Mirror-reflective algebras and Tachikawa’s second conjecture
In memory of Professor Hiroyuki Tachikawa (1930-2022)

Hongxing Chen, Ming Fang and Changchang Xi∗

†† ∗ Corresponding author. Email<EMAIL_ADDRESS>Fax: 0086 10
68903637.††2010 Mathematics Subject Classification: Primary 16E35, 18E30,
19D50; Secondary 16S10, 13B30, 20E06.††Keywords: Auslander algebra; Gendo-
symmetric algebra; Mirror-reflective algebra; Recollement; Symmetric algebra;
Tachikawa’s second conjecture.

###### Abstract

Given an algebra with an idempotent, we introduce two procedures to construct
families of new algebras, termed mirror-reflective algebras and reduced
mirror-reflective algebras. We then establish connections among these algebras
by recollements of derived module categories. In case of given algebras being
gendo-symmetric, we show that the (reduced) mirror-reflective algebras are
symmetric and provide new methods to construct systematically both higher
dimensional (minimal) Auslander-Gorenstein algebras and gendo-symmetric
algebras of higher dominant dimensions. This leads to a new formulation of
Tachikawa’s second conjecture for symmetric algebras in terms of idempotent
stratifications.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Standard recollements and stratified dimensions
    2. 2.2 Dominant dimensions and gendo-symmetric algebras
  3. 3 Mirror-reflective algebras
    1. 3.1 Mirror-reflective algebras and their derived recollements
    2. 3.2 Examples of mirror-reflective algebras: quivers with relations
  4. 4 Mirror-reflective algebras and gendo-symmetric algebras
    1. 4.1 Relations among mirror-reflective, symmetric and gendo-symmetric algebras
    2. 4.2 Mirror-reflective algebras and Auslander-Gorenstein algebras
    3. 4.3 Recollements of mirror-reflective algebras and Tachikawa’s second conjecture

##  1 Introduction

In the representation theory of algebras, the long-standing and not yet solved
Nakayama conjecture says that a finite-dimensional algebra with infinite
dominant dimension is self-injective [23]. This conjecture is related to the
so-called Tachikawa’s second conjecture [25]:

(TC2) Let $\Lambda$ be a finite-dimensional self-injective algebra and $M$ a
finitely generated $\Lambda$-module. Then $M$ is projective if it is
orthogonal, that is, ${\rm Ext}_{\Lambda}^{n}(M,M)=0$ for all $n\geq 1$.

By Müller’s characterization of dominant dimension in [22], (TC2) holds for a
self-injective algebra $\Lambda$ if and only if the Nakayama conjecture holds
for all endomorphism algebras ${\rm End}_{\Lambda}(\Lambda\oplus M)$ of
finitely generated generators over $\Lambda$. This suggests to consider the
algebras $A$ of the form ${\rm End}_{\Lambda}(\Lambda\oplus M)$ with $\Lambda$
a self-injective algebra and $M$ an arbitrary $\Lambda$-module. Such algebras
are called _Morita algebras_ [21]. In case, $\Lambda$ is symmetric, they are
called _gendo-symmetric algebras_ [14]. In [9], orthogonal generators over a
self-injective Artin algebra have been discussed systematically from the
viewpoint of recollements of (relative) stable module categories. In
particular, it is shown that the Nakayama conjecture holds true for
Gorenstein-Morita algebras [9, Corollary 1.4].

In the present paper, we mainly focus on developing general methods to
construct gendo-symmetric algebras. First, for an arbitrary algebra with an
arbitrary idempotent, we introduce two inductive procedures to construct
families of new algebras, called mirror-reflective algebras and reduced
mirror-reflective algebras. In the case of gendo-symmetric algebras, the
procedures produce systematically both higher dimensional (minimal) Auslander-
Gorenstein algebras and gendo-symmetric algebras of higher dominant
dimensions. Second, we show that these families of mirror-reflective algebras
are connected by recollements of their derived module categories. Moreover,
these recollements are standard in the sense that they are induced from strong
idempotent ideals (also called stratifying ideals) of algebras (see [10, 11,
2] for definitions). Finally, we give a new formulation of Tachikawa’s second
conjecture for symmetric algebras in terms of idempotent stratifications.

To state our results more precisely, we first introduce a few terminologies.

Let $A$ be an associative algebra over a commutative ring $k$, $e$ an
idempotent of $A$, and $\Lambda:=eAe$. For $\lambda\in Z(\Lambda)$, the center
of the algebra $\Lambda$, we introduce an associative algebra
$R(A,e,\lambda)$, called the _mirror-reflective algebra_ of $A$ at level
$(e,\lambda)$, which has the underlying $k$-module $A\oplus
Ae\otimes_{\Lambda}eA$, such that $Ae\otimes_{\Lambda}eA$ is an ideal in
$R(A,e,\lambda)$ (see Section 3.1 for details). The terminology “mirror-
reflective” can be justified by Example 3.11 in Section 3.2. Moreover, the
$k$-submodule of $R(A,e,\lambda)$

$S(A,e,\lambda):=(1-e)A(1-e)\oplus Ae\otimes_{\Lambda}eA$

is also an associative algebra with the multiplication induced from the one of
$R(A,e,\lambda)$. This algebra is called the _reduced mirror-reflective
algebra_ of $A$ at level $(e,\lambda)$. It has less simple modules than
$R(A,e,\lambda)$ does. The specialization of $R(A,e,\lambda)$ and
$S(A,e,\lambda)$ at $\lambda=e$ are called the mirror-reflective algebra and
reduced mirror-reflective algebra of $A$ at $e$, denoted as $R(A,e)$ and
$S(A,e)$, respectively. Moreover, $S(A,e)=e_{0}R(A,e)e_{0}$ for an idempotent
$e_{0}$ in $R(A,e)$.

Clearly, each $A$-module is an $R(A,e)$-module via the canonical surjective
homomorphism $R(A,e)\to A$ of algebras. Conversely, each $R(A,e)$-module
restricts to $A$-module via the canonical inclusion from $A$ into $R(A,e)$.
Remark that each module over $(1-e)A(1-e)$ can also be regarded as a module
over $S(A,e)$. So we can define two endomorphism algebras associated with
$(A,e)$:

$\mathcal{A}(A,e):={\rm End}_{R(A,e)}\big{(}R(A,e)\oplus
A(1-e)\big{)},\;\;\mathcal{B}(A,e):={\rm
End}_{S(A,e)}\big{(}S(A,e)\oplus(1-e)A(1-e)\big{)}.$

Now, assume that all algebras considered are finite-dimensional associative
$k$-algebras with identity over a field $k$. Further, assume that $A$ is a
gendo-symmetric algebra and $e$ is an idempotent of $A$ such that $Ae$ is a
faithful, projective-injective $A$-module. In this case, we write $(A,e)$ for
the gendo-symmetric algebra $A$. If $e^{\prime}$ is another idempotent of $A$
such that $Ae^{\prime}$ is a faithful, projective-injective $A$-module, then
$R(A,e)\simeq R(A,e^{\prime})$ as algebras (see Lemma 3.7(1)). Hence, up to
isomorphism of algebras, we can write $R(A)$ for $R(A,e)$ without referring to
$e$, and call it the _mirror-reflective algebra_ of the gendo-symmetric
algebra $A$.

Our first result reveals homological properties of mirror-reflective algebras
of gendo-symmetric algebras. Recall that $A$ is called an _$n$ -Auslander
algebra_ ($n\geq 0$) if ${\rm gl.dim}(A)\leq n+1\leq{\rm dom.dim}(A)$; an _$n$
-minimal Auslander-Gorenstein algebra_ if ${\rm idim}({{}_{A}}A)\leq
n+1\leq{\rm dom.dim}(A)$ (see [1, 19, 20, 4]), where ${\rm gl.dim}(A)$, ${\rm
dom.dim}(A)$ and ${\rm idim}(_{A}A)$ denote the global, dominant and left
injective dimensions of an algebra $A$, respectively. Clearly, $n$-Auslander
algebras are exactly $n$-minimal Auslander-Gorenstein algebras of finite
global dimension (see Subsection 2.2).

######  Theorem 1.1.

Let $(A,e)$ be a gendo-symmetric algebra. Then

$(1)$ $R(A,e,\lambda)$ is a symmetric algebra for $\lambda$ in the center of
$eAe$.

$(2)$ $\min\\{{\rm dom.dim}(\mathcal{A}(A,e)),{\rm
dom.dim}(\mathcal{B}(A,e))\\}$ $\geq{\rm dom.dim}(A)+2.$

$(3)$ $\mathcal{A}(A,e)$ is $(2n+3)$-minimal Auslander-Gorenstein if $A$ is
$n$-minimal Auslander-Gorenstein with $n>0$. Further, $\mathcal{A}(A,e)$ is an
$(2n+3)$-Auslander algebra if $A$ is an $n$-Auslander algebra.

According to Theorem 1.1, $\mathcal{A}(A,e)$ and $\mathcal{B}(A,e)$ are gendo-
symmetric. Thus the construction of (reduced) mirror-reflective algebras can
be done iteratively. Starting with a fixed gendo-symmetric algebra $(A,e)$, we
may define $4$ families of algebras: $R_{n}$, $S_{n}$, $A_{n}$ and $B_{n}$ for
$n>0$ (see Section 4.3 for details). They are called the $n$-th mirror
reflective, $n$-th reduced mirror-reflective, $n$-th gendo-symmetric and
$n$-th reduced gendo-symmetric algebras of $(A,e)$, respectively. By Theorem
1.1(2), these algebras have higher homological dimension: ${\rm
dom.dim}(A_{n+1})\geq{\rm dom.dim}(A_{n})+2$ and ${\rm
dom.dim}(B_{n+1})\geq{\rm dom.dim}(B_{n})+2$. Thus $\min\\{{\rm
dom.dim}(A_{n}),{\rm dom.dim}(B_{n})\\}\geq{\rm dom.dim}(A)+2(n-1)\geq 2n.$
Moreover, they are connected by derived recollements, as is shown in our
second result below. Here, ${\mathscr{D}}^{-}(A)$ and ${\mathscr{D}}(A)$
denote the bounded-above and unbounded derived categories of $A$,
respectively.

######  Theorem 1.2.

Let $(A,e)$ be a gendo-symmetric algebra and $n$ a positive integer. Then the
following hold.

$(1)$ There exist recollements of bounded-above derived categories of
algebras:

$\textstyle{{\mathscr{D}}^{-}(A_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{D}}^{-}(A_{n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{D}}^{-}(A_{n})\quad
and\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{{\mathscr{D}}^{-}(B_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{D}}^{-}(B_{n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{D}}^{-}(B_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

with $B_{0}:=(1-e)A(1-e)$.

$(2)$ Let $R_{0}=S_{0}:=eAe$. If ${\rm dom.dim}(A)=\infty$, then there are
recollements of unbounded derived categories of algebras induced by strong
idempotent ideals:

$\textstyle{{\mathscr{D}}(A_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{D}}(R_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{D}}(R_{n-1})\quad
and\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{{\mathscr{D}}(B_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{D}}(S_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{D}}(S_{n-1}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Motivated by Theorem 1.2(2), we introduce the _stratified dimension_ of an
algebra. It measures how many steps an algebra can be stratified by its
nontrivial strong idempotents (see Definition 2.7), or equivalently, the
derived category of the algebra can be stratified by nontrivial standard
recollements of derived module categories. We also define the _stratified
ratio_ of an algebra to be the ratio of its stratified dimension to the number
of isomorphism classes of simple modules (see Definition 2.10).

Our third result establishes a connection between ${\rm(TC2)}$ and stratified
dimensions of algebras.

######  Theorem 1.3.

The following are equivalent for a field $k$.

$(1)$ Tachikawa’s second conjecture holds for all symmetric algebras over $k$.

$(2)$ Each indecomposable symmetric algebra over $k$ has no nontrivial strong
idempotent ideals.

$(3)$ The supreme of stratified ratios of all indecomposable symmetric
algebras over $k$ is less than $1$.

As a consequence of Theorem 1.3, we shall provide a sufficient condition for
(TC2) on symmetric algebras to hold true. An algebra $\Lambda$ is said to be
_derived simple_ if its unbounded derived category ${\mathscr{D}}(\Lambda)$
admits no nontrivial recollements of unbounded derived module categories of
algebras. Examples of derived simple algebras include local algebras, blocks
of groups algebras and some indecomposable algebras with two simple modules.
One should not confuse the notion of derived simple algebras with the one of
_$\mathscr{D}^{\rm b}(\rm mod)$ -derived simple_ algebras in the sense that
the bounded derived categories (of finitely generated modules) are not
nontrivial recollements of bounded derived categories of any algebras (see
[18]). Derived simple algebras are $\mathscr{D}^{\rm b}(\rm mod)$-derived
simple, but the converse is not true in general. By [18, Theorem 3.2], each
indecomposable symmetric algebra is always $\mathscr{D}^{\rm b}(\rm
mod)$-derived simple.

If an algebra $\Gamma$ has a nontrivial strong idempotent ideal generated by
an idempotent $f$ (see Definition 2.2 below), then there is a nontrivial
recollement $({\mathscr{D}}(\Gamma/\Gamma
f\Gamma),{\mathscr{D}}(\Gamma),{\mathscr{D}}(f\Gamma f))$. Thus we obtain the
following corollary immediately from Theorem 1.3.

######  Corollary 1.4.

If each indecomposable symmetric algebra over a field $k$ is derived simple,
then Tachikawa’s second conjecture holds for all symmetric algebras over $k$.

The paper is structured as follows. In Section $2$, we recall the definitions
of strong idempotents, recollements as well as higher Auslander-Gorenstein and
Auslander algebras. Also, we present the definitions of stratified dimensions
and ratios of algebras (see Definitions 2.7 and 2.10), respectively. In
Section $3$, we define (reduced) mirror-reflective algebras by reflecting a
left (or right) ideal generated by an idempotent element. We then study the
derived module categories of these algebras. Also, an explicit description of
mirror-reflective algebras is demonstrated by quivers with relations. This
explains visually the terminology of mirror-reflective algebras. In Section
$4$, we first show Theorems 1.1 and 1.2. This relies on the fact that mirror-
reflective algebras of gendo-symmetric algebras at any levels are symmetric
(see Proposition 4.2). By iteration of forming (reduced) mirror-reflective
algebras from a gendo-symmetric algebra, a series of recollements of derived
module categories is established. This not only gives a proof of Theorem 1.3,
but also shows the relation between the numbers of simple modules over
different mirror-reflective algebras (see Corollary 4.10(2)-(3)).

##  2 Preliminaries

Throughout the paper, $k$ denotes a commutative ring, and all algebras
considered are associative $k$-algebras with identity.

Let $A$ be a $k$-algebra. We denote by $A\mbox{{\rm-Mod}}$ the category of all
left $A$-modules, and by $A\mbox{{\rm-mod}}$ the full subcategory of
$A\mbox{{\rm-Mod}}$ consisting of finitely generated $A$-modules. The _global
dimension_ of $A$, denoted by ${\rm gl.dim}(A)$, is defined to be the supreme
of projective dimensions of all $A$-modules. The _finitistic dimension_ of
$A$, denoted by ${\rm fin.dim}(A)$, is defined to be the supreme of projective
dimensions of those $A$-modules which have a finite projective resolution by
finitely generated projective modules. The _projective and injective
dimensions_ of an $A$-module $M$ are denoted by ${\rm pdim}(_{A}M)$ and ${\rm
idim}({{}_{A}}M)$, respectively. If $f:X\to Y$ and $g:Y\to Z$ are
homomorphisms of $A$-modules, we write $fg$ for the composite of $f$ with $g$,
and $(x)f$ for the image of $x\in X$ under $f$.

