox{{\rm-mod}}$ is called _minimal_ if it does not have a split exact
sequence as its direct summand, that is, there do not exist isomorphisms $u$,
$v$, $w$ such that the diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{u}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{v}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\oplus
X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\left(\begin{smallmatrix}f_{1}&0\\\
0&f_{2}\end{smallmatrix}\right)$$\textstyle{Y_{1}\oplus
Y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\left(\begin{smallmatrix}g_{1}&0\\\
0&g_{2}\end{smallmatrix}\right)$$\textstyle{Z_{1}\oplus
Z_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is exact commutative in $A\mbox{{\rm-mod}}$, where $Y_{2}\neq 0$ and the
sequence $0\rightarrow X_{2}\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}Y_{2}\stackrel{{\scriptstyle
g_{2}}}{{\longrightarrow}}Z_{2}\rightarrow 0$ splits. By definition, a minimal
exact sequence does not split.

######  Lemma 2.3.

([3, Theorem 7.5] or [5, Proposition 2.1]) Let
$H\in(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$ and $0\rightarrow
X\rightarrow Y\rightarrow Z\rightarrow 0$ be a minimal exact sequence in
$A\mbox{{\rm-mod}}$ such that the induced sequence

$0\longrightarrow{\rm\underline{Hom}}_{A}(-,X)\longrightarrow{\rm\underline{Hom}}_{A}(-,Y)\longrightarrow{\rm\underline{Hom}}_{A}(-,Z)\longrightarrow
H\longrightarrow 0$

of functors is exact. Then the following hold.

$(1)$ The induced exact sequence of functors

${\rm\underline{Hom}}_{A}(-,Y)\longrightarrow{\rm\underline{Hom}}_{A}(-,Z)\longrightarrow
H\longrightarrow 0$

is a minimal projective presentation of $H$ in
$(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$.

$(2)$ The induced exact sequence of functors

$0\longrightarrow H\longrightarrow{\rm Ext}_{A}^{1}(-,X)\longrightarrow{\rm
Ext}_{A}^{1}(-,Y)$

is a minimal injective copresentation of $H$ in
$(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$.

The following lemma is from [26, Lemma 1.6], while its proof is referred to
[5].

######  Lemma 2.4.

If $H\in(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$ has a minimal
projective presentation

$\textstyle{{\rm\underline{Hom}}_{A}(-,Y)\;\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm\underline{Hom}}_{A}(-,g)}$$\textstyle{\;\;{\rm\underline{Hom}}_{A}(-,Z)\longrightarrow
H\longrightarrow 0}$

with $Y,Z\in A\mbox{{\rm-mod}}_{\mathscr{P}}$, then there is a minimal exact
sequence

$0\longrightarrow X\longrightarrow Y\oplus P\stackrel{{\scriptstyle
g^{\prime}}}{{\longrightarrow}}Z\longrightarrow 0$

in $A\mbox{{\rm-mod}}$, where $\underline{g^{\prime}}=\underline{g}$ in
$A\mbox{{\rm-{\text@underline{mod}}}}$ and $P$ is a projective $A$-module.

The following generalization of [26, Theorem 1.7] shows that the functor $F$
possesses certain “exactness” property.

######  Lemma 2.5.

Let $0\rightarrow X\oplus X^{\prime}\rightarrow Y\oplus\bar{Y}\oplus I\oplus
P\oplus P^{\prime}\stackrel{{\scriptstyle g}}{{\rightarrow}}Z\rightarrow 0$ be
a minimal exact sequence in $A\mbox{{\rm-mod}}$ with $X,Y\in{\rm
add}(\bigtriangleup_{A}^{c})$, $X^{\prime}\in{\rm add}(\bigtriangleup_{A})$,
$\bar{Y}\in{\rm add}(\mathfrak{n}_{F}(A))$, $I\in{\rm
add}(\mathscr{I}(A)_{\mathscr{P}})$, $P\in{\rm
add}(\mathscr{P}(A)_{\mathscr{I}})$, $P^{\prime}\in A\mbox{{\rm-prinj}\,}$ and
$Z\in A\mbox{{\rm-mod}}_{\mathscr{P}}$. Then there exists a minimal exact
sequence

$0\longrightarrow F(X)\oplus F^{\prime}(X^{\prime})\longrightarrow
F(Y\oplus\bar{Y}\oplus I)\oplus Q\oplus Q^{\prime}\stackrel{{\scriptstyle
g^{\prime}}}{{\longrightarrow}}F(Z)\longrightarrow 0$

in $B\mbox{{\rm-mod}}$, where $Q$ lies in ${\rm
add}(\mathscr{P}(B)_{\mathscr{I}})$ and $Q^{\prime}$ belongs to $B\mbox{{\rm-
prinj}\,}$ such that $F(\bar{Y}\oplus I)\oplus Q\simeq
F^{\prime}(\bar{Y}\oplus P)\oplus J$ for some $J\in{\rm
add}(\mathscr{I}(B)_{\mathscr{P}})$ and
$\underline{g}^{\prime}=F(\underline{g})$ in
$B\mbox{{\rm-{\text@underline{mod}}}}$.

Proof. We provide a proof by using some idea in [9, Lemma 4.13]. Consider the
finitely presented functor $H$:

${\rm\underline{Hom}}_{A}(-,Y\oplus\bar{Y}\oplus I\oplus P\oplus
P^{\prime})\longrightarrow{\rm\underline{Hom}}_{A}(-,Z)\longrightarrow
H\longrightarrow 0$

induced from the given minimal exact sequence

$\qquad 0\longrightarrow X\oplus X^{\prime}\longrightarrow
Y\oplus\bar{Y}\oplus I\oplus P\oplus P^{\prime}\stackrel{{\scriptstyle
g}}{{\longrightarrow}}Z\longrightarrow 0$

in $A\mbox{{\rm-mod}}$ with $I\in{\rm add}(\mathscr{I}(A)_{\mathscr{P}})$,
$P\in{\rm add}(\mathscr{P}(A)_{\mathscr{I}})$ and $P^{\prime}\in A\mbox{{\rm-
prinj}\,}$. It follows from Lemma 2.3 that the sequence of functors

$\textstyle{{\rm\underline{Hom}}_{A}(-,Y\oplus\bar{Y}\oplus
I)\quad\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\tiny{\rm\underline{Hom}}_{A}(-,g)}}$$\textstyle{\quad{\rm\underline{Hom}}_{A}(-,Z)\longrightarrow
H\longrightarrow 0}$

is a minimal projective presentation of $H$ in
$(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$ and that the sequence
of functors

$0\longrightarrow H\longrightarrow{\rm Ext}_{A}^{1}(-,X\oplus
X^{\prime})\longrightarrow{\rm Ext}_{A}^{1}(-,Y\oplus\bar{Y}\oplus P)$

is a minimal injective copresentation of $H$ in
$(A\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$. Applying the
equivalence functor $\alpha$ to the above two sequences of functors, we see
that the sequence

$(\star)\qquad{\rm\underline{Hom}}_{B}(-,F(Y\oplus\bar{Y}\oplus
I))\longrightarrow{\rm\underline{Hom}}_{B}(-,F(Z))\longrightarrow\alpha(H)\longrightarrow
0$

is a minimal projective presentation of $\alpha(H)$ in
$(B\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$ and the sequence

$(*)\qquad 0\longrightarrow\alpha(H)\longrightarrow{\rm
Ext}_{B}^{1}(-,F^{\prime}(X)\oplus F^{\prime}(X^{\prime}))\longrightarrow{\rm
Ext}_{B}^{1}(-,F^{\prime}(Y\oplus\bar{Y}\oplus P))$

is a minimal injective copresentation of $\alpha(H)$ in
$(B\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$.

It follows from $(\star)$ and Lemma 2.4 that there is a minimal exact sequence

$(\diamondsuit)\qquad 0\longrightarrow W\longrightarrow F(Y\oplus\bar{Y}\oplus
I)\oplus Q\oplus Q^{\prime}\stackrel{{\scriptstyle
g^{\prime}}}{{\longrightarrow}}F(Z)\longrightarrow 0$

in $B\mbox{{\rm-mod}}$ with $Q\in{\rm add}(\mathscr{P}(B)_{\mathscr{I}})$,
$Q^{\prime}\in B\mbox{{\rm-prinj}\,}$ and
$\underline{g}^{\prime}=F(\underline{g})$ in
$B\mbox{{\rm-{\text@underline{mod}}}}$. The minimality of this sequence
implies $W\in B\mbox{{\rm-mod}}_{\mathscr{I}}$. Note that $(\star)$ is induced
from $(\diamondsuit)$. Now, by Lemma 2.3(2) and $(\diamondsuit)$, the exact
sequence

$({\dagger})\qquad 0\longrightarrow\alpha(H)\longrightarrow{\rm
Ext}_{B}^{1}(-,W)\longrightarrow{\rm Ext}_{B}^{1}(-,F(Y\oplus\bar{Y}\oplus
I)\oplus Q)$

of functors is a minimal injective copresentation of $\alpha(H)$ in
$(B\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$. Thus both
$({\dagger})$ and $(*)$ are minimal injective copresentations of $\alpha(H)$.
This implies that

$(**)\qquad{\rm Ext}_{B}^{1}(-,F^{\prime}(X)\oplus
F^{\prime}(X^{\prime}))\simeq{\rm Ext}_{B}^{1}(-,W)\;\mbox{ and}$
$\qquad\;\qquad({\ddagger})\qquad{\rm
Ext}_{B}^{1}(-,F^{\prime}(Y\oplus\bar{Y}\oplus P))\simeq{\rm
Ext}_{B}^{1}(-,F(Y\oplus\bar{Y}\oplus I)\oplus Q)$

in $(B\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-mod}}$. Since $X$ lies in
${\rm add}(\bigtriangleup_{A}^{c})$ and $X^{\prime}$ lies in ${\rm
add}(\bigtriangleup_{A})$, we know from Lemma 2.2 that $F^{\prime}(X)\in{\rm
add}(\bigtriangleup_{B}^{c})$ and $F^{\prime}(X^{\prime})\in{\rm
add}(\bigtriangleup_{B})$. In particular, $F^{\prime}(X)\oplus
F^{\prime}(X^{\prime})\in B\mbox{{\rm-mod}}_{\mathscr{I}}$. Thus
$F^{\prime}(X)\oplus F^{\prime}(X^{\prime})\simeq W$ as $B$-modules by Lemma
2.1 and $(**)$. It follows from $X\in{\rm add}(\bigtriangleup_{A}^{c})$ that
$F(X)\simeq F^{\prime}(X)$ and therefore $F(X)\oplus
F^{\prime}(X^{\prime})\simeq W$ as $B$-modules. Hence $(\diamondsuit)$ can be
written as