For an additive category $\mathcal{C}$, let ${\mathscr{C}}(\mathcal{C})$
denote the category of all complexes over $\mathcal{C}$ with chain maps, and
${\mathscr{K}}(\mathcal{C})$ the homotopy category of
${\mathscr{C}}(\mathcal{C})$. We denote by ${\mathscr{C}}^{b}(\mathcal{C})$
and ${\mathscr{K}}^{b}(\mathcal{C})$ the full subcategories of
${\mathscr{C}}(\mathcal{C})$ and ${\mathscr{K}}(\mathcal{C})$ consisting of
bounded complexes over $\mathcal{C}$, respectively. When $\mathcal{C}$ is
abelian, the _(unbounded) derived category_ of $\mathcal{C}$ is denoted by
${\mathscr{D}}(\mathcal{C})$, which is the localization of
${\mathscr{K}}(\mathcal{C})$ at all quasi-isomorphisms. The full subcategory
of ${\mathscr{D}}(\mathcal{C})$ consisting of bounded-above complexes over
$\mathcal{C}$ is denoted by ${\mathscr{D}}^{-}(\mathcal{C})$. As usual, we
simply write ${\mathscr{K}}(A)$ for ${\mathscr{K}}(A\mbox{{\rm-Mod}})$,
${\mathscr{D}}(A)$ for ${\mathscr{D}}(A\mbox{{\rm-Mod}})$, and
${\mathscr{D}}^{-}(A)$ for ${\mathscr{D}}^{-}(A\mbox{{\rm-Mod}})$. Also, we
identify $A\mbox{{\rm-Mod}}$ with the full subcategory of ${\mathscr{D}}(A)$
consisting of all stalk complexes in degree zero.

###  2.1 Standard recollements and stratified dimensions

In this section, we start with recalling recollements of triangulated
categories, introduced by Beilinson, Bernstein and Deligne in [3], and
introduce the notion of stratified dimensions of algebras.

######  Definition 2.1.

Let $\mathcal{T}$, $\mathcal{T^{\prime}}$ and $\mathcal{T^{\prime\prime}}$ be
triangulated categories. $\mathcal{T}$ admits a _recollement_ of
$\mathcal{T^{\prime}}$ and $\mathcal{T^{\prime\prime}}$ (or there is a
recollement among $\mathcal{T^{\prime\prime}},\mathcal{T}$ and
$\mathcal{T}^{\prime}$) if there are six triangle functors

$\textstyle{\mathcal{T^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}=i_{!}}$$\textstyle{\mathcal{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{!}=j^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{i^{*}}$$\textstyle{\mathcal{T^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{*}}$$\scriptstyle{j_{!}}$

among the three categories such that the $4$ conditions are satisfied:

$(1)$ $(i^{*},i_{*}),(i_{!},i^{!}),(j_{!},j^{!})$ and $(j^{*},j_{*})$ are
adjoint pairs.

$(2)$ $i_{*},j_{*}$ and $j_{!}$ are fully faithful functors.

$(3)$ $j^{!}i_{!}=0$ (and thus also $i^{!}j_{*}=0$ and $i^{*}j_{!}=0$).

$(4)$ For an object $X\in\mathcal{T}$, there are triangles $i_{!}i^{!}(X)\to
X\to j_{*}j^{*}(X)\to i_{!}i^{!}(X)[1]$ and $j_{!}j^{!}(X)\to X\to
i_{*}i^{*}(X)\to j_{!}j^{!}(X)[1]$ induced by the adjunctions of counits and
units, where [$1$] is the shift functor of $\mathcal{T}$.

Quasi-hereditary algebras, introduced by Cline, Parshall and Scott (see [10,
11]), provide a special class of recollements of derived module categories.
Recall that a _heredity ideal_ of an algebra $A$ is an ideal $I$ such that
$(i)$ $I$ is idempotent (i.e. $I^{2}=I$), $(ii)$ ${}_{A}I$ is projective as an
$A$-module and $(iii)$ ${\rm End}_{A}(_{A}I)$ is semisimple. For such an ideal
$I$, there holds always ${\rm Ext}_{A/I}^{i}(X,Y)\simeq{\rm Ext}_{A}^{i}(X,Y)$
for all modules $X,Y$ over $A/I$ and for all $i\geq 0$. A slight
generalisation of heredity ideals is the $n$-idempotent ideals defined in [2].

######  Definition 2.2.

[2] Let $A$ be an algebra, $I$ an ideal of $A$, and $n$ a positive integer.
The ideal $I$ of $A$ is said to be _$n$ -idempotent_ if, for
$X,Y\in(A/I)\mbox{{\rm-Mod}}$, the canonical homomorphism ${\rm
Ext}_{A/I}^{i}(X,Y)\to{\rm Ext}_{A}^{i}(X,Y)$ of $k$-modules is an isomorphism
for all $1\leq i\leq n$.

The ideal $I$ is said to be _strong idempotent_ if $I$ is $n$-idempotent for
all $n\geq 1$. In this case, if $I=AeA$ for an idempotent $e\in A$, then $e$
is called a _strong idempotent_ of $A$.

By a trivial strong idempotent of $A$ we mean the idempotent $0$ or an
idempotent $e$ with $AeA=A$. Note that an ideal $I$ is $1$-idempotent if and
only if $I$ is idempotent. Moreover, strong idempotent ideals are closely
related to homological ring epimorphisms. Recall that a ring homomorphism
$\lambda:A\to B$ is called a _homological ring epimorphism_ if the
multiplication map $B\otimes_{A}B\to B$ is an isomorphism and ${\rm
Tor}_{i}^{A}(B,B)=0$ for all $i\geq 1$. This is equivalent to saying that the
derived restriction functor
$D(\lambda_{*}):{\mathscr{D}}(B)\to{\mathscr{D}}(A)$, induced by the
restriction functor $\lambda_{*}:B\mbox{{\rm-Mod}}\to A\mbox{{\rm-Mod}}$, is
fully faithful. Note that strong idempotent ideals $I$ are also called
homological ideals, that is, the canonical surjection $A\to A/I$ is a
homological ring epimorphism.

Let us emphasize that strong idempotent ideals generated by idempotents are
exactly _stratifying ideals_ introduced in [11, Definition 2.1.1].

######  Lemma 2.3.

[2] Let $I=AeA$ for an idempotent $e$ in $A$.

$(1)$ Let $n$ be a positive integer. Then $I$ is $(n+1)$-idempotent if and
only if the multiplication map

$Ae\otimes_{eAe}eA\longrightarrow I,\;\;ae\otimes eb\mapsto aeb,\;\,a,b\in A$

is an isomorphism of $A$-$A$-bimodules and ${\rm Tor}_{i}^{eAe}(Ae,eA)=0$ for
all $1\leq i\leq n-1$.

$(2)$ If $I$ is $2$-idempotent, then

$\sup\\{n\in\mathbb{N}\mid{\rm Ext}_{A}^{i}(A/I,A/I)=0,1\leq i\leq
n\\}\geq\sup\\{n\in\mathbb{N}\mid{\rm Tor}_{i}^{eAe}(Ae,eA)=0,1\leq i\leq
n\\}+2.$

Proof. $(1)$ Although all the results in [2] are stated for finitely generated
modules over Artin algebras, many of them such as Theorem 2.1, Lemma 3.1 and
Propositions 2.4 and 3.7(b) hold for arbitrary modules over rings if we modify
$\mathbf{P}_{n}$ in [2, Definition 2.3] as follows:

Let $\mathbf{P}_{n}(Ae)$ be the full subcategory of $A\mbox{{\rm-Mod}}$
consisting of all modules $X$ such that there is an exact sequence
$P_{n}\to\cdots\to P_{1}\to P_{0}\to X\to 0$ of $A$-modules with $P_{i}\in{\rm
Add}(Ae)$ for $0\leq i\leq n$, where ${\rm Add}(Ae)$ is the full subcategory
of $A\mbox{{\rm-Mod}}$ consisting of direct summands of direct sums of copies
of $Ae$.

By [2, Theorem 2.1], $I:=AeA$ is $(n+1)$-idempotent if and only if
$I\in\mathbf{P}_{n}(Ae)$. In particular, $I$ is $2$-idempotent if and only if
$I\in\mathbf{P}_{1}(Ae)$. By [2, Lemma 3.1], the adjoint pair
$(Ae\otimes_{eAe}-,{\rm Hom}_{A}(Ae,-))$ between $(eAe)\mbox{{\rm-Mod}}$ and
$A\mbox{{\rm-Mod}}$ induces additive equivalences between $(eAe)\mbox{{\rm-
Mod}}$ and $\mathbf{P}_{1}(Ae)$. Note that ${\rm Hom}_{A}(Ae,I)\simeq eI=eA$.
Thus $I\in\mathbf{P}_{1}(Ae)$ if and only if the multiplication map
$Ae\otimes_{eAe}eA\to AeA$ is an isomorphism of $A$-$A$-bimodules. Now, assume
that $I$ is $2$-idempotent. By [2, Proposition 3.7(b)],
$I\in\mathbf{P}_{n}(Ae)$ if and only if ${\rm Tor}_{i}^{eAe}(Ae,eA)=0$ for all
$1\leq i<n$. This shows $(1)$.

$(2)$ If $I$ is $(n+1)$-idempotent, then ${\rm Ext}_{A}^{i}(A/I,A/I)\simeq{\rm
Ext}_{A/I}^{i}(A/I,A/I)=0$ for all $1\leq i\leq n+1$. Now, $(2)$ follows from
$(1)$. $\square$

######  Corollary 2.4.

$(1)$ Let $e$ and $f$ be idempotents of $A$ such that $ef=e=fe$. If $AeA$ is
an $(n+1)$-idempotent ideal of $A$ for a positive integer $n$, then $fAeAf$ is
an $(n+1)$-idempotent ideal of $fAf$. In particular, if $e$ is a strong
idempotent of $A$, then it is also a strong idempotent of $fAf$.

$(2)$ Let $\\{e,e_{1},e_{2}\\}$ be a set of orthogonal idempotents of $A$ such
that $e$ is a strong idempotent of $A$. Define $f:=e+e_{1}$,
$g:=e+e_{1}+e_{2}$ and $\overline{A}:=A/AeA$. Let $\overline{f}:=f+AeA$ denote
the image of $f$ in $\overline{A}$. If $\overline{f}$ is a strong idempotent
of $\overline{g}\overline{A}\overline{g}$, then $f$ is a strong idempotent of
$gAg$.

Proof. $(1)$ Transparently, $e\in fAf$, $efAfe=eAe$, $fAeAfe=fAe$ and
$efAeAf=eAf$. If $Ae\otimes_{eAe}eA\simeq AeA$, then
$fAe\otimes_{eAe}eAf\simeq fAeAf$. Since $Ae=fAe\oplus(1-f)Ae$ and
$eA=eAf\oplus eA(1-f)$, we see that the abelian group ${\rm
Tor}_{i}^{eAe}(fAe,eAf)$ is a direct summand of ${\rm Tor}_{i}^{eAe}(Ae,eA)$
for $i\in\mathbb{N}$. Now, (1) follows from Lemma 2.3(1).

$(2)$ Clearly, $AeA\subseteq AfA\subseteq AgA$, and
$\overline{g}\overline{A}\overline{g}\simeq gAg/gAeAg$ and
$\overline{g}\overline{A}\overline{g}/\overline{g}\overline{A}\overline{f}\overline{A}\overline{g}\simeq
gAg/gAfAg$ as algebras. Suppose that $\overline{f}$ is a strong idempotent of
$\overline{g}\overline{A}\overline{g}$. Then the canonical surjection
$\pi_{2}:gAg/gAeAg\to gAg/gAfAg$ is homological. Since $e$ is a strong
idempotent of $A$ and $ge=e=eg$, the canonical surjection $\pi_{1}:gAg\to
gAg/gAeAg$ is also homological by (1). Observe that compositions of
homological ring epimorphisms are again homological ring epimorphisms. Thus
$\pi_{1}\pi_{2}:gAg\to gAg/gAfAg$ is homological. This implies that $f$ is a
strong idempotent in $gAg$. $\square$

Let $e=e^{2}\in A$. If $AeA$ is a strong idempotent ideal in $A$, then the
recollement of derived module categories of algebras:

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is called a _standard recollement_ induced by $AeA$. If ${{}_{A}}AeA$ or
$AeA_{A}$ is projective (for example, $AeA$ is a heredity ideal in $A$), then
the ideal $AeA$ is strong idempotent. In the case that ${{}_{A}}AeA$ is
projective, the recollement restricts to a recollement
$({\mathscr{D}}^{-}(A/AeA),{\mathscr{D}}^{-}(A),{\mathscr{D}}^{-}(eAe))$ of
bounded-above derived categories.

A general method is given for constructing finitely generated (one-sided)
projective idempotent ideals of the endomorphism algebras of objects in
additive categories (see [5, Lemmas 3.2 and 3.4]). This implies the following.

######  Lemma 2.5.

Suppose that $R$ is an algebra and $I$ is an ideal of $R$.

$(1)$ Let $A:={\rm End}_{R}(R\oplus\;R/I)$ and $e^{2}=e\in A$ correspond to
the direct summand $R/I$ of the $R$-module $R\oplus\;R/I$. Then $AeA{{}_{A}}$
is finitely generated and projective, and there is a recollement
$({\mathscr{D}}(R/{\rm Ann}_{R^{\rm op}}(I)),$
${\mathscr{D}}(A),{\mathscr{D}}(R/I))$, with ${\rm Ann}_{R^{\rm
op}}(I):=\\{r\in R\mid Ir=0\\}$.

$(2)$ Let $B:={\rm End}_{R}(R\oplus I)$ and $f=f^{2}\in B$ correspond to the
direct summand $I$ of the $R$-module $R\oplus I$. If $I$ is idempotent, then
${{}_{B}}BfB$ is finitely generated and projective, and there is a recollement
$({\mathscr{D}}(R/I),{\mathscr{D}}(B),{\mathscr{D}}({\rm End}_{R}(I)))$.

Another way to produce finitely generated projective ideals comes from Morita
context algebras, as explained below.

Let $R$ be an algebra and let $I$ and $J$ be ideals of $R$ with $IJ=0$. Define

$M_{l}(R,I,J):=\left(\begin{array}[]{lc}R&I\\\
R/J&R/J\end{array}\right)\quad(\;\mbox{respectively,}\;\;M_{r}(R,I,J):=\left(\begin{array}[]{lc}R&R/I\\\
J&R/I\end{array}\right)\;)$

which is the _Morita context algebra_ with the bimodule homomorphisms given by
the canonical ones:

$I\otimes_{R/J}(R/J)\simeq I\hookrightarrow R,\quad(R/J)\otimes_{R}I\simeq
I/JI\twoheadrightarrow(I+J)/J\hookrightarrow R/J$

$(\;\mbox{respectively,}\;\;(R/I)\otimes_{R/I}J\simeq J\hookrightarrow R,\quad
J\otimes_{R}(R/I)\simeq J/JI\twoheadrightarrow(I+J)/I\hookrightarrow R/I).$
Note that $M_{r}(R,I,{\rm Ann}_{R^{\rm op}}(I))\simeq{\rm End}_{R}(R\oplus
R/I)$ as algebra. Moreover, if ${{}_{R}}R$ is injective and $I^{2}=I$, then
$M_{l}(R,I,{\rm Ann}_{R^{\rm op}}(I))\simeq{\rm End}_{R}(R\oplus I)$ as
algebras. This is due to ${\rm Hom}_{R}(I,R/I)=0.$

Let

$e:=\left(\begin{array}[]{lc}0&0\\\ 0&1+J\end{array}\right)\in
M_{l}(R,I,J),\;\;f:=\left(\begin{array}[]{lc}0&0\\\ 0&1+I\end{array}\right)\in
M_{r}(R,I,J).$

Then the next lemma is easy to verify.

######  Lemma 2.6.

Let $A:=M_{l}(R,I,J)$ and $B:=M_{r}(R,I,J)$. Then ${{}_{A}}AeA$ and $BfB_{B}$
are finitely generated and projective. Moreover, there are recollements
$({\mathscr{D}}(R/I),{\mathscr{D}}(A),{\mathscr{D}}(R/J))$ and
$({\mathscr{D}}(R/J),{\mathscr{D}}(B),{\mathscr{D}}(R/I))$.

Now, we introduce stratified dimensions of algebras, which measure how many
steps the given algebras can be stratified by their nontrivial strong
idempotents.

######  Definition 2.7.

By an _idempotent stratification_ of length $n$ of an algebra $A$, we mean a
set $\\{e_{i}\mid 0\leq i\leq n\\}$ of nonzero (not necessarily primitive)
orthogonal idempotents of $A$ satisfying the conditions:

$(a)$ $1=\sum_{j=0}^{n}e_{j}$ and $e_{i+1}\notin Ae_{\leq i}A$ (or
equivalently, $Ae_{\leq i}A\subsetneq Ae_{\leq(i+1)}A$) for all $0\leq i\leq
n-1$, where $e_{\leq m}:=\sum_{j=0}^{m}e_{j}$ for each $0\leq m\leq n$; and

$(b)$ $e_{\leq i}$ is a strong idempotent of the algebra
$e_{\leq(i+1)}Ae_{\leq(i+1)}$ for each $0\leq i\leq n-1$.