$0\longrightarrow F(X)\oplus F^{\prime}(X^{\prime})\longrightarrow
F(Y\oplus\bar{Y}\oplus I)\oplus Q\oplus Q^{\prime}\stackrel{{\scriptstyle
g^{\prime}}}{{\longrightarrow}}F(Z)\longrightarrow 0.$

To complete the proof, we have to show that $F(\bar{Y}\oplus I)\oplus Q\simeq
F^{\prime}(\bar{Y}\oplus P)\oplus J$ for some $J\in{\rm
add}(\mathscr{I}(B)_{\mathscr{P}})$. In fact, it follows from $Y\in{\rm
add}(\bigtriangleup_{A}^{c})$ that $F(Y)\simeq F^{\prime}(Y)$ as $B$-modules
and that both $F(Y)$ and $F^{\prime}(Y)$ lie in $B\mbox{{\rm-
mod}}_{\mathscr{I}}$. Since $\bar{Y}$ belongs to ${\rm
add}(\mathfrak{n}_{F}(A))$ and $I$ belongs to ${\rm
add}(\mathscr{I}(A)_{\mathscr{P}})$, it follows from Lemma 2.2 that
$F(\bar{Y}\oplus I)$ lies in ${\rm add}(\bigtriangledown_{B})$. Thus
$F(\bar{Y}\oplus I)\simeq F(\bar{Y})\oplus F(I)=V\oplus J$ for some $V\in{\rm
add}(\mathfrak{n}_{G}(B))$ and $J\in\mathscr{I}(B)_{\mathscr{P}}$. Therefore
we have the isomorphisms in $(B\mbox{{\rm-{\text@underline{mod}}}})\mbox{{\rm-
mod}}$:

$\begin{array}[]{rl}{\rm Ext}_{B}^{1}(-,F(Y\oplus\bar{Y}\oplus I)\oplus
Q)&=\;{\rm Ext}_{B}^{1}(-,F(Y)\oplus F(\bar{Y}\oplus I)\oplus Q)\\\
&\simeq\;{\rm Ext}_{B}^{1}(-,F(Y)\oplus V\oplus Q)\\\ &\simeq\;{\rm
Ext}_{B}^{1}(-,F^{\prime}(Y\oplus\bar{Y}\oplus P))\quad(\mbox{ by
}({\ddagger})).\end{array}$

As $\bar{Y}\in{\rm add}(\mathfrak{n}_{F}(A))$ and $P\in{\rm
add}(\mathscr{P}(A)_{\mathscr{I}})$, it follows from Lemma 2.2 that
$F^{\prime}(\bar{Y}\oplus P)$ is in ${\rm add}(\bigtriangleup_{B})$ and
$F^{\prime}(\bar{Y}\oplus P)$ is in $B\mbox{{\rm-mod}}_{\mathscr{I}}$. Now,
Lemma 2.1 shows that $F^{\prime}(Y\oplus\bar{Y}\oplus P)\simeq F(Y)\oplus
V\oplus Q$ and $F^{\prime}(\bar{Y})\oplus F^{\prime}(P)\simeq V\oplus Q$ as
$B$-modules. Thus $F(\bar{Y})\oplus F(I)\oplus Q\simeq V\oplus J\oplus Q\simeq
F^{\prime}(\bar{Y})\oplus F^{\prime}(P)\oplus J$ as $B$-modules. $\square$

The following special case of Lemma 2.5 is often used in our proofs.

######  Corollary 2.6.

Let $0\rightarrow X\oplus X^{\prime}\rightarrow
P^{\prime}\stackrel{{\scriptstyle g}}{{\rightarrow}}Z\rightarrow 0$ be a
minimal exact sequence in $A\mbox{{\rm-mod}}$ such that $X\in{\rm
add}(\bigtriangleup_{A}^{c})$, $X^{\prime}\in{\rm add}(\bigtriangleup_{A})$,
$P^{\prime}\in A\mbox{{\rm-prinj}\,}$ and $Z\in A\mbox{{\rm-
mod}}_{\mathscr{P}}$. Then there exists a minimal exact sequence

$0\longrightarrow F(X)\oplus F^{\prime}(X^{\prime})\longrightarrow
Q^{\prime}\stackrel{{\scriptstyle
g^{\prime}}}{{\longrightarrow}}F(Z)\longrightarrow 0$

of $B$-modules with $Q^{\prime}\in B\mbox{{\rm-prinj}\,}$.

##  3 Proofs of the statements

In this section, we prove all results mentioned in Introduction. We keep the
notation introduced in the previous sections.

Let $A$ be an Artin algebra over a commutative Artin ring $k$. Following [17],
a projective $A$-module $P$ is said to be _$\nu$ -stably projective_ if
$\nu_{A}^{i}P$ is projective for all $i>0$. Here $\nu_{A}$ is the Nakayama
functor $D{\rm Hom}_{A}(-,A)$ of $A$. Let $U$ be the direct sum of all non-
isomorphic indecomposable $\nu$-stably projective $A$-modules. Since
$\nu_{A}(U)$ is $\nu$-stably projective, we have $U\simeq\nu_{A}(U)$ and ${\rm
top}(U)\simeq{\rm soc}(U)$, where ${\rm soc}(U)$ is the socle of the
$A$-module $U$. Clearly, ${\rm soc}(U)\simeq\Omega_{A}(U/{\rm soc}(U))\oplus
Q$ for some projective $A$-module $Q$. Thus ${\rm del}({\rm top}(U))={\rm
del}({\rm soc}(U))=0$ by definition. Let $V$ be the direct sum of all non-
isomorphic indecomposable projective $A$-modules that are neither simple nor
$\nu$-stably projective. Then ${\rm del}(A)={\rm del}({\rm top}(U\oplus
V))=\max\\{{\rm del}({\rm top}(U)),{\rm del}({\rm top}(V))\\}={\rm del}({\rm
top}(V))$.

###  3.1 Proof of Theorem 1.1

Let $A$ and $B$ be Artin $k$-algebras that have neither nodes nor semisimple
direct summands. Assume that
$F:A\mbox{{\rm-{\text@underline{mod}}}}\rightarrow
B\mbox{{\rm-{\text@underline{mod}}}}$ is an equivalence of $k$-categories.

Under these assumptions, $\mathfrak{n}_{F}(A)=\varnothing$,
$\mathfrak{n}_{F^{-1}}(B)=\varnothing$ and there is a bijection
$F^{\prime}:\mathscr{P}(A)_{\mathscr{I}}\rightarrow\mathscr{P}(B)_{\mathscr{I}}$
(see Lemma 2.2). Further, Lemma 2.5 can be specified as follows.

######  Lemma 3.1.

[26, Theorem 1.7] Let $0\rightarrow X\oplus P_{1}\stackrel{{\scriptstyle
f}}{{\longrightarrow}}Y\oplus P\oplus P^{\prime}\stackrel{{\scriptstyle
g}}{{\longrightarrow}}Z\rightarrow 0$ be a minimal exact sequence of
$A$-modules, where $X,Y,Z\in A\mbox{{\rm-mod}}_{\mathscr{P}}$,
$P_{1},P\in\mathscr{P}(A)_{\mathscr{I}}$ and $P^{\prime}$ is a projective-
injective $A$-module. Then there is a minimal exact sequence

$0\longrightarrow F(X)\oplus F^{\prime}(P_{1})\stackrel{{\scriptstyle
f^{\prime}}}{{\longrightarrow}}F(Y)\oplus F^{\prime}(P)\oplus
Q\stackrel{{\scriptstyle g^{\prime}}}{{\longrightarrow}}F(Z)\longrightarrow 0$

in $B\mbox{{\rm-mod}}$ with $Q$ a projective-injective $B$-module and
$\underline{g^{\prime}}=F(\underline{g})$. In particular,
$\Omega_{B}F(Z)\simeq F\Omega_{A}(Z)$ in
$B\mbox{{\rm-{\text@underline{mod}}}}$ for $Z\in A\mbox{{\rm-
mod}}_{\mathscr{P}}$.

######  Lemma 3.2.

For $X\in A\mbox{{\rm-mod}}_{\mathscr{P}}$, we have ${\rm del}(X)={\rm
del}(F(X))$.

Proof. We show ${\rm del}(F(X))\leq{\rm del}(X)$. In fact, we may assume
$d:={\rm del}(X)<\infty$. Then, by the definition of delooping levels, there
exists $M\in A\mbox{{\rm-mod}}_{\mathscr{P}}$ such that
$\Omega_{A}^{d}(X)\in{\rm add}(_{A}A\oplus\Omega_{A}^{d+1}(M))$. Thus
$F(\Omega_{A}^{d}(X))\in{\rm add}({{}_{B}}B\oplus F\Omega_{A}^{d+1}(M))$ by
the additivity of the functor $F$. On the other hand, it follows from Lemma
3.1 that $\Omega_{B}F(X)\simeq F\Omega_{A}(X)$ and $\Omega_{B}F(M)$ $\simeq
F\Omega_{A}(M)$ in $B\mbox{{\rm-{\text@underline{mod}}}}$. Then
$\Omega_{B}^{i}F(X)\simeq F\Omega_{A}^{i}(X)$ and $\Omega_{B}^{i}F(M)\simeq
F\Omega_{A}^{i}(M)$ in $B\mbox{{\rm-{\text@underline{mod}}}}$ for $i\geq 0$.
Thus $\Omega_{B}^{d}F(X)\in{\rm add}({{}_{B}}B\oplus\Omega_{A}^{d+1}F(M))$,
and therefore ${\rm del}(F(X))\leq d={\rm del}(X)<\infty$. Similarly, we show
${\rm del}(X)\leq{\rm del}(F(X)).$ Thus ${\rm del}(X)={\rm del}(F(X))$.
$\square$

Proof of Theorem 1.1. Suppose that $A$ and $B$ are stably equivalent Artin
algebras without nodes.

(i) ${\rm del}(A)={\rm del}(B).$ In fact, the delooping levels of algebras
involve only simple modules. If $A$ or $B$ has a semisimple direct summand,
then the simple modules belonging to the semisimple direct summand have
delooping levels $0$, and therefore do not contribute to the delooping levels
of the considered algebra. So we may remove all semisimple direct summands
from both algebras $A$ and $B$. Of course, the resulting algebras are still
stably equivalent. Thus we assume that both $A$ and $B$ do not have any
semisimple direct summands. Let $V$ be the direct sum of all non-isomorphic
indecomposable projective $A$-modules that are neither simple nor $\nu$-stably
projective, and let $V^{\prime}$ be the direct sum of all non-isomorphic
indecomposable projective $B$-modules which are neither simple nor
$\nu$-stably projective. It follows from [26, Lemma 2.5], which is true also
for Artin algebras, that $F({\rm top}(V))\simeq{\rm top}(V^{\prime})$ as
$B$-modules. Note that ${\rm top}(V)$ does not have any nonzero projective
direct summands. By Lemma 3.2, we have ${\rm del}({\rm top}(V))={\rm del}({\rm
top}(V^{\prime}))$. Thus ${\rm del}(A)={\rm del}({\rm top}(V))={\rm del}({\rm
top}(V^{\prime}))={\rm del}(B)$.