The _stratified dimension_ of $A$, denoted by ${\rm st.dim}(A)$, is defined to
be the supreme of the lengths of all idempotent stratifications of $A$.

Clearly, ${\rm st.dim}(A)=0$ if and only if $A$ has no nontrivial strong
idempotent ideals. If ${\rm st.dim}(A)=n>0$, then there are iterated
nontrivial standard recollements
$\big{(}{\mathscr{D}}(A_{i}/I_{i}),{\mathscr{D}}(A_{i}),{\mathscr{D}}(A_{i-1})\big{)}$
for all $1\leq i<n+1$, where $A_{0}:=e_{0}Ae_{0}$, $A_{i}:=e_{\leq i}Ae_{\leq
i}$ and $I_{i}:=e_{\leq i}Ae_{\leq(i-1)}Ae_{\leq i}$ in Definition 2.7.
Moreover, for any two algebras $A_{1}$ and $A_{2}$, ${\rm st.dim}(A_{1}\times
A_{2})={\rm st.dim}(A_{1})+{\rm st.dim}(A_{2})+1.$ This implies that the
stratified dimension of the direct product of $\mathbb{N}$-copies of a field
$k$ is infinite.

Stratifications of algebras in the sense of Cline, Parshall and Scott are
idempotent stratifications. But the converse is not true. Following [11,
Chapter 2], a _stratification_ of length $(n+1)$ of an algebra $A$ is a chain
of ideals, $0=U_{-1}\subsetneq U_{0}\subsetneq U_{1}\subsetneq\cdots\subsetneq
U_{n-1}\subsetneq U_{n}=A$, generated by idempotents such that $U_{i}/U_{i-1}$
is a strong idempotent ideal of $A/U_{i-1}$ for $0\leq i\leq n$. In this case,
$A$ is said to be _CPS-stratified_. If $\\{e_{i}\mid 0\leq i\leq n\\}$ is a
complete set of nonzero primitive orthogonal idempotents of $A$ and
$U_{i}=Ae_{\leq i}A$ for $0\leq i\leq n$, then $A$ is called a _fully CPS-
stratified_ algebra. Standardly stratified algebras with respect to an order
of simple modules are fully CPS-stratified.

######  Lemma 2.8.

Let $\\{e_{i}\mid 0\leq i\leq n\\}$ be a set of nonzero orthogonal idempotents
of $A$ satisfying the condition $(a)$ in Definition 2.7. Define
$U_{i}:=Ae_{\leq i}A$ for $0\leq i\leq n$ and $U_{-1}:=0$. If $U_{i}/U_{i-1}$
is a strong idempotent ideal of $A/U_{i-1}$ for $0\leq i\leq n$, then the
condition $(b)$ in Definition 2.7 holds.

Proof. Since $U_{i}/U_{i-1}$ is a strong idempotent ideal of $A/U_{i-1}$ by
assumption, the canonical surjection $A/U_{i-1}\to A/U_{i}$ is homological. As
the composition of homological ring epimorphisms is still a homological ring
epimorphism, the canonical surjection $A\to A/U_{i}$ is homological. This
implies that $e_{\leq i}$ is a strong idempotent of $A$. By Corollary 2.4,
$e_{\leq i}$ is a strong idempotent of $e_{\leq(i+1)}Ae_{\leq(i+1)}$. Thus
Definition 2.7(b) holds. $\square$

For an Artin algebra $A$, we denote by $\\#(A)$ the number of isomorphism
classes of simple $A$-modules.

######  Proposition 2.9.

Let $A$ be an Artin algebra over a commutative Artin ring $k$. Then

$(1)$ ${\rm st.dim}(A)\leq\\#(A)-1$.

$(2)$ If $A$ has a stratification of length $n+1$ with $n\in\mathbb{N}$, then
${\rm st.dim}(A)\geq n$. In particular, if $A$ is a fully CPS-stratified
algebra, then ${\rm st.dim}(A)=\\#(A)-1$.

$(3)$ If ${\rm st.dim}(A)\geq 1$, then ${\rm st.dim}(A)=\sup_{e\in A}\\{{\rm
st.dim}(eAe)+{\rm st.dim}(A/AeA)+1\\}$, where $e$ runs over all nonzero strong
idempotents of $A$ with $AeA\neq A$.

$(4)$ If $k$ is a field and $B$ is a finite-dimensional $k$-algebra, then

${\rm st.dim}(A\otimes_{k}B)\geq({\rm st.dim}(A)+1)({\rm st.dim}(B)+1)-1.$

Proof. $(1)$ This is clear by Definition 2.7(a).

$(2)$ The first part of $(2)$ follows from Lemma 2.8. If $A$ is a fully CPS-
stratified algebra, then it has a stratification of length $\\#(A)-1$. By
$(1)$, we obtain ${\rm st.dim}(A)=\\#(A)-1$.

$(3)$ An Artin algebra always has only finitely many nonisomorphic,
indecomposable, finitely generated projective modules. This implies

$(\ast)$ If $f$ is an idempotent of $A$ and $I$ is an idempotent ideal of $A$
such that $AfA\subseteq I$, then there is an idempotent $f^{\prime}$ of $A$
which is orthogonal to $f$ such that $I=A(f+f^{\prime})A$.

Now, let $n:={\rm st.dim}(A)\geq 1$. On the one hand, since $e_{\leq n-1}$ in
Definition 2.7(b) is a strong idempotent of $A$, we have ${\rm st.dim}(A)={\rm
st.dim}(e_{\leq n-1}Ae_{\leq n-1})+1$ and ${\rm st.dim}(A/Ae_{\leq n-1}A)=0$
by $(\ast)$ and Corollary 2.4(2). On the other hand, for each nonzero strong
idempotent $e$ of $A$ with $AeA\neq A$, it follows again from $(\ast)$ and
Corollary 2.4(2) that ${\rm st.dim}(eAe)+{\rm st.dim}(A/AeA)+1\leq n$. Thus
$(3)$ holds.

$(4)$ Let $m:={\rm st.dim}(B)$ and $\ell:=n+m$. If $\ell=0$ (i.e. $n=0=m$),
then the inequality holds obviously. Let $\ell\geq 1$. Without loss of
generality, suppose $n\geq 1$. By the proof of $(3)$, there is a nonzero
strong idempotent $e$ of $A$ with $AeA\neq A$ such that ${\rm
st.dim}(eAe)=n-1$ and ${\rm st.dim}(A/AeA)=0$. Then the canonical surjection
$\pi:A\to A/AeA$ is homological. Note that, for homological epimorphisms
$\lambda_{i}:R_{i}\to S_{i}$ of algebras over the field $k$ with $i=1,2$, the
algebra homomorphism
$\lambda_{1}\otimes_{k}\lambda_{2}:R_{1}\otimes_{k}R_{2}\to
S_{1}\otimes_{k}S_{2}$ is still a homological ring epimorphism. This is due to
the isomorphism

${\rm
Tor}_{j}^{R_{1}\otimes_{k}R_{2}}(S_{1}\otimes_{k}S_{2},S_{1}\otimes_{k}S_{2})\simeq\bigoplus_{p+q=j}{\rm
Tor}_{p}^{R_{1}}(S_{1},S_{1})\otimes_{k}{\rm
Tor}^{R_{2}}_{q}(S_{2},S_{2})\;\mbox{ for all }j\in\mathbb{N}.$

Now, let $C:=A\otimes_{k}B$ and $e^{\prime}:=e\otimes 1\in C$. Then the
surjection $\pi\otimes 1:C\to A/AeA\otimes_{k}B$ is homological. Clearly,
there are algebra isomorphisms $(A/AeA)\otimes_{k}B\simeq
C/(AeA\otimes_{k}B)\simeq C/Ce^{\prime}C$. It follows that the canonical
surjection $C\to C/Ce^{\prime}C$ is homological, and therefore $e^{\prime}$ is
a nonzero strong idempotent of $C$ with $Ce^{\prime}C\neq C$. By $(3)$, ${\rm
st.dim}(C)\geq{\rm st.dim}(eAe\otimes_{k}B)+{\rm
st.dim}((A/AeA)\otimes_{k}B)+1$. Moreover, by induction, ${\rm
st.dim}(eAe\otimes_{k}B)\geq({\rm st.dim}(eAe)+1)({\rm st.dim}(B)+1)-1$ and
${\rm st.dim}((A/AeA)\otimes_{k}B)\geq{\rm st.dim}(B)$. Thus ${\rm
st.dim}(C)\geq(n+1)(m+1)-1$. $\square$

######  Definition 2.10.

Let $A$ be an Artin algebra over a commutative Artin ring $k$. The rational
number $\frac{{\rm st.dim}(A)}{\\#(A)}$ is called the _stratified ratio_ of
$A$ and denoted by ${\rm sr}(A)$.

By Proposition 2.9(1), ${\rm sr}(A)\in\mathbb{Q}\cap[0,1)$. Let $A^{n}$ denote
the product of $n$-copies of $A$. Then

$\lim\limits_{n\to\infty}{\rm sr}(A^{n})=\lim\limits_{n\to\infty}\frac{n({\rm
st.dim}(A))+n-1}{n\;\\#(A)}=\frac{{\rm st.dim}(A)+1}{\\#(A)}\leq 1.$

In particular, if ${\rm st.dim}(A)=\\#(A)-1$ (for example, $A$ is quasi-
hereditary or local), then $\lim\limits_{n\to\infty}{\rm sr}(A^{n})=1$. In
Section 4, for a gendo-symmetric algebra with infinite dominant dimension, we
construct a series of indecomposable symmetric algebras $S_{n}$ such that
$\lim\limits_{n\to\infty}{\rm sr}(S_{n})=1$ (see Corollary 4.12).

###  2.2 Dominant dimensions and gendo-symmetric algebras

Let $A$ be a finite-dimensional algebra over a field $k$.

######  Definition 2.11.

The _dominant dimension_ of an algebra $A$, denoted by ${\rm dom.dim}(A)$, is
the maximal natural number $n$ or $\infty$ such that the first $n$ terms
$I_{0},I_{1},\cdots,I_{n-1}$ in a minimal injective resolution
$0\rightarrow{}_{A}A\rightarrow I^{0}\rightarrow
I^{1}\rightarrow\cdots\rightarrow I^{i-1}\rightarrow I^{i}\rightarrow\cdots$
of $A$ are projective.

Recall that a module $M\in A\mbox{{\rm-mod}}$ is called a _generator_ if
$A\in{\rm add}(M)$; a _cogenerator_ if $D(A)\in{\rm add}(M)$; a _generator-
cogenerator_ if it is a generator and cogenerator. By [22, Lemma 3], if
${}_{A}M\in A\mbox{{\rm-mod}}$ is a generator-cogenerator, then ${\rm
dom.dim}({\rm End}_{A}(M))=\sup\\{n\in\mathbb{N}\mid{\rm
Ext}_{A}^{i}(M,M)=0,1\leq i\leq n\\}+2.$

Algebras of the form ${\rm End}_{A}(A\oplus M)$ with $A$ an algebra and $M$ an
$A$-module has double centralizer property and has been studied for a long
time. Following [14], such an algebra is called a _gendo-symmetric_ algebra if
the algebra $A$ is symmetric. Note that if $A$ is a symmetric algebra, then so
is $eAe$ for $e=e^{2}\in A$.

######  Lemma 2.12.

[13, Theorem 3.2] The following are equivalent for an algebra $A$ over a
field.

$(1)$ $A$ is a gendo-symmetric algebra.

$(2)$ ${\rm dom.dim}(A)\geq 2$ and $D(Ae)\simeq eA$ as $eAe$-$A$-bimodules,
where $e\in A$ is an idempotent such that $Ae$ is a faithful projective-
injective $A$-module.

$(3)$ ${\rm Hom}_{A}(D(A),A)\simeq A$ as $A$-$A$-bimodules.

$(4)$ $D(A)\otimes_{A}D(A)\simeq D(A)$ as $A$-$A$-bimodules.

In the rest of the paper, we always write $(A,e)$ for a gendo-symmetric
algebra with $e$ an idempotent in $A$ such that $Ae$ is a _faithful
projective-injective_ $A$-module. Note that ${\rm add}(Ae)$ coincides with the
full subcategory of $A\mbox{{\rm-mod}}$ consisting of projective-injective
$A$-modules.

An algebra $A$ is called an _Auslander algebra_ if ${\rm gl.dim}(A)\leq
2\leq{\rm dom.dim}(A)$. This is equivalent to saying that $A$ is the
endomorphism algebra of an additive generator of a representation-finite
algebra over a field (see [1]). A generalization of Auslander algebras is the
so-called $n$-Aulslander algebras. Let $n$ be a positive integer. Following
[1, 19, 20], $A$ is called an _$n$ -Auslander algebra_ if ${\rm gl.dim}(A)\leq
n+1\leq{\rm dom.dim}(A)$; an _$n$ -minimal Auslander-Gorenstein algebra_ if
${\rm idim}({{}_{A}}A)\leq n+1\leq{\rm dom.dim}(A)$. Clearly, $n$-Auslander
algebras are $n$-minimal Auslander-Gorenstein, while $n$-minimal Auslander-
Gorenstein algebras of finite global dimension are $n$-Auslander. Moreover,
these algebras can be characterized in terms of left or right perpendicular
categories. For each $M\in A\mbox{{\rm-mod}}$ and $m\in\mathbb{N}$, we define

${}^{\perp_{m}}M:=\\{X\in A\mbox{{\rm-mod}}\mid{\rm Ext}_{A}^{i}(X,M)=0,1\leq
i\leq m~{}\\},\;\;M^{\perp_{m}}:=\\{X\in A\mbox{{\rm-mod}}\mid{\rm
Ext}_{A}^{i}(M,X)=0,1\leq i\leq m~{}\\}.$

Recall that an $A$-module $N$ is said to be _maximal $(n-1)$-orthogonal_ or
_$n$ -cluster tilting_ if ${\rm
add}({{}_{\Lambda}}N)={}^{\perp_{n-1}}N=N^{\perp_{n-1}}$. A generator-
cogenerator $A$-module $M$ is said to be _$(n-1)$ -ortho-symmetric_ or _$n$
-precluster tilting_ if ${\rm
add}(_{\Lambda}M)\subseteq{}^{\perp_{n-1}}M=M^{\perp_{n-1}}$. The algebra $A$
is $n$-Auslander if and only if there is an algebra $\Lambda$ and a maximal
$(n-1)$-orthogonal $\Lambda$-module ${{}_{\Lambda}}N$ such that $A={\rm
End}_{\Lambda}(N)$ by [19, Proposition 2.4.1], and is $n$-minimal Auslander-
Gorenstein if and only if there is an algebra $\Lambda$ and an $(n-1)$-ortho-
symmetric generator-cogenerator ${{}_{\Lambda}}N$ such that $A={\rm
End}_{\Lambda}(N)$ by [20, Theorem 4.5] or [4, Corollary 3.18]. Moreover, by
[20, Proposition 4.1], if $A$ is $n$-minimal Auslander-Gorenstein, then either
$A$ is self-injective or ${\rm idim}({{}_{A}}A)=n+1={\rm dom.dim}(A)$. In the
latter case, we have ${\rm idim}(A{{}_{A}})=n+1={\rm dom.dim}(A)$ and thus $A$
is $(n+1)$-Gorenstein.

An $A$-module $M$ is said to be _$m$ -rigid_ if ${\rm Ext}_{A}^{i}(M,M)=0$ for
all $1\leq i\leq m$. Over symmetric algebras, ortho-symmetric modules have
been characterized as follows.

######  Lemma 2.13.

[4, Corollary 5.4] Let $A$ be a symmetric algebra and $N$ a basic $A$-module
without any nonzero projective direct summands. For a natural number $m$. the
$A$-module $A\oplus N$ is $m$-ortho-symmetric if and only if $N$ is $m$-rigid
and $\Omega_{A}^{m+2}(N)\cong N$.

##  3 Mirror-reflective algebras

In this section, we introduce (reduced) mirror-reflective algebras and prove
that these algebras can be linked by recollements of their derived module
categories (see Proposition 3.6).