(ii) $\phi\dim(A)=\phi\dim(B)$ and $\psi\dim(A)=\psi\dim(B)$. For Artin
algebras $A_{1}$ and $A_{2}$, there hold $\phi\dim(A_{1}\times
A_{2})=\max\\{\phi\dim(A_{1}),\phi\dim(A_{2})\\}$ and $\psi\dim(A_{1}\times
A_{2})=\max\\{\psi\dim(A_{1}),\psi\dim(A_{2})\\}$. Since $\phi$\- and
$\psi$-dimensions of semisimple algebras are $0$, we may remove semisimple
direct summands from $A$ and $B$ if they have any. Then the resulting algebras
are still stably equivalent. So we may assume that both algebras $A$ and $B$
do not have any semisimple direct summands. Let
$F:A\mbox{{\rm-{\text@underline{mod}}}}\rightarrow
B\mbox{{\rm-{\text@underline{mod}}}}$ defines a stable equivalence between $A$
and $B$. We also denote by $F$ the correspondence from $A$-modules to
$B$-modules, which takes projective $A$-module to $0$. As a functor of
$k$-categories, $F$ is additive and commutes with finite direct sums in
$A$-mod. Thus the map $\tilde{F}:K(A)\rightarrow K(B)$ given by
$\tilde{F}([X]):=[F(X)]$, is a well-defined homomorphism of Grothendieck
groups. It is actually an isomorphism of abelian groups. By Lemma 3.1 we have
$\Omega_{B}(F(X))\simeq F(\Omega_{A}(X))$ in $B$-mod for $X\in A\mbox{{\rm-
mod}}$. Let $\langle X\rangle$ be the $\mathbb{Z}$-submodule of $K(A)$
generated by the isomorphism classes of indecomposable, non-projective direct
summands of $X$. For $n\geq 0$, the following diagrams are commutative

$\textstyle{K(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{F}}$$\scriptstyle{\Omega_{A}}$$\textstyle{K(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Omega_{B}}$$\textstyle{\Omega_{A}^{n}\langle
X\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{F}_{res}}$$\scriptstyle{\Omega_{A}}$$\textstyle{\Omega_{B}^{n}\langle
F(X)\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Omega_{B}}$$\textstyle{K(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{F}}$$\textstyle{K(B),}$$\textstyle{\Omega_{A}^{n+1}\langle
X\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{F}_{res}}$$\textstyle{\Omega_{B}^{n+1}\langle
F(X)\rangle}$

where $\tilde{F}_{res}$ is the restriction of $\tilde{F}$. Since
$\tilde{F}:K(A)\rightarrow K(B)$ is an isomorphism of abelian groups, the
$\mathbb{Z}$-module homomorphism $\Omega_{A}:\Omega_{A}^{n}\langle
X\rangle\rightarrow\Omega_{A}^{n+1}\langle X\rangle$ is isomorphic for $n\geq
0$ if and only if so is the $\mathbb{Z}$-homomorphism
$\Omega_{B}:\Omega_{B}^{n}\langle
F(X)\rangle\rightarrow\Omega_{B}^{n+1}\langle F(X)\rangle$ for $n\geq 0$. By
the definition of $\phi$-dimensions, $\phi(X)=\phi(F(X))$ and
$\phi\dim(A)\leq\phi\dim(B)$. Similarly, $\phi\dim(B)\leq\phi\dim(A)$. Thus
$\phi\dim(A)=\phi\dim(B)$. For $Y\in A\mbox{{\rm-mod}}$, since
$\Omega_{B}(F(Y))\simeq F(\Omega_{A}(Y))$ in $B$-mod and $F$ is an
equivalence, we get ${\rm projdim}(_{B}F(Y))={\rm projdim}(_{A}Y)$. Then
$\psi(X)=\psi(F(X))$ and $\psi\dim(A)=\psi\dim(B)$. $\square$

Remark that Theorem 1.1 may fail if Artin algebras have nodes. This can be
seen by the following examples.

######  Example 3.3.

(1) Let $A_{1}$ be the algebra over a field $k$, given by the quiver with a
relation:

$\textstyle{1\,\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\;,}$$\textstyle{\quad\alpha^{2}=0.}$

Clearly, $A_{1}$ has a node and is stably equivalent to the path algebra
$A^{\prime}_{1}$ of the quiver $1\bullet\leftarrow\bullet 2$. Note that
$A^{\prime}_{1}$ has no nodes and its Frobenius part is $0$. In this case,
both $A_{1}$ and $A^{\prime}_{1}$ have only $1$ non-projective simple module.
Clearly, ${\rm del}(A_{1})=\phi\dim(A_{1})=\psi\dim(A_{1})=0<1={\rm
del}(A^{\prime}_{1})=\phi\dim(A^{\prime}_{1})=\psi\dim(A^{\prime}_{1}).$
Remark that $A_{1}$ and $A^{\prime}_{1}$ are never stably equivalent of Morita
type by Lemma 3.9 below.

(2) Let $A_{2}$ be the algebra over a field $k$, given by the quiver with a
relation:

$\textstyle{1\,\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\bullet
2\;,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\alpha\beta=0}$

(see [31, Example 4.9] for more general situations). In this case, we consider
the $2$ almost split sequences in $A_{2}\mbox{{\rm-mod}}$

$0\longrightarrow S(1)\longrightarrow P(2)\longrightarrow S(2)\longrightarrow
0\;\mbox{ and }\;0\longrightarrow S(2)\longrightarrow I(2)\longrightarrow
S(1)\longrightarrow 0,$

where $P(i)$, $I(i)$ and $S(i)$ are the projective, injective and simple
modules corresponding to the vertex $i$, respectively. Clearly, $A_{2}$ has
the Frobenius part isomorphic to $A_{1}$, and a unique node $S(1)$. Let $I$ be
the trace of $S(1)$ in $A_{2}$ and $J$ be the left annihilator of $I$ in
$A_{2}$. Then $I=\\{r_{1}\beta\alpha+r_{2}\beta\mid r_{1},r_{2}\in k\\}$ and
$J=\\{r_{1}e_{2}+r_{2}\alpha+r_{3}\beta\mid r_{i}\in k,1\leq i\leq 3\\}$.
Define $A^{\prime}_{2}$ to be the triangular matrix algebra

$A^{\prime}_{2}=\begin{pmatrix}A_{2}/I&0\\\
I&A_{2}/J\end{pmatrix}\simeq\begin{pmatrix}k&0&0\\\ k&k&0\\\
k&k&k\end{pmatrix}.$

Then $A^{\prime}_{2}$ has no nodes and its Frobenius part is $0$. By Lemma
3.13(2), $A_{2}$ and $A^{\prime}_{2}$ are stably equivalent. Particularly,
they have $2$ non-isomorphic, non-projective simple modules, and ${\rm
del}(A_{2})=\phi\dim(A_{2})=\psi\dim(A_{2})=2>1$ = ${\rm
del}(A^{\prime}_{2})=\phi\dim(A^{\prime}_{2})=\psi\dim(A^{\prime}_{2}).$

Note that neither delooping levels nor $\phi$-dimensions are preserved by
tilting. For example, the path algebra $A$ (over a field) of the quiver
$\bullet\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\bullet\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}\bullet$
can be tilted to the quotient algebra $B:=A/(\alpha\beta)$. In this case, $A$
has no nodes, but $B$ has a node, while we have ${\rm
del}(A)=\phi\dim(A)=\psi\dim(A)=1<2={\rm del}(B)=\phi\dim(B)=\psi\dim(B)$.
This shows that in general derived equivalences do not have to preserve the
delooping levels and the $\phi$-dimensions of algebras. Nevertheless, we will
show that almost $\nu$-stable derived equivalences do preserve delooping
levels, $\phi$-dimensions and $\psi$-dimensions.

######  Proposition 3.4.

Let $A$ and $B$ be arbitrary finite-dimensional algebras over a field. If $A$
and $B$ are almost $\nu$-stable derived equivalent, then ${\rm del}(A)={\rm
del}(B)$, $\phi\dim(A)=\phi\dim(B)$ and $\psi\dim(A)=\psi\dim(B)$. In
particular, if $A$ is a self-injective algebra over a field and $X\in
A\mbox{{\rm-mod}}$, then

$(1)$ ${\rm del}\big{(}{\rm End}_{A}(A\oplus X)\big{)}={\rm del}\big{(}{\rm
End}_{A}(A\oplus\Omega_{A}(X))\big{)}={\rm del}\big{(}{\rm End}_{A}(A\oplus
D{\rm Tr}(X))\big{)}.$

$(2)$ $\phi\dim\big{(}{\rm End}_{A}(A\oplus X)\big{)}=\phi\dim\big{(}{\rm
End}_{A}(A\oplus\Omega_{A}(X))\big{)}=\phi\dim\big{(}{\rm End}_{A}(A\oplus
D{\rm Tr}(X))\big{)}.$

$(3)$ $\psi\dim\big{(}{\rm End}_{A}(A\oplus X)\big{)}=\psi\dim\big{(}{\rm
End}_{A}(A\oplus\Omega_{A}(X))\big{)}=\psi\dim\big{(}{\rm End}_{A}(A\oplus
D{\rm Tr}(X))\big{)}.$

Before starting with the proof of this proposition, we recall two definitions.

######  Definition 3.5.

(see [16]) A derived equivalence $F$ of bounded derived module categories
between arbitrary Artin algebras $A$ and $B$ with a quasi-inverse $G$ is said
to be _almost $\nu$-stable_ if the associated radical tilting complexes
$T^{\bullet}$ over $A$ to $F$ and $\bar{T}^{\bullet}$ over $B$ to $G$ are of
the form

$T^{\bullet}:0\longrightarrow T^{-n}\longrightarrow\cdots
T^{-1}\longrightarrow T^{0}\longrightarrow 0\;\mbox{ and
}\,\bar{T}^{\bullet}:0\longrightarrow\bar{T}^{0}\longrightarrow\bar{T}^{1}\longrightarrow\cdots\longrightarrow\bar{T}^{n}\longrightarrow
0,$

respectively, such that ${\rm add}(\bigoplus_{i=1}^{n}T^{-i})={\rm
add}(\bigoplus_{i=1}^{n}\nu_{A}(T^{-i}))$ and ${\rm
add}(\bigoplus_{i=1}^{n}\bar{T}^{i})={\rm
add}(\bigoplus_{i=1}^{n}\nu_{B}(\bar{T}^{i})).$

Almost $\nu$-stable derived equivalences induce special stable equivalences
(see [16, Theorem 1.1(2)]), namely stable equivalences of Morita type.