###  3.1 Mirror-reflective algebras and their derived recollements

Throughout this section, assume that $A$ is an algebra over a commutative ring
$k$. Let $M$ be an $A$-$A$-bimodule and $\alpha:{}_{A}M\otimes_{A}M\rightarrow
M$ be a homomorphism of $A$-$A$-bimodules, such that the associative law holds

$(\heartsuit)\qquad\big{(}(x\otimes y)\alpha\otimes
z\big{)}\alpha=\big{(}x\otimes(y\otimes z)\alpha\big{)}\alpha\;\mbox{ for
}x,y,z\in M.$

We define a multiplication on the underlying abelian group $A\oplus M$ by
setting

$(a,m)\cdot(b,n):=(ab,an+mb+(m\otimes n)\alpha)\;\mbox{ for }a,b\in A,\,m,n\in
M.$

Then $A\oplus M$ becomes an associative algebra with the identity $(1,0)$,
denoted by $R(A,M,\alpha)$. In the following, we identify $A$ with $(A,0)$,
and $M$ with $(0,M)$ in $R(A,M,\alpha)$. Thus $A$ is a subalgebra of
$R(A,M,\alpha)$ with the same identity, and $M$ is an ideal of $R(A,M,\alpha)$
such that $R(A,M,\alpha)/M\simeq A$.

Now, we consider a special case of the above construction. Let $e=e^{2}\in A$,
$\Lambda:=eAe\;\;\mbox{and}\;\;Z(\Lambda)$ be the center of $\Lambda$. For
$\lambda\in Z(\Lambda)$, let $\omega_{\lambda}$ be the composite of the
natural maps:

$(Ae\otimes_{\Lambda}eA)\otimes_{A}(Ae\otimes_{\Lambda}eA)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}Ae\otimes_{\Lambda}(eA\otimes_{A}Ae)\otimes_{\Lambda}eA\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}Ae\otimes_{\Lambda}\Lambda\otimes_{\Lambda}eA\stackrel{{\scriptstyle{\rm
Id}\otimes(\cdot\lambda)\otimes{\rm
Id}}}{{\longrightarrow}}Ae\otimes_{\Lambda}\Lambda\otimes_{\Lambda}eA\rightarrow
Ae\otimes_{\Lambda}eA,$

where $(\cdot\lambda):\Lambda\to\Lambda$ is the multiplication map by
$\lambda$. Then $\omega_{\lambda}$ satisfies the associative law
$(\heartsuit)$.

Let $R(A,e,\lambda):=R(A,Ae\otimes eA,\omega_{\lambda})$. Then the elements of
$R(A,e,\lambda)$ are of the form

$a+\sum_{i=1}^{n}a_{i}e\otimes eb_{i}\;\;\mbox{for}\;\;a,a_{i},b_{i}\in
A,1\leq i\leq n\in\mathbb{N}.$

The multiplication, denoted by $\ast$, is explicitly given by

$(a+be\otimes ec)\ast(a^{\prime}+b^{\prime}e\otimes
ec^{\prime}):=aa^{\prime}+(ab^{\prime}e\otimes ec^{\prime}+be\otimes
eca^{\prime}+becb^{\prime}e\otimes\lambda ec^{\prime})$

for $a,b,c,a^{\prime},b^{\prime},c^{\prime}\in A$, and can be extended
linearly to elements of general form. Particularly,

$(\diamondsuit)\;\quad(ae\otimes eb)\ast(a^{\prime}e\otimes
eb^{\prime})=aeba^{\prime}e\lambda\otimes eb^{\prime}=ae\otimes\lambda
eba^{\prime}eb^{\prime}.$

######  Definition 3.1.

The algebra $R(A,e,\lambda)$ defined above is called the _mirror-reflective
algebra_ of $A$ at level $(e,\lambda)$. The algebra $R(A,e,e)$ is then called
the _mirror-reflective algebra_ of $A$ at $e$, denoted by $R(A,e)$.

The algebra $S(A,e,e):=(1-e)A(1-e)\oplus Ae\otimes_{\Lambda}eA$ with the
multiplication induced from the one of $R(A,e,e)$ is called the _reduced
mirror-reflective algebra_ of $A$ at $e$, denoted by $S(A,e)$.

Compared with $R(A,e)$, $S(A,e)$ has a fewer number of simple modules. So it
is termed the reduced mirror-reflective algebra. The following lemma is
obvious.

######  Lemma 3.2.

$(1)$ There is an algebra isomorphism
$R(A,e,\lambda)/(Ae\otimes_{\Lambda}eA)\simeq A$.

$(2)$ If $\mu\in Z(\Lambda)$ is an invertible element, then
$R(A,e,\lambda)\simeq R(A,e,\lambda\mu)$ as algebras.

For simplicity, let
$R:=R(A,e),\;\;S:=S(A,e)\;\;\mbox{and}\;\;\bar{e}:=e\otimes e\in R.$ Then
$\bar{e}=\bar{e}^{2}$, $e\bar{e}=\bar{e}=\bar{e}e$, and
$\\{\bar{e},e-\bar{e},1-e\\}$ is a set of orthogonal idempotents in $R$. Now,
we define

$\pi_{1}:R\longrightarrow A,\;\;a+\sum_{i=1}^{n}a_{i}\bar{e}b_{i}\mapsto
a,\mbox{ and }\;\pi_{2}:R\longrightarrow
A,\;\;a+\sum_{i=1}^{n}a_{i}\bar{e}b_{i}\mapsto a+\sum_{i=1}^{n}a_{i}eb_{i}$

for $a,a_{i},b_{i}\in A$ and $1\leq i\leq n.$ Then $\pi_{1}$ and $\pi_{2}$ are
surjective homomorphisms of algebras. Let

$I:={\rm Ker}(\pi_{1}),\;\;J:={\rm
Ker}(\pi_{2})\;\;\mbox{and}\;\;e_{0}:=(1-e)+\overline{e}.$

######  Lemma 3.3.

$(1)$ $I=R\bar{e}R$, $J=R(e-\bar{e})R$, $IJ=0=JI$, $I+J=ReR$ and
$S=e_{0}Re_{0}$.

$(2)$ As an $A$-$A$-bimodule, ${}_{A}R_{A}$ has two decompositions: $R=A\oplus
I=A\oplus J.$

$(3)$ There is an algebra isomorphism $\phi:R\rightarrow
R,\;\;a+\sum_{i=1}^{n}a_{i}\bar{e}b_{i}\mapsto
a+\sum_{i=1}^{n}a_{i}(e-\bar{e})b_{i}.$ Moreover, $\phi^{2}={\rm Id}_{R}$ and
the restriction of $\phi$ to $I$ induces an isomorphism $I\to J$ of
$A$-$A$-bimodules.

$(4)$ $\pi_{2}=\phi\pi_{1}$ and both $\pi_{1}$ and $\pi_{2}$ induce surjective
homomorphisms of algebras

$\pi_{1}^{\prime}:S\longrightarrow(1-e)A(1-e)\quad\mbox{and}\quad\pi_{2}^{\prime}:S\longrightarrow
A,$

respectively. Moreover, ${\rm Ker}(\pi_{1}^{\prime})=I$ and ${\rm
Ker}(\pi_{2}^{\prime})=(1-e)J(1-e)=J\cap S$.

Proof. $(1)$ Clearly, $I=Ae\otimes_{\Lambda}eA=A\bar{e}A=R\bar{e}R$. Since
$(e-\bar{e})\pi_{2}=0$, we have $e-\bar{e}\in{\rm Ker}(\pi_{2})=J$ and
$R(e-\bar{e})R\subseteq J$. Conversely, if
$r:=a+\sum_{i=1}^{n}a_{i}\bar{e}b_{i}\in J$, then
$a+\sum_{i=1}^{n}a_{i}eb_{i}=(r)\pi_{2}=0$, that is,
$a=-\sum_{i=1}^{n}a_{i}eb_{i}$. Consequently,
$r=-\sum_{i=1}^{n}a_{i}eb_{i}+\sum_{i=1}^{n}a_{i}\bar{e}b_{i}=-\sum_{i=1}^{n}a_{i}(e-\bar{e})b_{i}\in
R(e-\bar{e})R$. Thus $J=R(e-\bar{e})R=A(e-\bar{e})A$. Note that
$I+J=R\bar{e}R+R(e-\bar{e})R=ReR$. For any $x,y,x^{\prime},y^{\prime}\in A$,
since
$(x\bar{e}y)(x^{\prime}(e-\bar{e})y^{\prime})=x\bar{e}yx^{\prime}ey^{\prime}-x\bar{e}yx^{\prime}ey^{\prime}=0$,
we have $IJ=0$. Similarly, $(x^{\prime}(e-\bar{e})y^{\prime})(x\bar{e}y)=0$,
and therefore $JI=0$. Since $I$ is an ideal of $R$ and $IJ=JI=0$, it follows
that $S=e_{0}Re_{0}$.

$(2)$ $R$ contains $A$ as a subalgebra with the same identity, and the
composite of the inclusion $A\subseteq R$ with $\pi_{i}$ for $i=1,2$, is the
identity map of $A$. Thus $(2)$ follows.

$(3)$ By $(2)$, $I\simeq R/A\simeq J$ as $A$-$A$-bimodules. In fact, the
isomorphism from $I$ to $J$ is given by

$\varphi^{\prime}:\;I\longrightarrow
J,\;\;\sum_{i=1}^{n}a_{i}\bar{e}b_{i}\mapsto\sum_{i=1}^{n}a_{i}(e-\bar{e})b_{i}.$

Then the map $\phi:R=A\oplus I\to R=A\oplus J$ induced from $\varphi^{\prime}$
is a well-defined isomorphism of $A$-$A$-bimodules. Moreover, $\phi$ preserves
the multiplication of $R$ and $\phi^{2}={\rm Id}_{R}$. Thus $\phi$ is an
automorphism of algebras.

$(4)$ The first equality in $(4)$ follows from the definitions of $\phi$,
$\pi_{1}$ and $\pi_{2}$. To see other statements in $(4)$, we apply the left
and right multiplications by $e_{0}$ to $\pi_{1}$ and $\pi_{2}$, and then use
$(1)$. $\square$

Under some conditions, the associated ideals $I$ and $J$ are related by
annihilators of modules. Recall that the _annihilator_ of an $R$-module $M$ is
defined as ${\rm Ann}_{R}(M):=\\{r\in R\mid rM=0\\}$, which is an ideal of
$R$.

######  Lemma 3.4.

$(1)$ If $eA_{A}$ is a faithful right module, then $J={\rm Ann}_{R^{\rm
op}}(I)$. Dually, if ${}_{A}Ae$ is faithful, then $J={\rm Ann}_{R}(I)$.

$(2)$ The map $\pi_{2}$ induces isomorphisms of abelian groups:

$R\bar{e}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}Ae,\;\;\bar{e}R\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}eA\;\;\mbox{and}\;\;\bar{e}R\bar{e}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}eAe.$

The map $\pi_{2}^{\prime}:S\to A$ in Lemma 3.3(4) induces isomorphisms of
abelian groups:

$S\bar{e}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}Ae,\;\;\bar{e}S\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}eA\;\;\mbox{and}\;\;\bar{e}S\bar{e}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}eAe.$

$(3)$ The map $\pi_{1}$ induces isomorphisms of abelian groups:

$R(e-\bar{e})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}Ae,\;\;(e-\bar{e})R\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}eA\;\;\mbox{and}\;\;(e-\bar{e})R(e-\bar{e})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}eAe.$

Proof. $(1)$ Since $IJ=0$ by Lemma 3.3(1), $J\subseteq{\rm Ann}_{R^{\rm
op}}(I)$. Let $x:=a+\sum_{i=1}^{n}a_{i}\bar{e}b_{i}\in{\rm Ann}_{R^{\rm
op}}(I)$ with $a,a_{i},b_{i}\in A$ and $1\leq i\leq n\in\mathbb{N}$. Since
$J={\rm Ker}(\pi_{2})$, it suffices to show $y:=(x)\pi_{2}=0$. In fact, by
$Ix=0$, we have $0=(Ix)\pi_{2}=(I)\pi_{2}y=AeAy$. This implies $eAy=0$. In
other words, $y\in{\rm Ann}_{A^{\rm op}}(eA)$. Since $eA_{A}$ is faithful,
${\rm Ann}_{A^{\rm op}}(eA)=0$. Thus $y=0$. This shows $J={\rm Ann}_{R^{\rm
op}}(I)$. Similarly, we show the second identity.

$(2)$ Since $(\bar{e})\pi_{2}=e$, the restriction $f_{2}:R\bar{e}\to Ae$ of
$\pi_{2}$ to $R\bar{e}$ is surjective. As ${\rm Ker}(f_{2})=R\bar{e}\cap
J\subseteq JI=0$ by Lemma 3.3(1), $f_{2}$ is an isomorphism. Dually, the
restriction $\bar{e}R\to eA$ of $\pi_{2}$ to $\bar{e}R$ is also an
isomorphism. Consequently, $\pi_{2}$ induces an algebra isomorphism from
$\bar{e}R\bar{e}$ to $eAe$.

Since $IJ=JI=0$ by Lemma 3.3(1), we have $S\overline{e}=R\overline{e}$ and
$\overline{e}S=\overline{e}R$. Clearly,
$\overline{e}S\overline{e}=\overline{e}R\overline{e}$. Thus the second
statement in $(2)$ holds.

$(3)$ This follows from $(2)$ and Lemma 3.3(3)-(4). $\square$

Consequently, Lemma 3.3(1) and Lemma 3.4(2) imply $\\#(R)=\\#(A)+\\#(eAe)$.

######  Proposition 3.5.

Let $A_{2}:={\rm End}_{R}(R\oplus R/I)$ and $B_{2}:={\rm End}_{S}(S\oplus
S/I)$. Suppose that the right $A$-module $eA_{A}$ is faithful. Then the
following hold true.

$(1)$ There are standard recollements of derived module categories

$\textstyle{\mathscr{D}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{D}(A_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{D}(A),\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\mathscr{D}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{D}(B_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{D}((1-e)A(1-e))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

induced by finitely generated, right-projective idempotent ideals of $A_{2}$
and $B_{2}$, respectively.

$(2)$ ${\rm st.dim}(A_{2})\geq 2{\rm st.dim}(A)+1$ and ${\rm
st.dim}(B_{2})\geq{\rm st.dim}(A)+{\rm st.dim}((1-e)A(1-e))+1$.

$(3)$ ${\rm gl.dim}(A^{\rm op})\leq{\rm gl.dim}(A_{2}^{\rm op})\leq 2{\rm
gl.dim}(A^{\rm op})+2,$ ${\rm fin.dim}(A^{\rm op})\leq{\rm fin.dim}(A_{2}^{\rm
op})\leq 2{\rm fin.dim}(A^{\rm op})+2.$

Proof. $(1)$ Since $eA_{A}$ is faithful, $J={\rm Ann}_{R^{\rm op}}(I)$ by
Lemma 3.4(1). Note that $I$ is an ideal of $S$ and ${\rm Ann}_{S^{\rm
op}}(I)=S\cap{\rm Ann}_{R^{\rm op}}(I)=S\cap J$. By Lemma 3.3(4), there are
algebra isomorphisms $A\simeq R/I\simeq R/J\simeq S/(S\cap J)$ and
$S/I\simeq(1-e)A(1-e)$. Now, Proposition 3.5 follows from Lemma 2.5(1).

$(2)$ This follows from $(1)$ and Proposition 2.9(3).

$(3)$ This is a consequence of Proposition 3.5(1) and [8, Corollary 3.12 and
Theorem 3.17]. $\square$

Now, we consider $n$-idempotent and strong idempotent ideals of mirror-
reflective algebras.

######  Proposition 3.6.

$(1)$ The ideals $I$ and $J$ of $R$ are $2$-idempotent.

$(2)$ Let $n\geq 1$ be an integer. Then $I$ is $(n+2)$-idempotent if and only
if so is $J$ if and only if ${\rm Tor}_{i}^{eAe}(Ae,eA)=0$ for all $1\leq
i\leq n$.

$(3)$ If ${\rm Tor}_{i}^{eAe}(Ae,eA)=0$ for all $i\geq 1$, then there are
standard recollements of derived module categories induced by $I:=R\bar{e}R$:

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Proof. $(1)$ There is a commutative diagram

$\textstyle{R\overline{e}\otimes_{\overline{e}R\overline{e}}\overline{e}R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\pi_{2}\otimes\pi_{2}}$$\textstyle{R\overline{e}R=Ae\otimes_{eAe}eA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\textstyle{Ae\otimes_{eAe}eA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\prime}}$$\textstyle{AeA}$

where $\mu$ and $\mu^{\prime}$ are given by multiplications. By Lemma 3.4(2),
$\pi_{2}\otimes\pi_{2}$ is an isomorphism. Note that the composition of the
inverse of $\pi_{2}\otimes\pi_{2}$ with $\mu$ is the identity of
$Ae\otimes_{eAe}eA$. Thus $\mu$ is an isomorphism. This shows that $I$ is
$2$-idempotent by Lemma 2.3(1). Similarly, we can show that $J$ is
$2$-idempotent by using the idempotent $e-\overline{e}$ and the algebra
homomorphism $\pi_{1}$.