######  Definition 3.6.

Let $A$ and $B$ be arbitrary finite-dimensional algebras over a field $k$.

$(1)$ $A$ and $B$ are said to be _stably equivalent of Morita type_ (see [8])
if there exist bimodules ${}_{A}M_{B}$ and ${}_{B}N_{A}$ such that

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

(ii) $M\otimes_{B}N\simeq A\oplus P$ 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 $Q$.

$(2)$ $A$ and $B$ are said to be _stably equivalent of adjoint type_ (see
[30]) if the bimodules $M$ and $N$ in $(1)$ provide additionally two adjoint
pairs $(M\otimes_{B}-,N\otimes_{A}-)$ and $(N\otimes_{A}-,M\otimes_{B}-)$ of
functors on module categories.

Stable equivalences of adjoint type have some nice properties.

######  Lemma 3.7.

Let $A$ and $B$ be arbitrary finite-dimensional algebras over a field. Suppose
$A$ and $B$ are stably equivalent of adjoint type induced by ${}_{A}M_{B}$ and
${}_{B}N_{A}$. Write ${}_{A}M\otimes_{B}N_{A}\simeq A\oplus P$ and
${}_{B}N\otimes_{A}M_{B}\simeq B\oplus Q$ as bimodules. Then the following
hold:

$(1)$ ${\rm add}(\nu_{A}P)={\rm add}(_{A}P)$ and ${\rm add}(\nu_{B}Q)={\rm
add}(_{B}Q)$, where $\nu_{A}$ is the Nakayama functor of $A$.

$(2)$ If $S$ is a simple $A$-module with ${\rm Hom}_{A}(P,S)=0$, then
$N\otimes_{A}S$ is a simple $B$-module with ${\rm
Hom}_{B}(Q,N\otimes_{A}S)=0$.

$(3)$ For an $A$-module $X$ and a $B$-module $Y$, we have
$\Omega_{B}^{i}(N\otimes_{A}X)\simeq N\otimes_{A}\Omega_{A}^{i}(X)$ in
$B\mbox{{\rm-{\text@underline{mod}}}}$ and
$\Omega_{A}^{i}(M\otimes_{B}Y)\simeq M\otimes_{B}\Omega_{B}^{i}(Y)$ in
$A\mbox{{\rm-{\text@underline{mod}}}}\,$ for $i\geq 0$.

$(4)$ For an $A$-module $X$ and a $B$-module $Y$, there hold ${\rm
del}(_{A}M\otimes_{B}N\otimes_{A}X)={\rm del}({{}_{B}}N\otimes_{A}X)={\rm
del}(_{A}X)$ and ${\rm del}(_{B}N\otimes_{A}M\otimes_{B}Y)={\rm
del}({{}_{A}}M\otimes_{B}Y)={\rm del}(_{B}Y)$.

Proof. (1) and (2) follow from [17, Lemma 3.1], while (3) and (4) can be
deduced easily. $\square$

######  Lemma 3.8.

Let $A$ and $B$ be arbitrary Artin algebras with no separable direct summands.
If $A$ and $B$ are stably equivalent of Morita type, then they are even stably
equivalent of adjoint type.

Proof. Since $A$ and $B$ are stably equivalent of Morita type and have no
separable direct summands, it follows from [22, Proposition 2.1 and Theorem
2.2] which are valid also for Artin algebras by checking the argument there,
that $A$ and $B$ have the same number of indecomposable direct summands (as
two-sided ideals) and that we may write $A=A_{1}\times A_{2}\times\cdots\times
A_{s}$ and $B=B_{1}\times B_{2}\times\cdots\times B_{s}$ as products of
indecomposable algebras, such that the blocks $A_{i}$ and $B_{i}$ are stably
equivalent of Morita type for $1\leq i\leq s$. Suppose that $M^{(i)}$ and
$N^{(i)}$ define a stable equivalence of Morita type between $A_{i}$ and
$B_{i}$. Observe that the two results [11, Lemma 2.1 and Corollary 3.1] hold
true for indecomposable and non-separable Artin algebras. Thus, by [11, Lemma
2.1], we may assume that ${}_{A_{i}}{M^{(i)}}_{B_{i}}$ and
${}_{B_{i}}{N^{(i)}}_{A_{i}}$ are indecomposable and non-projective bimodules.
Since the algebra $A_{i}$ is indecomposable and non-separable by assumption,
it follows from [11, Corollary 3.1] that
$(M^{(i)}\otimes_{B_{i}}-,N^{(i)}\otimes_{A_{i}}-)$ and
$(N^{(i)}\otimes_{A_{i}}-,M^{(i)}\otimes_{B_{i}}-)$ are adjoint pairs between
$A_{i}\mbox{{\rm-mod}}$ and $B_{i}\mbox{{\rm-mod}}$. Let $M:=\bigoplus_{1\leq
j\leq s}M^{(j)}$ and $N:=\bigoplus_{1\leq j\leq s}N^{(j)}$. Then ${}_{A}M_{B}$
and ${}_{B}N_{A}$ define a stable equivalence of adjoint type between $A$ and
$B$. $\square$

For stable equivalences of Morita type, the requirement that algebras
considered have no nodes can be eliminated.

######  Lemma 3.9.

Let $A$ and $B$ be arbitrary finite-dimensional algebras over a field $k$. If
$A$ and $B$ are stably equivalent of Morita type, then

$(1)$ ${\rm del}(A)={\rm del}(B)$.

$(2)$ $\phi\dim(A)=\phi\dim(B)$ and $\psi\dim(A)=\psi\dim(B).$

Proof. (1) Let $A=A_{0}\times A_{1}$ and $B=B_{0}\times B_{1}$, where $A_{0}$
and $B_{0}$ are separable algebras, and where $A_{1}$ and $B_{1}$ are algebras
without separable direct summands. Since $A$ and $B$ are stably equivalent of
Morita type, it follows from the proof of [23, Theorem 4.7] that $A_{1}$ and
$B_{1}$ are stably equivalent of Morita type. By Lemma 3.8, $A_{1}$ and
$B_{1}$ are stably equivalent of adjoint type. Suppose that the adjoint type
between $A_{1}$ and $B_{1}$ is defined by two bimodules ${}_{A_{1}}M_{B_{1}}$
and ${}_{B_{1}}N_{A_{1}}$. By Definition 3.6, we write
${{}_{A_{1}}}M\otimes_{B_{1}}N_{A_{1}}\simeq A_{1}\oplus P$ and
${}_{B_{1}}N\otimes_{A_{1}}M_{B_{1}}\simeq B_{1}\oplus Q$. It follows from
Lemma 3.7(1) that ${\rm add}(\nu_{A_{1}}P)={\rm add}(_{A_{1}}P)$. Then ${\rm
del}({\rm top}(P))={\rm del}({\rm soc}(P))=0$ and

${\rm del}(A_{1})=\max\\{{\rm del}(_{A_{1}}S)\mid{{}_{A_{1}}}S\mbox{ is simple
with }{\rm Hom}_{A_{1}}(P,S)=0\\}.$

Let $S$ be a simple $A_{1}$-module with ${\rm Hom}_{A_{1}}(P,S)=0$. By Lemma
3.7(2), ${}_{B_{1}}N\otimes_{A_{1}}S$ is a simple $B_{1}$-module. It follows
from Lemma 3.7(4) that ${\rm del}(_{A_{1}}S)={\rm
del}(_{B_{1}}N\otimes_{A_{1}}S)\leq{\rm del}(B_{1})$, and therefore ${\rm
del}(A_{1})\leq{\rm del}(B_{1})$. Similarly, we prove ${\rm
del}(B_{1})\leq{\rm del}(A_{1})$. Thus ${\rm del}(A_{1})={\rm del}(B_{1})$.
Since the delooping levels of separable blocks are $0$, we have ${\rm
del}(A)=\max\\{{\rm del}(A_{0}),{\rm del}(A_{1})\\}={\rm del}(A_{1})={\rm
del}(B_{1})=\max\\{{\rm del}(B_{0}),{\rm del}(B_{1})\\}={\rm del}(B)$.

(2) Suppose that $A$ and $B$ are stably equivalent of Morita type defined by
${}_{A}M_{B}$ and ${}_{B}N_{A}$. Since ${}_{B}N$ is projective, the functor
$F:=N\otimes_{A}-:A\mbox{{\rm-mod}}\rightarrow B\mbox{{\rm-mod}}$ takes
projective $A$-modules to projective $B$-modules, and commutes with finite
direct sums. Thus $F$ induces an equivalence:
$A\mbox{{\rm-{\text@underline{mod}}}}\rightarrow B$-mod. As in the proof of
Theorem 1.1(ii), we obtain $\phi\dim(A)=\phi\dim(B)$ and
$\psi\dim(A)=\psi\dim(B).$ $\square$

Proof of Proposition 3.4. Suppose that there is an almost $\nu$-stable derived
equivalence between finite-dimensional algebras $A$ and $B$ over a field $k$.
It follows from [16, Theorem 1.1(2)] that $A$ and $B$ are stably equivalent of
Morita type. By Lemma 3.9, ${\rm del}(A)={\rm del}(B)$,
$\phi\dim(A)=\phi\dim(B)$ and $\psi\dim(A)=\psi\dim(B)$.

Let $A$ be a self-injective algebra over a field and $X\in A\mbox{{\rm-mod}}$.
Then ${\rm End}_{A}(A\oplus X)$ and ${\rm End}_{A}(A\oplus\Omega_{A}(X))$ are
almost $\nu$-stable derived equivalent by [18, Corollary 3.14] (see also the
remark at the end of Section 3 in [16]), and therefore they are stably
equivalent of Morita type. Hence

$(\sharp)\quad{\rm del}\big{(}{\rm End}_{A}(A\oplus X)\big{)}={\rm
del}\big{(}{\rm End}_{A}(A\oplus\Omega_{A}(X))\big{)}.$

As $\nu_{A}$ is an auto-equivalence of $A\mbox{{\rm-mod}}$ and $D{\rm
Tr}(Y)\simeq\Omega^{2}(\nu_{A}(Y))$ in $A$-mod for $Y\in A\mbox{{\rm-mod}}$,
we have

$\begin{array}[]{rl}{\rm del}\big{(}{\rm End}_{A}(A\oplus X)\big{)}=&{\rm
del}\big{(}{\rm End}_{A}(\nu_{A}(A\oplus X))\big{)}={\rm del}\big{(}{\rm
End}_{A}(A\oplus\nu_{A}(X))\big{)}\\\ =&{\rm del}\big{(}{\rm
End}_{A}(A\oplus\Omega(\nu_{A}(X))\big{)}\;\;(\mbox{ by }(\sharp))\\\ =&{\rm
del}\big{(}{\rm End}_{A}(A\oplus\Omega^{2}(\nu_{A}(X))\big{)}\;\;(\mbox{ by
}(\sharp))\\\ =&{\rm del}\big{(}{\rm End}_{A}(A\oplus D{\rm
Tr}(X))\big{)}.\end{array}$

Note that the last equality is due to the fact that ${\rm
End}_{A}(A\oplus\Omega^{2}(\nu_{A}(X))$ and ${\rm End}_{A}(A\oplus D{\rm
Tr}(X))$ are Morita equivalent. Similarly, we can prove the equalities for
$\phi$-dimensions and $\psi$-dimensions by Lemma 3.9(2). $\square$

Proposition 3.4 can distinguish almost $\nu$-stable derived equivalences out
of derived equivalences. For instance, let $A$ and $B$ be algebras given by
the following quivers $Q_{A}$ and $Q_{B}$ with relations, respectively:

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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\alpha\delta\alpha=\gamma\delta=\delta\alpha-\beta\gamma=0\;;&&\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\alpha^{\prime}=\gamma^{\prime}\alpha^{\prime}\beta^{\prime}\gamma^{\prime}=0.\\\
\end{array}$

It was shown in [19, Example 4.10] that $A$ and $B$ are derived equivalent.
One can check that both algebras have no nodes and ${\rm del}(A)=2\neq 1={\rm
del}(B)$. Thus $A$ and $B$ are neither almost $\nu$-stable derived equivalent
by Proposition 3.4 nor stably equivalent by Theorem 1.1.

###  3.2 Proof of Theorem 1.3

Now we turn to the proof of Theorem 1.3. We first recall the notion of
$\nu$-dominant dimensions.

Let $A$ be an Artin algebra. We denote by $A$-prinj the full subcategory of
$A\mbox{{\rm-mod}}$ consisting of those $A$-modules that are both projective
and injective. For an $A$-module $M\in A\mbox{{\rm-mod}}$, we consider its
minimal injective resolution

$0\longrightarrow{}_{A}M\longrightarrow I_{0}\longrightarrow
I_{1}\longrightarrow I_{2}\longrightarrow\cdots.$

Let $I$ be an injective $A$-module and $0\leq n\leq\infty$. If $n$ is maximal
such that all modules $I_{j}$ are in ${\rm add}(I)$ for $j<n$, then $n$ is
called the _$I$ -dominant dimension_ of $M$, denoted by $I$-${\rm
dom.dim}(M)$. Dually, we consider its minimal projective resolution

$\cdots\longrightarrow P_{2}\longrightarrow P_{1}\longrightarrow
P_{0}\longrightarrow{}_{A}M\longrightarrow 0.$

Let $P$ be a projective $A$-module and $0\leq m\leq\infty$. If $m$ is maximal
such that all modules $P_{j}$ are in ${\rm add}(P)$ for $j<m$, then $m$ is
called the _$P$ -codominant dimension_ of $M$, denoted by $P$-${\rm
codom.dim}(M)$. Clearly, the codominant dimension of $M$ and the dominant
dimension of $A^{{}^{\rm op}}$-module $D(M)$ are equal. Now, if ${\rm
add}(I)={\rm add}(P)=A\mbox{{\rm-prinj}\,}$, then we define the _dominant
dimension_ of $M$ to be $I$-${\rm dom.dim}(M)$, denoted by ${\rm dom.dim}(M)$;
the _codominant dimension_ of $M$ to be $P$-${\rm codom.dim}(M)$, denoted by
${\rm codom.dim}(M)$, and the _dominant dimension_ of the algebra $A$ to be
${\rm dom.dim}(_{A}A)$, denoted by ${\rm dom.dim}(A)$. Note that ${\rm
dom.dim}(A)={\rm dom.dim}(A^{{}^{\rm op}})$ (see [27, Theorem 4] or [15]). It
is clear that ${\rm dom.dim}(A)=\min\\{{\rm dom.dim}(P)\mid P\in{\rm
add}(_{A}A)\\}$. If ${\rm add}(I)=A{\mbox{\rm-stp}}$, then $I$-${\rm
dom.dim}(M)$ is called the $\nu$-_dominant dimension_ of $M$, denoted by
$\nu\mbox{-}{\rm dom.dim}(M)$. The $\nu$-_dominant dimension of the algebra
$A$_ is defined to be $\nu\mbox{-}{\rm dom.dim}(_{A}A)$.

######  Lemma 3.10.

Let $A$ be an Artin algebra with $\nu\mbox{-}{\rm dom.dim}(A)\geq 1$. Then

$(1)$ $A{\mbox{\rm-stp}}=A\mbox{{\rm-prinj}\,}$ and $\nu\mbox{-}{\rm
dom.dim}(A)={\rm dom.dim}(A)$.

$(2)$ The projective cover of a simple module ${}_{A}S$ is injective if and
only if the injective envelope of $S$ is projective.

$(3)$ If the projective cover of a simple module ${}_{A}S$ is not injective,
then $S$ itself is neither projective nor injective.

Proof. (1) and (2) are trivial. We prove (3). Let $P$ and $I$ be the
projective cover and injective envelope of $S$, respectively. By assumption,
$P\not\in A\mbox{{\rm-prinj}\,}$. By (2), $I\not\in A\mbox{{\rm-prinj}\,}$. If
$S$ is injective, then it follows from ${\rm codom.dim}(D(A_{A}))={\rm
dom.dim}(A)=\nu\mbox{-}{\rm dom.dim}(A)\geq 1$ that ${\rm codom.dim}(S)\geq
1$. This implies $P\in A\mbox{{\rm-prinj}\,}$, a contradiction. Thus $S$ is
not an injective module. Suppose that $S$ is projective. It follows from ${\rm
dom.dim}(A)\geq 1$ that ${\rm dom.dim}(S)\geq 1$ and $I\in A\mbox{{\rm-
prinj}\,}$, again a contradiction. Thus $S$ is not a projective module.
$\square$

######  Lemma 3.11.

Let $F:A\mbox{{\rm-{\text@underline{mod}}}}\rightarrow
B\mbox{{\rm-{\text@underline{mod}}}}$ define a stable equivalence between
Artin algebras $A$ and $B$, and let $G$ be a quasi-inverse of $F$. If
$\nu\mbox{-}{\rm dom.dim}(A)\geq 1$ and $\nu\mbox{-}{\rm dom.dim}(B)\geq 1$,
then there exist bijections

$F:\mathscr{I}(A)_{\mathscr{P}}\longrightarrow\mathscr{I}(B)_{\mathscr{P}},\;F:\mathfrak{n}_{F}(A)\longrightarrow\mathfrak{n}_{G}(B),\;F^{\prime}:\mathscr{P}(A)_{\mathscr{I}}\longrightarrow\mathscr{P}(B)_{\mathscr{I}}\mbox{
and }F^{\prime}:\mathfrak{n}_{F}(A)\longrightarrow\mathfrak{n}_{G}(B).$

Proof. Suppose $I\in\mathscr{I}(A)_{\mathscr{P}}$, we show
$F(I)\in\mathscr{I}(B)_{\mathscr{P}}$. Indeed, let $S$ be the socle of $I$. By
Lemma 3.10(2)-(3), $S$ is not injective. Thus $S\not\simeq I$ and the natural
projection $\pi:I\rightarrow I/S$ is an irreducible map. Since $I$ is not a
projective module, we have $I/S\in A\mbox{{\rm-mod}}_{\mathscr{P}}$. Thus
$0\neq\pi\in A\mbox{{\rm-{\text@underline{mod}}}}$ and $0\neq F(\pi)\in
B\mbox{{\rm-{\text@underline{mod}}}}$. By [6, Chapter X, Proposition 1.3],
$F(\pi):F(I)\rightarrow F(I/S)$ is irreducible. By Lemma 2.2, we have
$F(I)\in\bigtriangledown_{A}$, namely $F(I)\in\mathfrak{n}_{G}(B)$ or
$F(I)\in\mathscr{I}(B)_{\mathscr{P}}$. Suppose $F(I)\in\mathfrak{n}_{G}(B)$.
Then $F(I)$ is a node and there is an almost split sequence $0\rightarrow
F(I)\rightarrow Q\rightarrow{\rm Tr\,}D(F(I))\rightarrow 0$ with ${}_{B}Q$
projective. Since $F(\pi)$ is irreducible and ${}_{B}F(I)$ is indecomposable,
by [6, Chapter V, Theorem 5.3], we get $F(I/S)\in{\rm add}(_{B}Q)$. Thus
$F(I/S)$ is a projective $B$-module and $F(\pi)=0$ in
$B\mbox{{\rm-{\text@underline{mod}}}}$. This is a contradiction and shows
$F(I)\in\mathscr{I}(B)_{\mathscr{P}}$.

Similarly, we show that $G(J)$ lies in $\mathscr{I}(A)_{\mathscr{P}}$ for
$J\in\mathscr{I}(B)_{\mathscr{P}}$. By Lemma 2.2,
$F:\mathscr{I}(A)_{\mathscr{P}}\rightarrow\mathscr{I}(B)_{\mathscr{P}}$ and
$F:\mathfrak{n}_{F}(A)\rightarrow\mathfrak{n}_{G}(B)$ are bijections.

Let $P\in\mathscr{P}(A)_{\mathscr{I}}$ with $S$ as its top. By Lemma 3.10(3),
$S$ is not projective. Now, Lemma 2.5 implies $F^{\prime}(P)\in B\mbox{{\rm-
proj}}$. Thus $F^{\prime}(P)\in\bigtriangleup_{A}$ by Lemma 2.2, and therefore
$F^{\prime}(P)\in\mathscr{P}(B)_{\mathscr{I}}$. Similarly,
$G^{\prime}(Q)\in\mathscr{P}(A)_{\mathscr{I}}$ for
$Q\in\mathscr{P}(B)_{\mathscr{I}}$. Thus Lemma 2.2 yields the bijections
$F^{\prime}:\mathscr{P}(A)_{\mathscr{I}}\rightarrow\mathscr{P}(B)_{\mathscr{I}}$
and $F^{\prime}:\mathfrak{n}_{F}(A)\rightarrow\mathfrak{n}_{G}(B)$. $\square$

Before starting the proof of Theorem 1.3, we introduce a few notation.