$(2)$ By Lemma 3.3(3), $I$ is $(n+2)$-idempotent if and only if so is $J$.
Since $I$ is $2$-idempotent by $(1)$, it follows from Lemma 2.3(1) that $I$ is
$(n+2)$-idempotent if and only if ${\rm
Tor}_{i}^{\overline{e}R\overline{e}}(R\overline{e},\overline{e}R)=0$ for
$1\leq i\leq n$. By Lemma 3.4(2), $\pi_{2}$ induces isomorphisms of abelian
groups ${\rm
Tor}_{i}^{\overline{e}R\overline{e}}(R\overline{e},\overline{e}R)\simeq{\rm
Tor}_{i}^{eAe}(Ae,eA)$ for all $i\in\mathbb{N}$. Thus $I$ is
$(n+2)$-idempotent if and only if ${\rm Tor}_{i}^{eAe}(Ae,eA)=0$ for $1\leq
i\leq n$.

$(3)$ By $(2)$, $I$ is a strong idempotent ideal of $R$ if and only if ${\rm
Tor}_{i}^{eAe}(Ae,eA)=0$ for all $i\geq 1$. According to Corollary 2.4(1), if
$I$ is a strong idempotent ideal of $R$, then $e_{0}Ie_{0}$ is a strong
idempotent ideal of $S$. By Lemma 3.3 and Lemma 3.4(2), $e_{0}Ie_{0}=I$,
$S/I\simeq(1-e)A(1-e)$, $R/I\simeq A$ and $\overline{e}R\overline{e}\simeq
eAe\simeq\overline{e}S\overline{e}$. Thus the recollements in $(3)$ exist.
$\square$

To discuss the decomposition of $R$ as an algebra and to lift algebra
homomorphisms, we show the following result. For a homomorphism
$\alpha:A\to\Gamma$ of algebras, denote by ${\rm Hom}_{\alpha\mbox{-}\rm
Alg}(R,\Gamma)$ the set of all algebra homomorphisms $\beta:R\to\Gamma$ such
that the restriction of $\beta$ to $A$ coincides with $\alpha$.

######  Lemma 3.7.

$(1)$ If $u=u^{2}\in A$ such that ${\rm add}({{}_{A}}Au)={\rm
add}({{}_{A}}Ae)$, then $R\simeq R(A,u,u)$ as algebras.

$(2)$ If ${{}_{A}}Ae$ is a generator, then $R\simeq A\times A$ as algebras.

$(3)$ If $\Gamma$ is an algebra and $\alpha:A\to\Gamma$ is an algebra
homomorphism, then there is a bijection

${\rm Hom}_{\alpha\mbox{-}\rm
Alg}(R,\Gamma)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\\{x\in(e)\alpha\;\Gamma\;(e)\alpha\mid
x^{2}=x,\;(c)\alpha\,x=x\,(c)\alpha\;\;\mbox{ for
}\;\;c\in\Lambda\\},\;\;\overline{\alpha}\mapsto(\overline{e})\overline{\alpha}.$

Proof. $(1)$ Let $U:=uAu$. We keep the notation in the proof of Lemma 2.3 and
identify ${\rm Hom}_{A}(Au,-):A\mbox{{\rm-Mod}}\to U\mbox{{\rm-Mod}}$ with the
functor $u\cdot:A\mbox{{\rm-Mod}}\to U\mbox{{\rm-Mod}}$, given by the left
multiplication of $u$. Let $\mu:Au\otimes_{U}u(-)\to{\rm Id}$ be the counit
adjunction of the adjoint pair $(Au\otimes_{U}-,u\cdot)$. Then, for an
$A$-module $X$, the map $\mu_{X}$ is an isomorphism if and only if
$X\in\mathbf{P}_{1}(Au)$. Applying $Ae\otimes_{\Lambda}-$ to a projective
presentation of ${{}_{\Lambda}}eA$, we obtain an exact sequence $P_{1}\to
P_{0}\to Ae\otimes_{\Lambda}eA\to 0$ of $A$-modules with $P_{1},P_{0}\in{\rm
Add}(Ae)$. This shows $Ae\otimes_{\Lambda}eA\in\mathbf{P}_{1}(Ae)$. Due to
${\rm add}({{}_{A}}Au)={\rm add}({{}_{A}}Ae)$, we have
$Ae\otimes_{\Lambda}eA\in\mathbf{P}_{1}(Au)$, and therefore
$\mu_{Ae\otimes_{\Lambda}eA}:Au\otimes_{U}u(Ae\otimes_{\Lambda}eA)\to
Ae\otimes_{\Lambda}eA$ is an isomorphism of $A$-$A$-modules. Since the
multiplication map $\rho:Ae\otimes_{\Lambda}eA\to A,ae\otimes eb\mapsto aeb$
for $a,b\in A$, satisfies $e{\rm Ker}(\rho)=0=e{\rm Coker}(\rho)$, it follows
from ${\rm add}({{}_{A}}Au)={\rm add}({{}_{A}}Ae)$ that $u{\rm
Ker}(\rho)=0=u{\rm Coker}(\rho)$. Then $u\rho:u(Ae\otimes_{\Lambda}eA)\to uA$
is an isomorphism of $U$-$A$-bimodules, and $u\rho
u:u(Ae\otimes_{\Lambda}eA)u\to uAu$ is an isomorphism of $U$-$U$-bimodules.
Consequently, there is an isomorphism of $A$-$A$-bimodules

$Au\otimes_{U}u\rho:\;\;Au\otimes_{U}u(Ae\otimes_{\Lambda}eA)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}Au\otimes_{U}uA.$

Thus
$\psi:=(Au\otimes_{U}u\rho)^{-1}\mu_{Ae\otimes_{\Lambda}eA}:Au\otimes_{U}uA\to
Ae\otimes_{\Lambda}eA$ is an isomorphism of $A$-$A$-bimodules. In fact, if
$x_{i}\in uAe$ and $y_{i}\in eAu$ with $1\leq i\leq n$ such that
$\sum_{i=1}^{n}x_{i}y_{i}=u$, then $(a(u\otimes
u)b)\psi=a(\sum_{i=1}^{n}x_{i}\otimes y_{i})b.$ This induces an isomorphism of
$A$-$A$-bimodules:

$({\rm Id}_{A},\psi):R(A,u,u)=A\oplus Au\otimes_{U}uA\longrightarrow R=A\oplus
Ae\otimes_{\Lambda}eA,$ $(a,x\otimes y)\mapsto(a,(x\otimes y)\psi)\mbox{ for
}a\in A,x\in Au,y\in uA.$

A verification shows that this is an algebra isomorphism.

$(2)$ Suppose that ${{}_{A}}Ae$ is a generator. Then ${\rm add}(_{A}Ae)={\rm
add}(_{A}A)$. Let $B:=R(A,1,1)$. By $(1)$, $R\simeq B$ as algebras. Now,
identifying $A\otimes_{A}A$ with $A$, we then get $B=A\oplus A$ with the
multiplication given by

$(a_{1},a_{2})(b_{1},b_{2}):=(a_{1}b_{1},a_{1}b_{2}+a_{2}b_{1}+a_{2}b_{2})\;\mbox{
for }a_{1},a_{2},b_{1},b_{2}\in A.$

Clearly, $(1,0)$ is the identity of $B$ and $(1,-1)$ is a central idempotent
of $B$. Thus the map $B\to A\times A$,
$(a_{1},a_{2})\mapsto(a_{1},a_{1}+a_{2})$, is an algebra isomorphism. Thus
$R\simeq B\simeq A\times A$ as algebras.

$(3)$ Note that $\Gamma$ can be regarded as an $A$-$A$-bimodule via $\alpha$
and that any $A$-$A$-bimodule can be considered as a module over the
enveloping algebra $A^{e}:=A\otimes_{k}A^{{}^{\rm op}}$. Define
$F=Ae\otimes_{\Lambda}-\otimes_{\Lambda}eA:\Lambda^{e}\mbox{{\rm-Mod}}\to
A^{e}\mbox{{\rm-Mod}}$ and $G=e(-)e:A^{e}\mbox{{\rm-
Mod}}\to\Lambda^{e}\mbox{{\rm-Mod}}.$ Then there are isomorphisms of
$k$-modules

${\rm Hom}_{A^{\rm e}}(Ae\otimes_{\Lambda}eA,\Gamma)\simeq{\rm
Hom}_{A^{e}}(F(\Lambda),\Gamma)\simeq{\rm
Hom}_{\Lambda^{e}}(\Lambda,G(\Gamma))={\rm
Hom}_{\Lambda^{e}}\big{(}\Lambda,(e)\alpha\,\Gamma\,(e)\alpha\big{)}$
$\quad\quad\quad\quad\quad=\\{y\in(e)\alpha\;\Gamma\,(e)\alpha\mid(c)\alpha\,y=y\,(c)\alpha\;\;\mbox{for
any}\;\;c\in\Lambda\\}=:\Gamma^{\prime}.$

Let $\overline{\alpha}\in{\rm Hom}_{\alpha\mbox{-}\rm Alg}(R,\Gamma)$ and
$x:=(\overline{e})\overline{\alpha}\in\Gamma$. Since the restriction of
$\overline{\alpha}$ to $A$ equals $\alpha$, the restriction of
$\overline{\alpha}$ to $Ae\otimes_{\Lambda}eA$ is an homomorphism of
$A$-$A$-bimodules. By $\overline{e}^{2}=\overline{e}$, we have
$x^{2}=x\in\Gamma^{\prime}$ and $(ae\otimes
eb)\overline{\alpha}=(a)\alpha\,x\,(b)\alpha$ for any $a,b\in A$. This means
that $\overline{\alpha}$ is determined by $\alpha$ and $x$.

Conversely, let $y\in\Gamma^{\prime}$ and let
$h:Ae\otimes_{\Lambda}eA\to\Gamma$ be the homomorphism of $A$-$A$-bimodules
sending $ae\otimes eb$ to $(a)\alpha\;y\;(b)\alpha$. Define
$\overline{h}:=(\alpha,h):R\to\Gamma$. Then $\overline{h}$ is an algebra
homomorphism if and only if $((ae\otimes eb)\ast(a^{\prime}e\otimes
eb^{\prime}))h=(ae\otimes eb)h(a^{\prime}e\otimes eb^{\prime})h$ for any
$a,a^{\prime},b,b^{\prime}\in A$ if and only if
$y(ba^{\prime})\alpha\;y=(eba^{\prime})\alpha\;y$ for any $b,a^{\prime}\in A$.
Now, suppose $y^{2}=y$. Since $\alpha$ is an algebra homomorphism and
$(e)\alpha\;y=y=y\;(e)\alpha$, we see that
$(eba^{\prime})\alpha\;y=(eba^{\prime}e)\alpha\;y=(eba^{\prime}e)\alpha\;y^{2}=y\;(eba^{\prime}e)\alpha
y=y(ba^{\prime})\alpha\,y.$ Thus $\overline{h}$ is an algebra homomorphism. As
$y=(\overline{e})h$, the bijection in $(3)$ is clear. $\square$

######  Proposition 3.8.

Let $A$ be an indecomposable algebra. Then

$(1)$ $R$ is a decomposable algebra if and only if ${{}_{A}}Ae$ is a
generator. In this case, $R\simeq A\times A$ as algebras.

$(2)$ If ${\rm add}(Ae)\cap{\rm add}(A(1-e))=0$ and $(1-e)A(1-e)$ is an
indecomposable algebra, then $S$ is an indecomposable algebra.

Proof. $(1)$ If ${{}_{A}}Ae$ is a generator, then $R\simeq R(A,1,1)\simeq
A\times A$ as algebras by Lemma 3.7(2), and therefore $R$ is decomposable.
Conversely, assume that $R$ is a decomposable algebra. Then there is an
idempotent $z$ in the center $Z(R)$ of $R$ such that $z\neq 0,1$. Since
$\pi_{1}:R\to A$ is a surjective homomorphism of algebras, it restricts to an
algebra homomorphism $Z(R)\to Z(A)$. This implies $(z)\pi_{1}\in Z(A)$. Since
$A$ is indecomposable, $(z)\pi_{1}=0$ or $1$. If $(z)\pi_{1}=0$, then $z\in
I={\rm Ker}(\pi_{1})$. If $(z)\pi_{1}=1$, then $1-z\in I$. So, without loss of
generality, we can assume $z\in I$. Similarly, $z\in J$ or $1-z\in J$ by
$\pi_{2}$. If $z\in J$, then $z=z^{2}\in IJ=0$ by Lemma 3.3(1), a
contradiction. Thus $1-z\in J$, and $1=z+(1-z)\in I+J=ReR$ by Lemma 3.3(1).
This shows $ReR=R$ and implies $AeA=A$ by $\pi_{1}$. Hence ${{}_{A}}Ae$ is a
generator.

$(2)$ Let $J_{1}:=S\cap J$. In the proof of $(1)$, we replace $\pi_{1}$ and
$\pi_{2}$ with $\pi_{1}^{\prime}:S\to(1-e)A(1-e)$ and $\pi_{2}^{\prime}:S\to
A$ (see Lemma 3.3(4)), respectively, and show similarly that if $(1-e)A(1-e)$
is indecomposable and $S$ is decomposable, then $S=I+J_{1}$. In this case, the
equality $A=AeA$ still holds because $\pi_{2}^{\prime}$ is surjective with
${\rm Ker}(\pi_{2}^{\prime})=J_{1}$ and $(\overline{e})\pi_{2}^{\prime}=e$.
Consequently, ${{}_{A}}Ae$ is a generator, and therefore the assumption ${\rm
add}(Ae)\cap{\rm add}(A(1-e))=0$ forces $e=1$. Thus $S=I\simeq A$ as algebras.
This contradicts to $A$ being indecomposable. $\square$

###  3.2 Examples of mirror-reflective algebras: quivers with relations

In this subsection, we describe explicitly the mirror-reflective algebras for
algebras presented by quivers with relations. This explains the terminology
“mirror-reflective algebras” (see Example 3.11 below).

Let $Q:=(Q_{0},Q_{1})$ be a quiver with the vertex set $Q_{0}$ and arrow set
$Q_{1}$. For an arrow $\alpha:i\to j$, we denote by $s(\alpha)$ and
$t(\alpha)$ the starting vertex $i$ and the terminal vertex $j$, respectively.
Composition of an arrow $\alpha:i\to j$ with an arrow $\beta:j\to m$ is
written as $\alpha\beta$. A _path_ of length $n\geq 0$ in $Q$ is a sequence
$p:=\alpha_{1}\cdots\alpha_{n}$ of $n$ arrows $\alpha_{i}$ in $Q_{1}$ such
that $t(\alpha_{i})=s(\alpha_{i+1})$ for $1\leq i<n\in\mathbb{N}$. Set
$s(p)=s(\alpha_{1})$ and $t(p)=t(\alpha_{n})$. In case of $n=0$, we understand
the trivial path as an vertex $i\in Q_{0}$, denote by $e_{i}$ and set
$s(e_{i})=i=t(e_{i})$. We write $\mathscr{P}(Q)$ for the set of all paths of
finite length in $Q$. For a field $k$, we write $kQ$ for the path algebra of
$Q$ over $k$. Clearly, it has $\mathscr{P}(Q)$ as a $k$-basis.

A _relation_ $\sigma$ on $Q$ over $k$ is a $k$-linear combination of paths
$p_{i}$ of length at least $2$. We may assume that all paths in a relation
have the same starting vertex and terminal vertex, and define
$s(\sigma)=s(p_{i})$ and $t(\sigma)=t(p_{i})$. If $\rho=\\{\sigma_{i}\\}_{i\in
T}$ is a set of relations on $Q$ over $k$ with $T$ an index set, the pair
$(Q,\rho)$ is called a _quiver with relations_ over $k$. In this case, we have
a $k$-algebra $k(Q,\rho):=kQ/\langle\rho\rangle$, the quotient algebra of the
path algebra $kQ$ modulo the ideal $\langle\rho\rangle$ generated by the
relations $\sigma_{i},i\in T$.

######  Lemma 3.9.

Let $B$ be a $k$-algebra, $\\{f_{i}\mid i\in Q_{0}\\}$ a set of orthogonal
idempotents in $B$ with $1_{B}=\sum_{i\in Q_{0}}f_{i}$, and
$\\{f_{\alpha}\mid\alpha\in Q_{1}\\}$ a set of elements in $B$. If
$f_{s(\alpha)}f_{\alpha}=f_{\alpha}=f_{\alpha}f_{t(\alpha)}$ for $\alpha\in
Q_{1}$, then there is a unique algebra homomorphism $f:kQ\to B$ which sends
$e_{i}\mapsto f_{i}$ and $\alpha\mapsto f_{\alpha}$.

Let $Q^{\prime}:=(Q^{\prime}_{0},Q^{\prime}_{1})$ be a full subquiver of $Q$,
that is, $Q^{\prime}_{0}\subseteq Q_{0}$ and $Q^{\prime}_{1}=\\{\alpha\in
Q_{1}\mid s(\alpha),t(\alpha)\in Q^{\prime}_{0}\\}$. Define

$A:=k(Q,\rho),\;\;V_{0}:=Q_{0}\setminus
Q^{\prime}_{0}\;\;\mbox{and}\;\;e:=\sum_{i\in V_{0}}e_{i}\in A.$

We shall describe the quiver and relations for the mirror-reflective algebra
$R(A,e)$ explicitly.

Let $\overline{Q}$ be a copy of the quiver $Q$, say
$\overline{Q}_{0}=\\{\bar{i}\mid i\in Q_{0}\\}$ and
$\overline{Q}_{1}=\\{\bar{\alpha}\mid\alpha\in Q_{1}\\}$, with
$s(\bar{\alpha})=\bar{i}$ and $t(\bar{\alpha})=\bar{j}$ if $s(\alpha)=i$ and
$t(\alpha)=j$. Consider $Q^{\prime}$ as a full subquiver of $\overline{Q}$ by
identifying $\bar{i}$ with $i$ for $i\in Q^{\prime}_{0}$, and $\bar{\alpha}$
with $\alpha$ for $\alpha\in Q^{\prime}_{1}$. So
$Q_{0}\cap\overline{Q}_{0}=Q^{\prime}_{0}$ and
$Q_{1}\cap\overline{Q}_{1}=Q^{\prime}_{1}$. Let
$\Delta:=(\Delta_{0},\Delta_{1})$ be the pullback of $Q$ and $\overline{Q}$
over $Q^{\prime}$, that is,

$\Delta_{0}:=Q_{0}\dot{\cup}(\overline{Q_{0}}\setminus
Q^{\prime}_{0})\quad\mbox{and}\quad\Delta_{1}:=Q_{1}\dot{\cup}(\overline{Q_{1}}\setminus
Q^{\prime}_{1}).$

We define a map $(-)^{+}:\\{e_{i}\mid i\in Q_{0}\\}\cup Q_{1}\to k\Delta$ by

$e_{i}^{+}:=\left\\{\begin{array}[]{ll}e_{i},&\hbox{$i\in Q^{\prime}_{0}$},\\\
e_{i}+e_{\overline{i}},&\hbox{$i\in
V_{0}$},\end{array}\right.\quad\alpha^{+}:=\left\\{\begin{array}[]{ll}\alpha,&\hbox{$\alpha\in
Q^{\prime}_{1}$,}\\\ \alpha+\overline{\alpha},&\hbox{$\alpha\in Q_{1}\setminus
Q^{\prime}_{1}$.}\end{array}\right.$

Since $e_{s(\alpha)}^{+}\alpha^{+}=\alpha^{+}=\alpha^{+}e_{t(\alpha)}^{+}$ for
any $\alpha\in Q_{1}$, it follows from Lemma 3.9 that $(-)^{+}$ can be
extended to an algebra homomorphism

$(-)^{+}:\quad kQ\longrightarrow k\Delta,\quad p\mapsto
p^{+}:=\alpha_{1}^{+}\cdots\alpha_{n}^{+}\mbox{ for
}p=\alpha_{1}\cdots\alpha_{n}\in\mathscr{P}(Q).$

Given a relation $\sigma:=\sum_{i=1}^{n}a_{i}p_{i}$ on $Q$ with $a_{i}\in k$,
$p_{i}\in\mathscr{P}(Q)$ for $1\leq i\leq n\in\mathbb{N}$, and
$s(\sigma),t(\sigma)\in Q^{\prime}_{0}$, we define

$\sigma_{+}:=\sum_{1\leq j\leq
n,\;p_{j}\in\mathscr{P}(Q^{\prime})}a_{j}p_{j}\;+\sum_{1\leq i\leq
n,\;p_{i}\notin\mathscr{P}(Q^{\prime})}a_{i}(p_{i}+\overline{p_{i}})=\sigma\;+\sum_{1\leq
i\leq n,\;p_{i}\notin\mathscr{P}(Q^{\prime})}a_{i}\overline{p_{i}}.$

Now, let $\psi:=\psi_{1}\cup\psi_{2}\cup\psi_{3}\cup\psi_{4}$ with

$\begin{array}[]{ll}\psi_{1}:=&\\{\overline{a}pb,ap\overline{b}\mid a,b\in
Q_{1},s(a),t(b)\in
V_{0},p\in\mathscr{P}(Q^{\prime}),apb\in\mathscr{P}(Q)\\},\\\
\psi_{2}:=&\\{\sigma\in\rho\mid s(\sigma)\in
V_{0}\;\;\mbox{or}\;\;t(\sigma)\in V_{0}\\},\\\
\psi_{3}:=&\\{\overline{\sigma}\mid\sigma\in\psi_{2}\\},\;\mbox{and}\\\
\psi_{4}:=&\\{\sigma_{+}\mid\sigma\in\rho,s(\sigma),t(\sigma)\in
Q^{\prime}_{0}\\}.\\\ \end{array}$

Then $\psi$ is a set of relations on $\Delta$ over $k$, and we consider the
$k$-algebra $k(\Delta,\psi)$.

######  Proposition 3.10.

$(1)$ The homomorphism $(-)^{+}:kQ\to k\Delta$ of algebras is injective and
induces an injective homomorphism $\mu:A\to k(\Delta,\psi)$ of algebras.

$(2)$ There exists an isomorphism
$\theta:R(A,e)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}k(\Delta,\psi)$
of algebras such that $(e_{i}\otimes e_{i})\theta=e_{\overline{i}}$ for $i\in
V_{0}$, and the restriction of $\theta$ to $A$ coincides with $\mu$ in _(1)_.

Proof. $(1)$ For $\mathcal{U}\subseteq k\Delta$, we denote by
$\langle\mathcal{U}\rangle$ the ideal of $k\Delta$ generated by $\mathcal{U}$.
Let $E:=\\{e_{\overline{i}}\mid i\in V_{0}\\}$. Then $k\Delta/\langle
E\rangle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}kQ$ as algebras. Let
$\delta:k\Delta\to k\Delta/\langle E\rangle$ be the canonical surjection. Then
we have the homomorphisms of algebras

$kQ\stackrel{{\scriptstyle(-)^{+}}}{{\longrightarrow}}k\Delta\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}k\Delta/\langle
E\rangle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}kQ$

such that their composition is the identity map of $kQ$. This shows that
$(-)^{+}$ is injective. Applying the map $(-)^{+}$, we define

$\rho^{+}:=\\{\sigma^{+}\mid\sigma\in\rho\\}\quad\mbox{and}\quad\psi^{\prime}:=\rho^{+}\cup\big{(}\bigcup_{i,j\in
V_{0}}(e_{i}k\Delta e_{\overline{j}}\cup e_{\overline{j}}k\Delta
e_{i})\big{)}.$

We shall show $\langle\psi^{\prime}\rangle=\langle\psi\rangle$ in $k\Delta$.

In fact, let $\varphi=\bigcup_{i,j\in V_{0}}(e_{i}k\Delta e_{\overline{j}}\cup
e_{\overline{j}}k\Delta e_{i})\subseteq\psi^{\prime}$. Clearly,
$\langle\varphi\rangle=\langle\psi_{1}\rangle$. Now, let us consider the image
of a path under $(-)^{+}$.

$(i)$ For $p\in\mathscr{P}(Q)$ of length at least $1$, we have

1) If $p\in\mathscr{P}(Q^{\prime}),$ then $p^{+}=p.$

2) If $p\not\in\mathscr{P}(Q^{\prime}),$ then
$p^{+}=p+\overline{p}+p^{\prime}$ with $p^{\prime}$ in the $k$-space
$k\varphi$ generated by elements of $\varphi$.

$(ii)$ For $\sigma\in\rho$, we write
$\sigma=\sum_{i=1}^{s}a_{i}p_{i}+\sum_{j=s+1}^{n}a_{j}p_{j}$ such that
$p_{i}\in\mathscr{P}(Q^{\prime})$ for $1\leq i\leq s$ and
$p_{j}\not\in\mathscr{P}(Q^{\prime})$ for $s+1\leq j\leq n$. It follows from
$(i)$ that

$(\ast)\quad\sigma^{+}=\sum_{i=1}^{s}a_{i}p^{+}_{i}+\sum_{j=s+1}^{n}a_{j}p^{+}_{j}=\sum_{i=1}^{s}a_{i}p_{i}+\sum_{j=s+1}^{n}a_{j}(p_{j}+\overline{p_{j}}+p^{\prime}_{j})=\sigma+\sum_{j=s+1}^{n}a_{j}\overline{p_{j}}+\sum_{j=s+1}^{n}a_{j}p^{\prime}_{j}$

If $\sigma\in\psi_{2}$, then $s=0$ and
$\sigma^{+}=\sigma+\overline{\sigma}+\sum_{j=1}^{n}a_{j}p^{\prime}_{j}$ with
$\overline{\sigma}\in\psi_{3}$, and therefore
$\sigma^{+}\in\langle\psi\rangle$. If $\sigma\not\in\psi_{2}$, that is
$s(\sigma),t(\sigma)\in Q_{0}^{\prime}$, then $\sigma_{+}\in\psi_{4}$ and
$\sigma^{+}=\sigma_{+}+\sum_{j=s+1}^{n}a_{j}p^{\prime}_{j}\in\langle\psi\rangle$.
Thus $\langle\psi^{\prime}\rangle\subseteq\langle\psi\rangle$ in $k\Delta$.

Conversely, pick up $\tau\in\psi$, we show
$\tau\in\langle\psi^{\prime}\rangle.$ If $\tau=\sigma_{+}\in\psi_{4}$, then
$\tau=\sigma^{+}-\sum_{j=s+1}^{n}a_{j}p^{\prime}_{j}\in\langle\psi^{\prime}\rangle.$
If $\tau=\sigma\in\psi_{2}$ and $s(\sigma)\in V_{0}$, then
$e_{s(\sigma)}\overline{\sigma}=0$ and therefore
$\sigma=e_{s(\sigma)}\sigma=e_{s(\sigma)}\sigma^{+}-e_{s(\sigma)}\sum_{j=1}^{n}a_{j}p^{\prime}_{j}\in\langle\psi^{\prime}\rangle.$
If $\tau=\sigma\in\psi_{2}$ and $t(\sigma)\in V_{0}$, then
$\overline{\sigma}e_{t(\sigma)}=0$ and $\sigma=\sigma
e_{t(\sigma)}=\sigma^{+}e_{t(\sigma)}-\sum_{j=1}^{n}a_{j}p^{\prime}_{j}e_{t(\sigma)}\in\langle\psi^{\prime}\rangle.$
If $\tau=\overline{\sigma}\in\psi_{3}$ with $\sigma\in\psi_{2}$, then
$\overline{\sigma}=\sigma^{+}-\sigma-\sum_{j=1}^{n}a_{j}p^{\prime}_{j}$. By
what we have just proved, $\sigma\in\langle\psi^{\prime}\rangle$, and
therefore $\overline{\sigma}\in\langle\psi^{\prime}\rangle$. Thus
$\langle\psi\rangle\subseteq\langle\psi^{\prime}\rangle,$ and therefore
$\langle\psi^{\prime}\rangle=\langle\psi\rangle$ and
$k(\Delta,\psi^{\prime})=k(\Delta,\psi)$.

Since $\varphi\subseteq\langle E\rangle$, it is clear that
$\langle\psi^{\prime}\rangle\subseteq\langle\rho^{+}\cup E\rangle$. By the
third equality in $(\ast)$ and the fact that
$\sum_{j=s+1}^{n}a_{j}\overline{p_{j}}$ and
$\sum_{j=s+1}^{n}a_{j}p^{\prime}_{j}$ belong to $\langle E\rangle$, we obtain
$\langle\rho^{+}\cup E\rangle=\langle\rho\cup E\rangle$ in $k\Delta.$ Thus
$k\Delta/\langle\rho^{+}\cup E\rangle=k\Delta/\langle\rho\cup E\rangle\simeq
kQ/\langle\rho\rangle=A$ as algebras. Moreover, since
$\langle\rho^{+}\rangle\subseteq\langle\psi^{\prime}\rangle\subseteq\langle\rho^{+}\cup
E\rangle\subseteq k\Delta,$ the homomorphisms $(-)^{+}$ and $\delta$ induce
algebra homomorphisms $\mu:A\to k\Delta/\langle\psi^{\prime}\rangle$ and
$\overline{\delta}:k\Delta/\langle\psi^{\prime}\rangle\to
k\Delta/\langle\rho^{+}\cup E\rangle$, respectively. Now, we identify
$k\Delta/\langle\rho^{+}\cup E\rangle$ with $A$. Then
$\mu\overline{\delta}={\rm Id}_{A}$ and $\mu$ is injective.

$(2)$ We first construct a map $\theta$ by applying Lemma 3.7(3). For
simplicity, let

$R:=R(A,e),\;S:=k(\Delta,\psi),\;x:=\sum_{i\in V_{0}}e_{\overline{i}}\in S.$

Then $x^{2}=x$. By $(1)$, $(e)\mu=e^{+}=\sum_{j\in
V_{0}}(e_{j}+e_{\overline{j}})$. Since
$e^{+}e_{\overline{i}}=e_{\overline{i}}=e_{\overline{i}}e^{+}$, we have
$e^{+}x=x=xe^{+}$ and $x\in e^{+}Se^{+}$. Recall that
$e_{j}Se_{\overline{i}}=e_{\overline{i}}Se_{j}=0$ for $i,j\in V_{0}$, due to
the relation set $\psi_{1}$. Thus, for $s\in S$, we have

$e^{+}se^{+}x=e^{+}sx=\sum_{j\in V_{0}}\sum_{i\in
V_{0}}(e_{j}+e_{\overline{j}})se_{\overline{i}}=\big{(}\sum_{j\in
V_{0}}e_{\overline{j}}\big{)}s\big{(}\sum_{i\in
V_{0}}e_{\overline{i}}\big{)},$ $xe^{+}se^{+}=xse^{+}=\sum_{i\in
V_{0}}\sum_{j\in
V_{0}}e_{\overline{i}}s(e_{j}+e_{\overline{j}})=\big{(}\sum_{i\in
V_{0}}e_{\overline{i}}\big{)}s\big{(}\sum_{j\in
V_{0}}e_{\overline{j}}\big{)}.$

This shows $e^{+}se^{+}x=xe^{+}se^{+}$. Since $\Lambda=eAe$ and
$(\Lambda)\mu\subseteq e^{+}Se^{+}$, we have $(c)\mu x=xe^{+}(c)\mu$ for any
$c\in\Lambda$. By Lemma 3.7(3), there is a unique algebra homomorphism
$\theta:R\to S$ such that the restriction of $\theta$ to $A$ equals $\mu$ and
$(\overline{e})\theta=x$. Let $\overline{e_{i}}:=e_{i}\otimes e_{i}\in R$.
Then $\overline{e_{i}}=e_{i}\overline{e}e_{i}$ and
$(\overline{e_{i}})\theta=e_{i}^{+}xe_{i}^{+}=e_{i}^{+}(\sum_{i\in
V_{0}}e_{\overline{i}})e_{i}^{+}=e_{\overline{i}}$.

Next, we prove that $\theta$ is surjective. It suffices to show that
$\Delta_{1}\subseteq{\rm Im}(\theta)$ and $e_{t}\in{\rm Im}(\theta)$ for
$t\in\Delta_{0}$.