Suppose that $X$ is an $A$-module such that $A{\mbox{\rm-stp}}={\rm
add}(_{A}X)$, where $A{\mbox{\rm-stp}}$ stands for the full subcategory of
$A\mbox{{\rm-mod}}$ consisting of all $\nu$-stably projective $A$-modules. By
$\mbox{pre}(X)$ we denote the full subcategory of $A\mbox{{\rm-mod}}$
consisting of all those $A$-modules $M$ that have a minimal projective
presentation $P_{1}\rightarrow P_{0}\rightarrow M\rightarrow 0$ with
$P_{1},P_{2}\in{\rm add}(X)$. Let $\Lambda:={\rm End}_{A}(X).$ It follows from
[6, Chapter II, Proposition 2.5] that the functor ${\rm
Hom}_{A}(X,-):\mbox{pre}(X)\rightarrow\Lambda\mbox{{\rm-mod}}$ is an
equivalence of additive categories with a quasi-inverse
$X\otimes_{\Lambda}-:\Lambda\mbox{{\rm-mod}}\rightarrow\mbox{pre}(X)$. Let
$R_{X}(M,N)$ be the $k$-submodule of ${\rm Hom}_{A}(M,N)$ consisting of all
those homomorphisms of $A$-modules that factorize through a module in ${\rm
add}(X)$. Then $R_{X}$ is an ideal of the $k$-category pre$(X)$. Note that
$R_{X}(N,M)=R_{A}(N,M)$ for all $N\in A\mbox{{\rm-mod}}$ and
$M\in\mbox{pre}(X)$ because a homomorphism $f:N\rightarrow M$ of $A$-modules
factorizes through a projective $A$-module must factorize through the
projective cover $P_{0}\rightarrow M$. We denote by
$\underline{\mbox{pre}}(X)$ the quotient category of $\mbox{pre}(X)$ modulo
the ideal $R_{X}$. Thus $\underline{\mbox{pre}}(X)$ is a full additive
subcategory of $A$-mod.

Proof of Theorem 1.3. Suppose that
$F:A\mbox{{\rm-{\text@underline{mod}}}}\rightarrow
B\mbox{{\rm-{\text@underline{mod}}}}$ defines a stable equivalence between
Artin algebras $A$ and $B$, where both algebras have positive $\nu$-dominant
dimensions. Then $A{\mbox{\rm-stp}}=A\mbox{{\rm-prinj}\,}$ and $B{\mbox{\rm-
stp}}=B\mbox{{\rm-prinj}\,}$ by Lemma 3.10(1). Let ${}_{A}X$ and ${}_{B}Y$ be
modules such that $A{\mbox{\rm-stp}}={\rm add}(_{A}X)$ and $B{\mbox{\rm-
stp}}={\rm add}(_{B}Y)$, respectively. We define $\Lambda:={\rm End}(X)$ and
$\Gamma:={\rm End}_{B}(Y)$. Then $\Lambda$ and $\Gamma$ are the Frobenius
parts of $A$ and $B$, respectively. Moreover, $\Lambda$ and $\Gamma$ are self-
injective algebras

To show that $\Lambda$ and $\Gamma$ are stably equivalent, it is enough to
show that $F$ induces an equivalence from $\underline{\mbox{pre}}(X)$ to
$\underline{\mbox{pre}}(Y)$. Since $F$ is an equivalence, we need only to show
that $F(M)$ lies in $\mbox{pre}(Y)$ for all $M\in\mbox{pre}(X)$.

In fact, take $M\in\mbox{pre}(X)$ and a minimal projective presentation:
$P_{1}\rightarrow P_{0}\rightarrow M\rightarrow 0$ with $P_{1},P_{0}\in{\rm
add}(_{A}X)$. We may assume that $M$ has no nonzero projective direct
summands. Then the exact sequence

$0\longrightarrow\Omega_{A}(M)\longrightarrow P_{0}\longrightarrow
M\longrightarrow 0$

is minimal and $\Omega_{A}(M)$ does not have any injective direct summands,
that is, $\Omega_{A}(M)\in A\mbox{{\rm-mod}}_{\mathscr{I}}$. Since $P_{1}$ is
a projective-injective $A$-module, we have $\Omega_{A}(M)\in A\mbox{{\rm-
mod}}_{\mathscr{P}}$. So we write $\Omega_{A}(M)\simeq K_{1}\oplus K_{2}$ with
$K_{1}\in{\rm add}(\bigtriangleup_{A}^{c})$ and $K_{2}\in{\rm
add}(\mathfrak{n}_{F}(A))$. By Corollary 2.6, we have a minimal exact sequence

$(*)\quad 0\longrightarrow F(K_{1})\oplus F^{\prime}(K_{2})\longrightarrow
Q_{0}\longrightarrow F(M)\longrightarrow 0$

in $B\mbox{{\rm-mod}}$ with $Q_{0}\in B\mbox{{\rm-prinj}\,}={\rm add}(Y)$.

Next, we investigate the projective covers of $F(K_{1})$ and
$F^{\prime}(K_{2})$. Let $P^{\prime}_{1}$ be the projective cover of $K_{1}$.
Then $P^{\prime}_{1}\in{\rm add}(_{A}P_{1})$ and $P^{\prime}_{1}\in
A\mbox{{\rm-prinj}\,}$. Note that $\Omega_{A}(K_{1})$ lies in $A\mbox{{\rm-
mod}}_{\mathscr{I}}$ and we can write $\Omega_{A}(K_{1})=L_{1}\oplus L_{2}$
with $L_{1}\in{\rm add}(\bigtriangleup_{A}^{c})$ and $L_{2}\in{\rm
add}(\Delta_{A})$. Applying Corollary 2.6 to the minimal exact sequence

$0\longrightarrow\Omega_{A}(K_{1})\longrightarrow
P^{\prime}_{1}\longrightarrow K_{1}\longrightarrow 0$

in $A\mbox{{\rm-mod}}$, we get a minimal exact sequence of $B$-modules

$0\longrightarrow F(K_{1}))\oplus F^{\prime}(L_{2})\longrightarrow
Q^{\prime}_{1}\longrightarrow F(K_{1})\longrightarrow 0$

with $Q^{\prime}_{1}\in B\mbox{{\rm-prinj}\,}$.

Now, we investigate the projective cover of $F^{\prime}(K_{2})$. By Lemma
3.11, it follows from $K_{2}\in{\rm add}(\mathfrak{n}_{F}(A))$ that
$F^{\prime}(K_{2})\in{\rm add}(\mathfrak{n}_{G}(B))$, where $G$ is the quasi-
inverse of the functor $F$. From the sequence $(*)$ and $Q_{0}\in B\mbox{{\rm-
prinj}\,}$, we infer that the injective envelope of $F^{\prime}(K_{2})$ is
projective. Since nodes are simple modules, it follows from Lemma 3.10(2) that
the projective cover ${}_{B}Q_{2}^{\prime}$ of $F^{\prime}(K_{2})$ is
projective-injective. Thus the minimal projective presentation of $F(M)$ is as
follows.

$Q_{1}^{\prime}\oplus Q_{2}^{\prime}\longrightarrow Q_{0}\longrightarrow
F(M)\longrightarrow 0$

with $Q_{1}^{\prime},Q_{2}^{\prime},Q_{0}\in B\mbox{{\rm-prinj}\,}$. This
yields $F(M)\in\mbox{pre}(Y)$. Similarly, we prove that the quasi-inverse $G$
of $F$ sends $N\in\mbox{pre}(Y)$ to $G(N)\in\mbox{pre}(X)$.

Finally, we reach to the commutative diagram of functors in stable module
categories:

$\textstyle{\underline{\mbox{pre}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F.}$$\textstyle{\underline{\mbox{pre}}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G.}$$\textstyle{A\mbox{{\rm-{\text@underline{mod}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{B\mbox{{\rm-{\text@underline{mod}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$

where $F.$ and $G.$ stand for the restrictions of $F$ and $G$ to
$\underline{\mbox{pre}}(X)$ and $\underline{\mbox{pre}}(Y)$ , respectively. It
follows from the equivalence of $F$ that $F.$ is an equivalence of
$k$-categories. $\square$

Let us now present an example which illustrates Theorem 1.3 and shows that the
assumption of $\nu$-dominant dimensions in Theorem 1.3 cannot be dropped.

######  Example 3.12.

Let $A$ and $B$ be algebras given by the quivers with relations:

$\begin{array}[]{ccc}A:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.1111pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.87215pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.1111pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.1111pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet
2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
42.2222pt\raise-18.22221pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\beta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
42.2222pt\raise-26.99998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{4\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-18.22221pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\delta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.1111pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet
3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.29904pt\raise-31.25693pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\gamma}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
9.1111pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\\\
\quad\beta\gamma=\delta\alpha=0,\end{array}\qquad\begin{array}[]{ccc}B:\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
8.77pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1^{\prime}\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.98106pt\raise 7.21527pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.0625pt\hbox{$\scriptstyle{\alpha^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.77pt\raise 2.15277pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.77pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet
2^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad
3^{\prime}\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.24034pt\raise-8.81943pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30556pt\hbox{$\scriptstyle{\beta^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
8.77002pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
80.71017pt\raise 7.89583pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.38196pt\hbox{$\scriptstyle{\gamma^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 98.07222pt\raise
2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
98.07222pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet
4^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
80.96669pt\raise-8.13887pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.98611pt\hbox{$\scriptstyle{\delta^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
74.07224pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\\\
\quad\beta^{\prime}\alpha^{\prime}=\delta^{\prime}\gamma^{\prime}=0.\\\
\vspace{0.85cm}\hfil\end{array}$

We denote by $P(i)$ and $I(i)$ the indecomposable projective and injective
modules corresponding to the vertex $i$, respectively. The indecomposable
projective $A$-modules and $B$-modules are displayed, respectively.

$\textstyle{P(1)}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3}$
$\textstyle{P(2)}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3}$
$\textstyle{P(3)}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$
$\textstyle{P(4)}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$
$\textstyle{P(1^{\prime})}$$\textstyle{1^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1^{\prime}}$
$\textstyle{P(2^{\prime})}$$\textstyle{2^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1^{\prime}}$
$\textstyle{P(3^{\prime})}$$\textstyle{3^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3^{\prime}}$
$\textstyle{P(4^{\prime})}$$\textstyle{4^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3^{\prime}}$

The indecomposable injective $A$-modules and $B$-modules are given as follows.

$\textstyle{I(1)}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$
$\textstyle{I(2)}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2}$
$\textstyle{I(3)}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3}$
$\textstyle{I(4)}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4}$
$\textstyle{I(1^{\prime})}$$\textstyle{1^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1^{\prime}}$
$\textstyle{I(2^{\prime})}$$\textstyle{1^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2^{\prime}}$
$\textstyle{I(3^{\prime})}$$\textstyle{3^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3^{\prime}}$
$\textstyle{I(4^{\prime})}$$\textstyle{3^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4^{\prime}}$

Then $A{\mbox{\rm-stp}}={\rm add}\big{(}P(1)\oplus P(3)\big{)}$, $B{\mbox{\rm-
stp}}={\rm add}\big{(}P(1^{\prime})\oplus P(3^{\prime})\big{)}$ and
$\nu\mbox{-}{\rm dom.dim}(A)=\nu\mbox{-}{\rm dom.dim}(B)=2$. The Frobenius
parts $\Lambda$ and $\Gamma$ of $A$ and $B$ are given by the quivers with
relations, respectively.