In fact, if $t\in Q^{\prime}_{0}$, then $(e_{t})\theta=(e_{t})\mu=e_{t}$; if
$t\in V_{0}$, then $(\overline{e_{t}})\theta=e_{\overline{t}}$ and
$(e_{t}-{\overline{e_{t}}})\theta=e_{t}+e_{\overline{t}}-e_{\overline{t}}=e_{t}$.
This implies that $e_{t}\in{\rm Im}(\theta)$ for any $t\in\Delta_{0}$. Now,
let $\alpha:u\to v\in Q_{1}$. If $u,v\in Q^{\prime}_{0}$, then
$(\alpha)\theta=\alpha$. If $u\in V_{0}$ or $v\in V_{0}$, then
$(\alpha)\theta=(\alpha)\mu=\alpha+\overline{\alpha}$. In case of $u\in
V_{0}$, we get

$(\overline{e_{u}}\alpha)\theta=(\overline{e_{u}})\theta(\alpha)\theta=e_{\overline{u}}(\alpha)\mu=e_{\overline{u}}(\alpha+\overline{\alpha})=\overline{\alpha}\quad\mbox{and}\quad(\alpha-\overline{e_{u}}\alpha)\theta=\alpha.$

In case of $v\in V_{0}$, we have
$(\alpha\overline{e_{v}})\theta=\overline{\alpha}$ and
$(\alpha-\alpha\overline{e_{v}})\theta=\alpha$. Thus $Q_{1}\subseteq{\rm
Im}(\theta)$ and $\overline{Q_{1}}\setminus Q^{\prime}_{1}\subseteq{\rm
Im}(\theta)$.

Finally, we construct an algebra homomorphism $\pi:S\to R$ such that
$\theta\pi={\rm Id}_{R}$, the identity map of $R$. This means that $\theta$ is
injective. Hence it is bijective.

We define a map $\\{e_{t}\mid t\in\Delta_{0}\\}\cup\Delta_{1}\to R$ by
$e_{i}\mapsto
e_{i}-\overline{e_{i}},\;\;e_{\overline{i}}\mapsto\overline{e_{i}}\;\mbox{ for
}i\in V_{0};\quad e_{j}\mapsto e_{j}\;\mbox{ for }j\in Q^{\prime}_{0};$ and
for $\alpha\in Q_{1}$,

$i\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}j\mapsto\left\\{\begin{array}[]{llll}\alpha&\hbox{$i,j\in
Q^{\prime}_{0}$},\\\ \alpha-\alpha\overline{e_{j}},&\hbox{$i\in Q_{0},j\in
V_{0}$},\\\ \alpha-\overline{e_{i}}\alpha,&\hbox{$i\in V_{0},j\in
Q_{0};$}\end{array}\right.\quad\overline{i}\stackrel{{\scriptstyle\overline{\alpha}}}{{\longrightarrow}}\overline{j}\mapsto\left\\{\begin{array}[]{ll}\alpha\overline{e_{j}},&\hbox{$i\in
Q_{0},j\in V_{0}$},\\\ \overline{e_{i}}\alpha,&\hbox{$i\in V_{0},j\in
Q_{0},$}\end{array}\right.\quad$

Note that $\overline{e_{i}}\alpha=e_{i}\otimes\alpha=\alpha\otimes
e_{j}=\alpha\overline{e_{j}}$ in $R$ for $i,j\in V_{0}$. By Lemma 3.9, the map
can be extended to a unique algebra homomorphism $\gamma:k\Delta\to R$.
Clearly, $\gamma$ preserves the idempotents corresponding to the vertices in
$Q^{\prime}_{0}$ and also the arrows in $Q^{\prime}_{1}$. Further, if $i\in
V_{0}$, then $(e_{i}^{+})\gamma=(e_{i}+e_{\overline{i}})\gamma=e_{i}$; if
$\alpha\in Q_{1}\setminus Q^{\prime}_{1}$, then
$(\alpha^{+})\gamma=(\alpha+\overline{\alpha})\gamma=\alpha$. This implies
$(\sigma^{+})\gamma=\sigma$ for any $\sigma\in\rho$. Moreover, by Lemma
3.3(1),

$(e_{i}k\Delta
e_{\overline{j}})\gamma\subseteq(e_{i}-\overline{e_{i}})R\overline{e_{j}}\subseteq(e-\overline{e})R\overline{e}=0\;\;\mbox{and}\;\;(e_{\overline{j}}k\Delta
e_{i})\varphi\subseteq\overline{e_{j}}R(e_{i}-\overline{e_{i}})\subseteq\overline{e}R(e-\overline{e})=0$

for any $i,j\in V_{0}$. Consequently, we have
$\langle\psi^{\prime}\rangle\subseteq{\rm Ker}(\gamma)$, and therefore
$\gamma$ induces an algebra homomorphism $\pi:S\to R$. Now, let
$g:=\theta\pi:R\to R$ and $h:=(-)^{+}\;\gamma:kQ\to R$. Since the restriction
of $\theta$ to $A$ equals $\mu$, the restriction $g|_{A}:A\to R$ of $g$ to $A$
is induced from $h.$ As $\gamma$ preserves the idempotents corresponding to
the vertices in $Q_{0}$ and also the arrows in $Q_{1}$, we see that $g|_{A}$
has its image in $A$ and factorizes through ${\rm Id}_{A}$. Since
$(\overline{e_{i}})g=(e_{\overline{i}})\pi=\overline{e_{i}}$ for $i\in V_{0}$
and $\overline{e}=\sum_{i\in V_{0}}\overline{e_{i}}$, we have
$(\overline{e})g=\overline{e}$. Thus $g={\rm Id}_{R}$ by Lemma 3.7(3).
$\square$

Now, let us illustrate the construction $R(A,e)$ by an example.

######  Example 3.11.

Suppose that $A$ is an algebra over a field $k$ presented by the quiver with
relations:

---
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\sigma}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\tau}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta\;,}$$\scriptstyle{\theta}$$\textstyle{\eta^{2}=\sigma\eta=\tau\eta=\alpha\gamma=\delta\beta\tau=0,\;\;\beta\gamma=\beta\tau\theta.}$$\textstyle{3}$

Let $Q^{\prime}$ be the full subquiver of $Q$ consisting of the vertex set
$\\{1,2,3\\}$ and let $e=e_{4}+e_{5}$. By Proposition 3.10(2), the algebra
$R(A,e)$ is isomorphic to the algebra presented by the following quiver with
relations:

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\\\ \\\ \\\ \\\
\delta\beta\overline{\tau}=\overline{\delta}\beta\tau=\delta\alpha\overline{\tau}=\overline{\delta}\alpha\tau=0,\\\
\eta^{2}=\sigma\eta=\tau\eta=\delta\beta\tau=0,\\\
\overline{\eta}^{2}=\overline{\sigma}\,\overline{\eta}=\overline{\tau}\,\overline{\eta}=\overline{\delta}\beta\,\overline{\tau}=0,\\\
\alpha\gamma=0,\quad\beta\gamma=\beta\tau\theta+\beta\overline{\tau}\,\overline{\theta}.\end{array}\\\
\end{array}$

This quiver is the mirror reflection of the one of $A$ along the full
subquiver $Q^{\prime}$ of $Q$.

##  4 Mirror-reflective algebras and gendo-symmetric algebras

This section is devoted to proofs of all results mentioned in the
introduction. We first show that mirror-reflective algebras of gendo-symmetric
algebras at any levels are symmetric (see Proposition 4.2). By iterating this
procedure, we construct not only gendo-symmetric algebras of increasing
dominant dimensions and higher minimal Auslander-Gorenstein (see Theorem 1.1),
but also recollements of derived module categories of these algebras (see
Theorem 1.2). The constructed recollements are then applied to give a new
formulation of the Tachikawa’s second conjecture for symmetric algebras in
terms of stratified dimensions and ratios (see Theorem 1.3). Consequently, a
sufficient condition is given for the conjecture to hold for symmetric
algebras (see Corollary 1.4).

Throughout this section, all algebras considered are finite-dimensional
algebras over a field $k$.

###  4.1 Relations among mirror-reflective, symmetric and gendo-symmetric
algebras

Let $A$ be an algebra, $e^{2}=e\in A$ and $\Lambda:=eAe$. Suppose that there
is an isomorphism $\iota:eA\to D(Ae)$ of $\Lambda$-$A$-bimodules. Let
$\iota_{e}:=(e)\iota\in D(Ae)={\rm Hom}_{k}(Ae,k).$ Then
$\iota_{e}=e\iota_{e}=\iota_{e}e$. Moreover, $\iota$ is nothing else than the
right multiplication map by $\iota_{e}$. Define
$\zeta:Ae\otimes_{\Lambda}eA\to k$ to be the composite of the maps

$Ae\otimes_{\Lambda}eA\stackrel{{\scriptstyle\;id\otimes\iota\;}}{{\longrightarrow}}Ae\otimes_{\Lambda}D(Ae)\stackrel{{\scriptstyle{\rm
ev}}}{{\longrightarrow}}k$

where ${\rm ev}$ stands for the evaluation map: $ae\otimes f\mapsto(ae)f$ for
$a\in A$ and $f\in D(Ae)$. Then $\zeta$ is given by $(ae\otimes
eb)\zeta=(bae)\iota_{e}=(ebae)\iota_{e}$ for $a,b\in A$. Fix an element
$\lambda\in Z(\Lambda)$, there are associated two maps

$\chi:\;\;R(A,e,\lambda)=A\oplus Ae\otimes_{\Lambda}eA\longrightarrow
k,\;\;a+\sum_{i=1}^{n}a_{i}e\otimes
eb_{i}\;\mapsto\;\sum_{i=1}^{n}(a_{i}e\otimes
eb_{i})\zeta=\sum_{i=1}^{n}(eb_{i}a_{i}e)\iota_{e}\mbox{ for }a_{i},b_{i}\in
A,$ $\gamma:\;\;Ae\otimes_{\Lambda}eA\longrightarrow D(A),\;\;ae\otimes
eb\;\mapsto\;[a^{\prime}\mapsto(eba^{\prime}ae)\iota_{e}\,\mbox{ for
}a,a^{\prime},b\in A]$

of $k$-spaces. They have the properties.

######  Lemma 4.1.

$(1)$ For any $r_{1},r_{2}\in R(A,e,\lambda)$, $(r_{1}\ast
r_{2})\chi=(r_{2}\ast r_{1})\chi$, where $\ast$ denotes the multiplication of
$R(A,e,\lambda)$.

$(2)$ The map $\gamma$ is a homomorphism of $A$-$A$-bimodules. It is an
isomorphism if and only if the map $(\cdot e):{\rm End}_{A^{\rm op}}(A)\to{\rm
End}_{\Lambda^{\rm op}}(Ae)$ induced from the right multiplication by $e$ is
an isomorphism of algebras.

$(3)$ If $\epsilon:D(A)\to k$ denotes the map sending $f\in D(A)$ to $(1)f$,
then $\zeta=\gamma\,\epsilon$.

Proof. $(1)$ It suffices to show $\big{(}(a_{1}+ae\otimes
eb)\ast(a_{2}+a^{\prime}e\otimes
eb^{\prime})\big{)}\chi=\big{(}(a_{2}+a^{\prime}e\otimes
eb^{\prime})\ast(a_{1}+ae\otimes eb)\big{)}\chi$ for any
$a,a^{\prime},b,b^{\prime},a_{1},a_{2}\in A$. However, this follows from
$\big{(}a^{\prime}(ae\otimes eb)\big{)}\zeta=\big{(}(ae\otimes
eb)a^{\prime}\big{)}\zeta$ and $\big{(}(ae\otimes
eb)\otimes(a^{\prime}e\otimes
eb^{\prime})\big{)}\omega_{\lambda}\zeta=\big{(}(a^{\prime}e\otimes
eb^{\prime})\otimes(ae\otimes eb)\big{)}\omega_{\lambda}\zeta,$ by the
definitions of $\zeta$ and $\omega_{\lambda}$ (see Section 3.1 for
definition).

$(2)$ Note that there is a canonical isomorphism
$\varphi:\;\;Ae\otimes_{\Lambda}D(Ae)\rightarrow D({\rm End}_{\Lambda^{\rm
op}}(Ae)),\;\;ae\otimes f\mapsto[g\mapsto(ae)gf]$ for $a\in A$, $f\in D(Ae)$
and $g\in{\rm End}_{\Lambda^{\rm op}}(Ae)$. Let $\vartheta:A\to{\rm
End}_{A^{\rm op}}(A)$ be the isomorphism which sends $a$ to $(a\cdot)$. Then
the composition of the maps

$Ae\otimes_{\Lambda}eA\stackrel{{\scriptstyle\;Ae\otimes\iota\;}}{{\longrightarrow}}Ae\otimes_{\Lambda}D(Ae)\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}D({\rm
End}_{\Lambda^{\rm op}}(Ae))\stackrel{{\scriptstyle D(\cdot
e)}}{{\longrightarrow}}D({\rm End}_{A^{\rm op}}(A))\stackrel{{\scriptstyle
D(\vartheta)}}{{\longrightarrow}}D(A)$

coincides with $\gamma$. Clearly, all the maps above are homomorphisms of
$A$-$A$-bimodules. Thus $\gamma$ is a homomorphism of $A$-$A$-bimodules. Since
$D:k\mbox{{\rm-mod}}\to k\mbox{{\rm-mod}}$ is a duality, $\gamma$ is an
isomorphism if and only if the map $(\cdot e)$ in $(2)$ is an isomorphism of
algebras.

$(3)$ This follows from $(ae\otimes eb)\zeta=(ebae)\iota_{e}$ for $a,b\in A$.
$\square$

From now on, let $(A,e)$ be a _gendo-symmetric_ algebra. Recall that ${\rm
add}(Ae)$ coincides with the full subcategory of $A\mbox{{\rm-mod}}$
consisting of projective-injective $A$-modules. If $e^{\prime}$ is another
idempotent of $A$ such that ${\rm add}(Ae)={\rm add}(Ae^{\prime})$, then the
mirror-reflective algebras $R(A,e)$ and $R(A,e^{\prime})$ are isomorphic as
algebras by Lemma 3.7(1). So, for simplicity, we write $R(A)$ for $R(A,e).$

In the following, we describe $R(A)$ as deformation of a trivial extension.
Let $\Lambda:=eAe$ and $\iota:eA\to D(Ae)$ be an isomorphism of
$\Lambda$-$A$-bimodules (see Lemma 2.12(2)). Then $\Lambda$ is symmetric and
$eA$ is a generator over $\Lambda$. Moreover, there are algebra isomorphisms
$A\simeq{\rm End}_{\Lambda}(eA)$ and $A^{\rm op}\simeq{\rm End}_{\Lambda^{\rm
op}}(Ae)$. By Lemma 4.1(2), there is an isomorphism of $A$-$A$-bimodules:
$\gamma:\;\;Ae\otimes_{\Lambda}eA\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}D(A).$
Since $A\simeq{\rm End}_{\Lambda}(eA)$ and $eA$ is a generator over $\Lambda$,
the functor $e(-)e:A^{e}\mbox{{\rm-Mod}}\to\Lambda^{e}\mbox{{\rm-Mod}}$
between the categories of bimodules induces an algebra isomorphism $Z(A)\to
Z(\Lambda)$. So, for $\lambda\in Z(\Lambda)$, there exists a unique element
$\lambda^{\prime}\in Z(A)$ such that $e\lambda^{\prime}e=\lambda$. Define
$\overline{\omega_{e}}:=(\gamma\otimes\gamma)^{-1}\omega_{e}\gamma:\;\;D(A)\otimes_{A}D(A)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}D(A)$
and $F=Ae\otimes_{\Lambda}-\otimes_{\Lambda}eA:\Lambda^{e}\mbox{{\rm-Mod}}\to
A^{e}\mbox{{\rm-Mod}}.$ We obtain the commutative diagram

$\textstyle{(Ae\otimes_{\Lambda}eA)\otimes_{A}(Ae\otimes_{\Lambda}eA)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega_{e}}$$\scriptstyle{\simeq}$$\scriptstyle{\gamma\otimes\gamma}$$\textstyle{Ae\otimes_{\Lambda}eA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F(\cdot\lambda)}$$\scriptstyle{\gamma}$$\textstyle{Ae\otimes_{\Lambda}eA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{D(A)\otimes_{A}D(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\omega_{e}}}$$\textstyle{D(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\cdot\lambda^{\prime})}$$\textstyle{D(A).}$

Define
$\overline{\omega_{\lambda}}:=\overline{\omega_{e}}(\cdot\lambda^{\prime}):\;\;D(A)\otimes_{A}D(A)\longrightarrow
D(A).$ Now, we extend $\overline{\omega_{\lambda}}$ to a multiplication on the
direct sum $A\oplus D(A)$ by setting

$(A\oplus D(A))\times(A\oplus D(A))\longrightarrow A\oplus
D(A),\;\;\big{(}(a,f),(b,g)\big{)}\mapsto\big{(}ab,ag+fb+(f\otimes
g)\overline{\omega_{\lambda}}\big{)}$

for $a,b\in A$ and $f,g\in D(A)$. Denote by $A\ltimes_{\lambda}D(A)$ the
abelian group $A\oplus D(A)$ with the above-defined multiplication. By Lemma
3.2(1), $A\ltimes_{\lambda}D(A)$ is an algebra with an algebra isomorphism

$\overline{\gamma}:=\left(\begin{array}[]{lc}{\rm Id}_{A}&0\\\
0&\gamma\end{array}\right):\;\;R(A,e,\lambda)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}A\ltimes_{\lambda}D(A).$

Compared with the trivial extension $A\ltimes D(A)$, the following result,
suggested by Kunio Yamagata, shows that $A\ltimes_{\lambda}D(A)$ is also a
symmetric algebra for any $\lambda$.