$\begin{array}[]{ccc}\Lambda:\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.1111pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.87215pt\raise 6.65971pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.1111pt\raise
2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.1111pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet
3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.29904pt\raise-7.34026pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\gamma}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 9.1111pt\raise-2.15277pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces&&\Gamma:\quad\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
8.77pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1^{\prime}\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{{}{{}{{}{{}}{}{{}{{}{{}{{}{{}}{}{{}}{}{{}}{}{{}}}}}}}}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}}\ignorespaces\ignorespaces{\hbox{\kern
19.0919pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.0625pt\hbox{$\scriptstyle{\alpha^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{}{{}{{}}{}{{}}{}{{}}{}{{}}{}{{}{{}}{}{{}{{}{{}{{}}{}{{}}{}{{}}}}}}}{\hbox{\kern
3.2935pt\raise-2.99599pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces\quad\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
8.77pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{3^{\prime}\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{{}{{}{{}{{}}{}{{}{{}{{}{{}{{}}{}{{}}{}{{}}{}{{}}}}}}}}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}}\ignorespaces\ignorespaces{\hbox{\kern
19.0919pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.38196pt\hbox{$\scriptstyle{\gamma^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{}{{}{{}}{}{{}}{}{{}}{}{{}}{}{{}{{}}{}{{}{{}{{}{{}}{}{{}}{}{{}}}}}}}{\hbox{\kern
3.2935pt\raise-2.99599pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces\\\
\qquad\quad\alpha\gamma=\gamma\alpha=0,&&\qquad\alpha^{\prime
2}=\gamma^{\prime 2}=0.\end{array}$

It follows from[24, Theorem 2.10] (see Lemma 3.13(2) below) that both $A$ and
$B$ are stably equivalent to the path algebra $C$ of the quiver

$C:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.1111pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
26.18292pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
26.18292pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
26.93292pt\raise 5.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{2}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 54.25475pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
54.25475pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet
5}$}}}}}}}\ignorespaces}}}}\ignorespaces\quad\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
9.1111pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{3\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
26.18292pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
26.18292pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
26.93292pt\raise 5.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{4}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 54.25475pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
54.25475pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
54.25475pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet
6}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Thus $A$ and $B$ are stably equivalent, and so are $\Lambda$ and $\Gamma$ by
Theorem 1.3. Now, we consider the stably equivalent algebras $A$ and $C$.
Clearly, $\nu\mbox{-}{\rm dom.dim}(C)=0$ and the Frobenius part of $C$ is $0$.
Thus the Frobenius part $\Lambda$ of $A$ is not stably equivalent to the
Frobenius part of $C$. This shows that the assumption on $\nu$-dominant
dimensions on Artin algebras in Theorem 1.3 cannot be omitted. Observe that
${\rm dom.dim}(C)=1$. This shows that the $\nu$-dominant dimensions in Theorem
1.3 cannot be weakened to dominant dimensions either.

###  3.3 Proof of Theorem 1.4

This section is devoted to the proof of Theorem 1.4.

In [24, Theorem 2.10], Martínez-Villa showed that any Artin algebra with nodes
is stably equivalent to an Artin algebra without nodes. The process of
removing nodes runs precisely as follows. Suppose that $A$ is an Artin algebra
with nodes. Let $\\{S(1),S(2),\cdots,S(n)\\}$ be a complete set of non-
isomorphic simple $A$-modules. Suppose that $P(i)=Ae_{i}$ has the top $S(i)$
with $e_{i}^{2}=e_{i}\in A$ for $1\leq i\leq n$. We may assume that
$\\{S(1),\cdots,S(m)\\}$ is a complete set of nodes of $A$ with $m\leq n$. Set
$S:=\bigoplus_{i=1}^{m}S(i)$. Let $I$ be the trace of $S$ in $A$. Then
$I\in{\rm add}(_{A}S)$. By the definition of nodes, $S\in{\rm add}({\rm
soc}(_{A}A))$ and $S\in{\rm add}(_{A}I)$. Thus ${\rm add}(_{A}I)={\rm
add}(_{A}S)$. Clearly, $I^{2}=0$ and ${\rm rad}(A)I=0$. Let
$J:=$ann${}_{l}(I)$ be the left annihilator of $I$. Then ${\rm
rad}(A)\subseteq J$ and $A/J$ is semisimple. Since ${}_{A}I$ has only
composition factors $S(i)$ for $1\leq i\leq m$, we have $e_{i}I\neq 0$ for
$1\leq i\leq m$ and $e_{j}I=0$ for $m+1\leq j\leq n$. This yields ${\rm
add}(_{A}A/J)={\rm add}(_{A}S)$. Note that $I$ is a two-sided ideal of $A$ and
$JI=0$. Thus $I$ is an $(A/J)$-$(A/I)$-bimodule. Let $A^{\prime}$ be the
triangular matrix algebra

$A^{\prime}:=\begin{pmatrix}A/I&0\\\ I&A/J\end{pmatrix}.$

The following lemma describes some common properties of $A$ and $A^{\prime}$.

######  Lemma 3.13.

$(1)$ The triangular matrix Artin algebra $A^{\prime}$ has no nodes.

$(2)$ $A$ and $A^{\prime}$ are stably equivalent.

$(3)$ $A$ and $A^{\prime}$ have the same numbers of non-isomorphic, non-
projective simples.

$(4)$ If $A$ is Frobenius-finite, then so is $A^{\prime}$.

Proof. The first two statements are taken from [24, Theorem 2.10], while we
prove (3) and (4).

(3) As is known, $A^{\prime}$-modules can be identified with triples
$(X,Y,f)$, where $X$ is an $A/I$-module, $Y$ is an $A/J$-module and
$f:I\otimes_{A/I}X\rightarrow Y$ is a homomorphism of $A/J$-modules. It
follows from $I^{2}=0$ that $I\subseteq{\rm rad}(A)$. Thus simple $A$-modules
coincide with simple $A/I$-modules, and therefore $A$ and $A/I$ have the same
number of non-isomorphic simple modules. Note that the projective cover of an
$A/I$-module $X$ is of the form $P/IP$ with $P$ being a projective cover of
the $A$-module ${}_{A}X$. Obviously, the simple $A^{\prime}$-modules are
either of the form $(T,0,0)$, where $T$ is a simple $A$-module, or of the form
$(0,T^{\prime},0)$, where $T^{\prime}$ is a simple $A/J$-module. The
indecomposable projective $A^{\prime}$-modules are either of the form
$\tilde{P}:=(P/IP,I\otimes_{A/I}P/IP,{\rm id})$ with $P$ an indecomposable
projective $A$-module or of the form $(0,T^{\prime},0)$ with $T^{\prime}$ an
indecomposable projective $A/J$-module. Thus $(0,T^{\prime},0)$ is a
projective simple $A^{\prime}$-module, and so the indecomposable non-
projective simple $A^{\prime}$-modules are of the form $(T,0,0)$, where $T$ is
a simple $A$-module.

We prove that $A$ and $A^{\prime}$ have the same number of non-isomorphic,
non-projective simples. Indeed, take a simple $A$-module $T$, then $IT=0$ and
$(T,0,0)$ is a simple $A^{\prime}$-module. If $T$ is a projective $A$-module,
then $T$ is also a projective $A/I$-module, and therefore $(T,0,0)$ is a
projective $A^{\prime}$-module. Thus $(T,0,0)$ is a projective simple
$A^{\prime}$-module. Suppose that $T$ is not a projective $A$-module. Let
$P(T)$ be a projective cover of ${}_{A}T$. Then $P(T)\not\simeq T$. If
$IP(T)=0$, then $P(T)$ is a projective cover of ${}_{A/I}T$. Thus $(P(T),0,0)$
is a projective cover of $(T,0,0)$ and $(T,0,0)$ is not a projective
$A^{\prime}$-module. If $IP(T)\neq 0$, then it follows that
$I\otimes_{A/I}\big{(}P(T)/IP(T)\big{)}\simeq
I\otimes_{A}\big{(}P(T)/IP(T)\big{)}\simeq
I\otimes_{A}(A/I)\otimes_{A}P(T)\simeq I\otimes_{A}P(T)\simeq IP(T)\neq 0$.
Thus the $A^{\prime}$-module $\big{(}P(T)/IP(T),I\otimes_{A/I}P(T)/IP(T),{\rm
id}\big{)}$ is a projective cover of $(T,0,0)$. This implies that $(T,0,0)$ is
not projective. Thus $A$ and $A^{\prime}$ have the same numbers of non-
isomorphic, non-projective simples.

(4) Suppose that $(_{A/I}X,_{A/J}Y,f)$ is an indecomposable
$A^{\prime}$-module in $A^{\prime}{\mbox{\rm-stp}}$. We show that
${}_{A/I}X\in A/I{\mbox{\rm-stp}}$ and ${}_{A/J}Y\in A/J{\mbox{\rm-stp}}$.
Indeed, $(_{A/I}X,_{A/J}Y,f)$ is projective-injective with
$\nu_{A^{\prime}}(_{A/I}X,_{A/J}Y,f)\in A^{\prime}{\mbox{\rm-stp}}$. It
follows from [6, Proposition 2.5, p.76] that there are two possibilities:

(a) ${}_{A/J}Y=0$ and ${}_{A/I}X$ is an indecomposable projective-injective
$A/I$-module with $I\otimes_{A/I}X=0$;

(b) ${}_{A/I}X=0$ and ${}_{A/J}Y$ is an indecomposable projective-injective
$A/J$-module with ${\rm Hom}_{A/J}(I,Y)=0$.

Suppose (a) holds. Let $T_{0}:={\rm top}(_{A/I}X)$. Then $\nu_{A/I}(X)$ is an
injective envelope of ${}_{A/I}T_{0}$. By [6, Proposition 2.5, p.76], we see
that ${}_{A^{\prime}}(T_{0},0,0)$ is a simple $A^{\prime}$-module, that
${}_{A^{\prime}}(X,0,0)$ is a projective cover of
${}_{A^{\prime}}(T_{0},0,0)$, and that ${}_{A^{\prime}}(\nu_{A/I}(X),0,0)$ is
an injective envelope of ${}_{A^{\prime}}(T_{0},0,0)$. Thus
$\nu_{A^{\prime}}(_{A/I}X,0,0)\simeq(\nu_{A/I}(X),0,0)\in
A^{\prime}{\mbox{\rm-stp}}$. This implies that $\nu^{i}_{A/I}(X)$ is
projective-injective for all $i\geq 0$, and $X\in A/I{\mbox{\rm-stp}}$.
Similarly, if (b) holds, then $Y\in A/J{\mbox{\rm-stp}}$.

Let $S$ be the direct sum of all non-isomorphic nodes of $A$. Since $I$ is the
trace of $S$ in $A$ and $J$ is the left annihilator of $I$, we have ${\rm
add}(_{A}I)={\rm add}(_{A}S)={\rm add}(_{A}A/J)$. Thus ${\rm
Hom}_{A}(I,Z)={\rm Hom}_{A/J}(I,Z)\neq 0$ for any $A/J$-module $Z\neq 0$.
Hence we can assume that $\\{(X_{1},0,0),\cdots(X_{r},0,0)\\}$ is a complete
set of all non-isomorphic indecomposable modules in $A^{\prime}{\mbox{\rm-
stp}}$ for some natural number $r$ and $X_{i}\in A/I{\mbox{\rm-stp}}$ with
$I\otimes_{A/I}X_{i}=0$ for $1\leq i\leq r$. Since each indecomposable
projective $A/I$-module is of the form $P/IP$ for some indecomposable
projective $A$-module $P$, we assume $X_{i}\simeq P_{i}/IP_{i}$ for some
indecomposable projective $A$-module $P_{i}$. Note that
$I\otimes_{A/I}\big{(}Q/IQ\big{)}\simeq I\otimes_{A}\big{(}Q/IQ\big{)}\simeq
I\otimes_{A}(A/I)\otimes_{A}Q\simeq I\otimes_{A}Q\simeq IQ$ for each
projective $A$-module $Q$. It follows from $I\otimes_{A/I}X_{i}=0$ that
$IP_{i}=0$, and therefore $X_{i}\simeq P_{i}$ as $A/I$-modules and ${\rm
soc}(_{A}P_{i})$ is a simple $A$-module. Since $P_{i}$ is a projective
$A$-module, the trace of $S$ in $P_{i}$ is equal to $IP_{i}$. It then follows
from $IP_{i}=0$ that ${\rm soc}(_{A}P_{i})$ has no nodes as its direct
summands for $1\leq i\leq r$. Set $U:=\bigoplus_{i=1}^{r}P_{i}$. Then ${\rm
soc}(_{A}U)$ has no nodes as its direct summands. Since
$\nu_{A^{\prime}}(U,0,0)$ is $\nu$-stably projective and
$\nu_{A^{\prime}}(U,0,0)\simeq(\nu_{A/I}(U),0,0)$, there hold
$U\simeq\nu_{A/I}(U)$ and ${\rm top}(_{A/I}U)\simeq{\rm soc}(_{A/I}U)$ as
$A/I$-modules. Thus ${\rm top}(_{A}U)$ is isomorphic to ${\rm soc}(_{A}U)$ and
has no nodes as its direct summands. In particular, ${\rm Hom}_{A}(U,I)=0$.
Applying ${\rm Hom}_{A}(U,-)$ to the exact sequence

$0\longrightarrow I\longrightarrow A\longrightarrow A/I\longrightarrow 0$

of $A$-$A$-bimodules, we get the exact sequence of $A^{{}^{\rm op}}$-modules

$0\longrightarrow{\rm Hom}_{A}(U,I)\longrightarrow{\rm
Hom}_{A}(U,A)\longrightarrow{\rm Hom}_{A}(U,A/I)\longrightarrow 0.$

It follows from ${\rm Hom}_{A}(U,I)=0$ that ${\rm Hom}_{A}(U,A)\simeq{\rm
Hom}_{A}(U,A/I)$. Clearly, ${\rm Hom}_{A}(U,A/I)={\rm Hom}_{A/I}(U,A/I)$ as
$A^{{}^{\rm op}}$-modules. Thus $D{\rm Hom}_{A}(U,A)\simeq D{\rm
Hom}_{A/I}(U,A/I)$ as $A$-modules. As $\nu_{A/I}(U)=D{\rm
Hom}_{A/I}(U,A/I)\simeq U$ as $A$-modules and $D{\rm Hom}_{A}(U,A)\simeq U$ as
$A$-modules, we get $U\in A{\mbox{\rm-stp}}$. Let $\Lambda$ be the Frobenius
part of $A$, and let $\Lambda^{\prime}$ be the Frobenius part of $A^{\prime}$.
Then ${\rm End}_{A}(U)$ is of the form $f\Lambda f$ for an idempotent
$f\in\Lambda$, and

$\Lambda^{\prime}:={\rm End}_{A^{\prime}}((U,0,0))\simeq{\rm
End}_{A/I}(U)\simeq{\rm End}_{A}(U).$

If $A$ is Frobenius-finite, then ${\rm End}_{A}(U)$ is representation-finite,
and therefore $A^{\prime}$ is Frobenius-finite. $\square$

######  Lemma 3.14.

Suppose that $A$ and $B$ are stably equivalent Artin algebras both of which
have no nodes and no semisimple direct summands. Let $\Lambda$ and $\Gamma$ be
the Frobenius parts of $A$ and $B$, respectively. Then

$(1)$ $\Lambda$ and $\Gamma$ are stably equivalent and have no semisimple
direct summands.

$(2)$ If $\Lambda$ and $\Gamma$ have the same number of non-isomorphic, non-
projective simples, then so do $A$ and $B$.

Proof. Let $X$ (respectively, $Y$) be an $A$-module (respectively, a
$B$-module) such that $A{\mbox{\rm-stp}}={\rm add}(_{A}X)$ (respectively,
$B{\mbox{\rm-stp}}={\rm add}(_{B}Y)$). Define $\Lambda:={\rm End}(X)$ and
$\Gamma:={\rm End}_{B}(Y)$. Then $\Lambda$ and $\Gamma$ are the Frobenius
parts of $A$ and $B$, respectively. Note that they are self-injective Artin
algebras.

(1) It follows from [26, Lemma 2.5] and the definition of Frobenius parts.

(2) Now, we show that $A$ and $B$ have the same number of non-isomorphic, non-
projective simples. Indeed, it follows from [26, Lemma 2.5] which holds true
also for Artin algebras, that $A$ and $B$ have the same number of non-
isomorphic, non-projective, simple modules whose projective covers are not
$\nu$-stably projective. It remains to show that $A$ and $B$ have the same
number of non-isomorphic, non-projective, simple modules whose projective
covers are $\nu$-stably projective. For a simple module whose projective cover
is $\nu$-stably projective, if it is projective, then it is projective-
injective. However, $A$ and $B$ have no projective-injective simple modules.
Thus we have to show that $A$ and $B$ have the same number of non-isomorphic
simple modules whose projective covers are $\nu$-stably projective. As
$A{\mbox{\rm-stp}}={\rm add}(_{A}X)$ and $B{\mbox{\rm-stp}}={\rm add}(_{B}Y)$,
we need to show that $\Lambda$ and $\Gamma$ have the same number of non-
isomorphic simples. Note that $\Lambda$ and $\Gamma$ do not have projective
simple modules by (1). By assumption, $\Lambda$ and $\Gamma$ have the same
number of non-isomorphic simples. Thus $A$ and $B$ have the same numbers of
non-isomorphic, non-projective simples. $\square$

Proof of Theorem 1.4. (1) Assume that $A$ and $B$ are Artin $k$-algebras.
Given a stable equivalence between $A$ and $B$, we get a stable equivalence
between $A^{\prime}$ and $B^{\prime}$ both of which have no nodes. We may
further assume that $A^{\prime}$ and $B^{\prime}$ have no semisimple direct
summands. By Lemma 3.14(1) or [26, Theorem 2.6], every stable equivalence of
Artin $k$-algebras with no nodes and no semisimple summands induces a stable
equivalence of their Frobenius parts. Let $\Lambda^{\prime}$ and
$\Gamma^{\prime}$ be the Frobenius parts of $A^{\prime}$ and $B^{\prime}$,
respectively. Then $\Lambda^{\prime}$ and $\Gamma^{\prime}$ are stably
equivalent.

Now, assume that $k$ is an algebraically closed field and that $A$ is
Frobenius-finite. Then $A^{\prime}$ is Frobenius-finite by Lemma 3.13(4), that
is, $\Lambda^{\prime}$ is representation-finite and therefore
$\Gamma^{\prime}$ is representation-finite. Since Auslander-Reiten conjecture
holds true for a stable equivalence between representation-finite $k$-algebras
over an algebraically closed field $k$ (see [25, Theorem 3.4]),
$\Lambda^{\prime}$ and $\Gamma^{\prime}$ have the same number of non-
isomorphic, non-projective simple modules. By Lemma 3.14(2), $A^{\prime}$ and
$B^{\prime}$ have the same number of non-isomorphic non-projective simple
modules, and therefore $A$ and $B$ have the same number of non-isomorphic non-
projective simple modules by Lemma 3.13(3).

(2) This follows from (1) and the fact that principal centralizer matrix
algebras over a field are Frobenius-finite by [32, Theorem 1.1]. $\square$

Recall that a finite-dimensional $k$-algebra $A$ over a field $k$ is called a
Morita algebra if $A$ is isomorphic to ${\rm End}_{H}(H\oplus M)$ for $H$ a
finite-dimensional self-injective $k$-algebra and $M$ a finitely generated
$H$-module. In case of $H$ being a symmetric algebra, the Marita algebra $A$
is then called a gendo-symmetric algebra (see [21] and [13], respectively). By
[12, Proposition 2.9], $A$ is a Morita algebra if and only if $\nu\mbox{-}{\rm
dom.dim}(A)\geq 2$.

The following result is an immediate consequence of Theorems 1.3 and 1.4(1).
Here we do not assume that algebras considered have no nodes.

######  Corollary 3.15.

Every stable equivalence of Morita $k$-algebras over a field $k$ induces a
stable equivalence of their Frobenius parts. In particular, if $A$ and $B$ are
stably equivalent Morita algebras over an algebraically closed field and if
one of $A$ and $B$ is Frobenius-finite, then $A$ and $B$ have the same number
of non-isomorphic, non-projective simples.

Finally, we suggest the following conjecture, though derived equivalences do
not have to preserve the delooping levels of algebras in general.

Conjecture. If $A$ and $B$ are derived equivalent noetherian rings, then ${\rm
del}(A)<\infty$ if and only if ${\rm del}(B)<\infty.$

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Changchang Xi, School of Mathematical Sciences, Capital Normal University,
100048 Beijing, P. R. China; and School of Mathematics and Statistics, Shaanxi
Normal University, 710119 Xi’an, P. R. China

Email<EMAIL_ADDRESS>(C.C.Xi)

Jinbi Zhang, School of Mathematical Sciences, Peking University, 100871
Beijing, P. R. China

Email<EMAIL_ADDRESS>(J.B.Zhang)