######  Proposition 4.2.

If $(A,e)$ is a gendo-symmetric algebra, then $R(A,e,\lambda)$ is symmetric
for $\lambda\in Z(\Lambda)$.

Proof. Let $R:=R(A,e,\lambda)$. Applying $\chi:R\to k$, we define a bilinear
form $\widetilde{\chi}:R\times R\to k$, $(r_{1},r_{2})\mapsto(r_{1}\ast
r_{2})\chi$ for $r_{1},r_{2}\in R$. By Lemma 4.1(1), $\widetilde{\chi}$ is
symmetric. To show that $R$ is a symmetric algebra, it suffices to show that
$\widetilde{\chi}$ is non-degenerate.

Let $T:=A\ltimes_{\lambda}D(A)$ and $\psi:=\overline{\gamma}^{-1}\chi:T\to k$.
Since $\overline{\gamma}:R\to T$ is an algebra isomorphism, $\psi$ induces a
symmetric bilinear form $\widetilde{\psi}:T\times T\to k$, $(t_{1},t_{1})\in
T\times T\mapsto(t_{1}t_{2})\psi$. Clearly, $\widetilde{\chi}$ is non-
degenerate if and only if so is $\widetilde{\psi}$. Further, by Lemma 4.1(3),
$\psi$ is given by $(a,f)\mapsto(1)f$ for $a\in A$ and $f\in D(A)$. This
implies that $\big{(}(a,f),(b,g)\big{)}\widetilde{\psi}=(a)g+(b)f+(1)(f\otimes
g)\overline{\omega_{\lambda}}$ for $b\in A$ and $g\in D(A)$. Now, we show that
$\widetilde{\psi}$ is non-degenerate.

Let $(a,f)\neq 0$. Then $a\neq 0$ or $f\neq 0$. If $f\neq 0$, then there is an
element $b\in A$ such that $(b)f\neq 0$, and therefore
$\big{(}(a,f),(b,0)\big{)}\widetilde{\psi}=(b)f\neq 0$. If $f=0$ and $a\neq
0$, then the canonical isomorphism $A\simeq DD(A)$ implies that there is an
element $g\in D(A)$ such that $(a)g\neq 0$. In this case,
$\big{(}(a,0),(0,g)\big{)}\widetilde{\psi}=(a)g\neq 0$. Thus
$\widetilde{\psi}$ is non-degenerate. $\square$

Compared with $R(A)$, the algebra $S(A,e)$ depends on the choice of $e$, that
is, if $f=f^{2}\in A$ such that $(A,f)$ is gendo-symmetric, then $S(A,e)$ and
$S(A,f)$ do not have to be isomorphic in general. The following result
collects basic homological properties of $S(A,e)$.

######  Proposition 4.3.

Let $S:=S(A,e)$ and $B_{0}:=(1-e)A(1-e)$. Then

$(1)$ $S$ is a symmetric algebra.

$(2)$ $B_{0}$ can be regarded as a $S$-module and contains no nonzero
projective direct summands.

$(3)$ If ${\rm add}({{}_{A}}Ae)\cap{\rm add}({{}_{A}}A(1-e))=0$, then
$\\#(S)=\\#(A)$. For instance, if $B_{0}$ is indecomposable as an algebra,
then so is $S$.

Proof. $(1)$ Let $R:=R(A)$, $\bar{e}:=e\otimes e\in R$ and
$e_{0}:=(1-e)+\overline{e}\in R$. Since $R$ is symmetric by Proposition 4.2(1)
and $S=e_{0}Re_{0}$ by Lemma 3.3(1), $S$ is symmetric.

$(2)$ Since $\pi_{1}$ induces a surjective algebra homomorphism
$\pi_{1}^{\prime}:S\to B_{0}$ such that $S/S\overline{e}S\simeq B_{0}$ (see
Lemma 3.3 for notation), $B_{0}$ can be regarded as an $S$-module. Assume that
the $S$-module $B_{0}$ contains an indecomposable projective direct summand
$X$. Then there is a primitive idempotent $f\in A$ such that
$1-e=f+f^{\prime}$ with $f$ and $f^{\prime}$ orthogonal idempotents in $A$,
and $X\simeq Sf$ as $S$-modules. Clearly, $S\overline{e}Sf=0$,
$(f)\pi_{2}^{\prime}=f$, $(1-e)\pi_{2}^{\prime}=1-e$ and
$(S\overline{e}Sf)\pi_{2}^{\prime}=AeAf$. Consequently, ${\rm
Hom}_{A}(Ae,Af)\simeq eAf=0$, and therefore ${\rm Hom}_{A}(Af,Ae)\simeq D{\rm
Hom}_{A}(Ae,Af)=0$. By Lemma 2.12(2), $Af$ can be embedded into $(Ae)^{n}$ for
some $n\geq 1$. This implies $Af=0$, a contradiction.

$(3)$ Since $\bar{e}S\bar{e}\simeq eAe$ by Lemma 3.4(2), it follows from $(2)$
that $\\#S(A)=\\#(eAe)+\\#(B_{0})$. Due to ${\rm add}(Ae)\cap{\rm
add}(A(1-e))=0$, we have $\\#(A)=\\#(eAe)+\\#(B_{0})$ and $\\#S(A)=\\#(A)$.
The second assertion in $(3)$ follows from Proposition 3.8(2). $\square$

###  4.2 Mirror-reflective algebras and Auslander-Gorenstein algebras

In the subsection, we construct new gendo-symmetric algebras from minimal
Auslander-Gorenstein algebras and then present a proof of Theorem 1.1. This is
based on study of mirror-reflective algebras.

Recall from Lemma 3.3 that we have an algebra automorphism $\phi:R(A)\to R(A)$
and two surjective algebra homomorphisms $\pi_{1},\pi_{2}:R(A)\to A$ such that
$\pi_{2}=\phi\pi_{1}$. Thus we regard $A$-modules as $R(A)$-modules via
$\pi_{1}$ in the following discussion. It turns out that $A\mbox{{\rm-mod}}$
is a Serre subcategory of $R(A)\mbox{{\rm-mod}}$, that is, it is closed under
direct summands, submodules, quotients and extensions in $R(A)\mbox{{\rm-
Mod}}$. Let

$\phi_{*}:R(A)\mbox{{\rm-mod}}\longrightarrow R(A)\mbox{{\rm-
mod}}\quad\mbox{and}\quad(\pi_{2})_{*}:A\mbox{{\rm-mod}}\longrightarrow
R(A)\mbox{{\rm-mod}}$

be the restriction functors induced by $\phi$ and $\pi_{2}$, respectively.
Then $\phi_{*}$ is an auto-equivalence and $\phi_{*}(X)=(\pi_{2})_{*}(X)$ for
each $A$-module $X$.

######  Lemma 4.4.

Suppose that $\Lambda$ is a symmetric algebra and $N$ is a basic
$\Lambda$-module without nonzero projective direct summands. Let $A:={\rm
End}_{\Lambda}(\Lambda\oplus N)$, $e$ an idempotent of $A$ corresponding to
the direct summand $\Lambda$ of $\Lambda\oplus N$, and $R:=R(A,e)$. If
${{}_{\Lambda}}N$ is $m$-rigid for a natural number $m$, then the following
hold.

$(1)$ The $R$-module $A(1-e)$ is $(m+2)$-rigid and there are isomorphisms of
$R$-modules:

$\Omega_{R}^{m+3}\big{(}A(1-e)\big{)}\simeq\Omega_{R}^{m+2}\big{(}\phi_{*}(Ae\otimes_{\Lambda}N)\big{)}\simeq\phi_{*}\big{(}{\rm
Hom}_{\Lambda}(eA,\Omega_{\Lambda}^{m+2}(N))\big{)}.$

$(2)$ If $\,\Omega_{\Lambda}^{m+2}(N)\simeq N$, then
$\Omega_{R}^{m+3}(A(1-e))\simeq\phi_{*}(A(1-e))$ and the $R$-module $A(1-e)$
is $(2m+4)$-rigid. In this case, $\Omega_{R}^{2m+6}(A(1-e))\simeq A(1-e)$.

Proof. $(1)$ By the proof of Proposition 3.6(2), $\pi_{2}$ induces an
isomorphism ${\rm
Tor}_{i}^{\overline{e}R\overline{e}}(R\overline{e},\overline{e}R)\simeq{\rm
Tor}_{i}^{\Lambda}(Ae,eA)$ for all $i\geq 1$. Since $\Lambda$ is symmetric and
$D(Ae)\simeq eA$ by Lemma 2.12(2), we have

$D{\rm Tor}_{i}^{\Lambda}(Ae,eA)\simeq{\rm
Ext}_{\Lambda}^{i}(eA,D(Ae))\simeq{\rm Ext}_{\Lambda}^{i}(eA,eA)={\rm
Ext}_{\Lambda}^{i}(\Lambda\oplus N,\Lambda\oplus N)\simeq{\rm
Ext}_{\Lambda}^{i}(N,N).$

As ${{}_{\Lambda}}N$ is $m$-rigid, there holds ${\rm
Tor}_{i}^{\overline{e}R\overline{e}}(R\overline{e},\overline{e}R)=0$ for
$1\leq i\leq m$. By Proposition 3.6(1), $I:=R\overline{e}R$ is $2$-idempotent.
Therefore $I$ is $(m+2)$-idempotent by Lemma 2.3(1). Further, it follows from
Lemma 2.3(2) that ${{}_{R}}R/I$ is $(m+2)$-rigid. Since $R/I\simeq A$ as
$R$-modules, ${}_{R}A$ is $(m+2)$-rigid. Note that ${{}_{R}}A\simeq
R(e-\overline{e})\oplus A(1-e)$ by Lemma 3.4(2). As $R$ is symmetric by
Proposition 4.2, we see that $R(e-\overline{e})$ is projective-injective.
Consequently, ${{}_{R}}A(1-e)$ is $(m+2)$-rigid.

The proof of Proposition 3.6(1) implies $I\simeq
R\overline{e}\otimes_{\overline{e}R\overline{e}}\overline{e}R$ as
$R$-$R$-bimodules. By Lemma 3.4(2), $\pi_{2}$ restricts to an algebra
isomorphism $\overline{e}R\overline{e}\to\Lambda$ and also an isomorphism
$R\overline{e}\to Ae$ of abelian groups. Via the algebra isomorphism, we can
regard $R\overline{e}$ as an $R$-$\Lambda$-bimodule. Then
$R\overline{e}\simeq(\pi_{2})_{*}(Ae)=\phi_{*}(Ae)$ as
$R$-$\Lambda$-bimodules. This gives a natural isomorphism
$R\overline{e}\otimes_{\Lambda}-\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\phi_{*}(Ae)\otimes_{\Lambda}-$
of functors from $\Lambda\mbox{{\rm-proj}}$ to $R\mbox{{\rm-proj}}$. Since $N$
has no nonzero projective direct summands, ${\rm add}({{}_{A}}Ae)\cap{\rm
add}({{}_{A}}A(1-e))=0$. From $A\otimes_{R}R\overline{e}\simeq Ae\simeq
R\overline{e}$ and $A\otimes_{R}R(1-e)\simeq A(1-e)$, we obtain ${\rm
add}(R\overline{e})\cap{\rm add}(R(1-e))=0$. Since $I(1-e)$ is isomorphic to
$R\overline{e}\otimes_{\overline{e}R\overline{e}}\overline{e}R(1-e)$ which is
a quotient module of $(R\overline{e})^{n}$ for some $n$, we deduce that
$I(1-e)$ does not contain nonzero direct summands in ${\rm add}(R(1-e))$. Thus
the surjection ${{}_{R}}R(1-e)\to A(1-e)$ induced by $\pi_{1}$ is a projective
cover of the $R$-module $A(1-e)$, and therefore $\Omega_{R}(A(1-e))=I(1-e)$.
Since $\pi_{2}$ induces an isomorphism $\overline{e}R\to eA$ and sends $1-e$
to $1-e$ by Lemma 3.4(2), we have $\overline{e}R(1-e)\simeq eA(1-e)$ and

$\Omega_{R}(A(1-e))\simeq
R\overline{e}\otimes_{\overline{e}R\overline{e}}eA(1-e)\simeq
R\overline{e}\otimes_{\Lambda}eA(1-e)\simeq\phi_{*}(Ae)\otimes_{\Lambda}N=\phi_{*}(Ae\otimes_{\Lambda}N).$

Let $\cdots\to
Q_{m+1}\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}Q_{m}\to\cdots\to
Q_{1}\to Q_{0}\to N\to 0$ be a minimal projective resolution of
${}_{\Lambda}N$. Then it follows from $eA(1-e)=N$ and ${\rm
Tor}_{i}^{\Lambda}(Ae,N)\simeq D{\rm Ext}_{\Lambda}^{i}(N,N)=0$ for $1\leq
i\leq m$ that the sequence

$Ae\otimes_{\Lambda}Q_{m+1}\stackrel{{\scriptstyle
Ae\otimes\partial}}{{\longrightarrow}}Ae\otimes_{\Lambda}Q_{m}\longrightarrow\cdots\longrightarrow
Ae\otimes_{\Lambda}Q_{1}\longrightarrow
Ae\otimes_{\Lambda}Q_{0}\longrightarrow Ae\otimes_{\Lambda}N\longrightarrow 0$

is exact. As the composition of ${{}_{A}}Ae\otimes_{\Lambda}-$ with $(e\cdot)$
is isomorphic to the identity functor of $\Lambda\mbox{{\rm-mod}}$, we have
$\Omega_{A}^{m+2}(Ae\otimes_{\Lambda}N)\simeq{\rm Ker}(Ae\otimes\partial)$.
Note that $Ae\otimes_{\Lambda}-\simeq{\rm
Hom}_{\Lambda}(eA,-):\Lambda\mbox{{\rm-
proj}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\rm add}({{}_{A}}Ae)$
since $Ae={\rm Hom}_{\Lambda}(\Lambda\oplus N,\Lambda)$. This shows ${\rm
Ker}(Ae\otimes\partial)\simeq{\rm Hom}_{\Lambda}(eA,{\rm Ker}(\partial))={\rm
Hom}_{\Lambda}(eA,\Omega_{\Lambda}^{m+2}(N))$, and therefore

$\Omega_{R}^{m+3}\big{(}A(1-e)\big{)}\simeq\Omega_{R}^{m+2}(\phi_{*}(Ae\otimes_{\Lambda}N))\simeq\phi_{*}(\Omega_{R}^{m+2}(Ae\otimes_{\Lambda}N))\simeq\phi_{*}({\rm
Hom}_{\Lambda}(eA,\Omega_{\Lambda}^{m+2}(N))).$

$(2)$ Let $X:=A(1-e)$. Suppose $\Omega_{\Lambda}^{m+2}(N)\simeq N$. Then
$\Omega_{R}^{m+3}\big{(}X\big{)}\simeq\phi_{*}({\rm Hom}_{\Lambda}(eA,eX)).$
Since the functor $(e\cdot):A\mbox{{\rm-mod}}\to\Lambda\mbox{{\rm-mod}}$
induces an algebra isomorphism ${\rm End}_{A}(A)\simeq{\rm
End}_{\Lambda}(eA)$, we have $X\simeq{\rm Hom}_{A}(A,X)\simeq{\rm
Hom}_{\Lambda}(eA,eX)$. It follows that
$\Omega_{R}^{m+3}\big{(}X\big{)}\simeq\phi_{*}(X)$. Note that $\phi$ is an
algebra isomorphism with $\phi^{2}={\rm Id}_{R}$ by Lemma 3.3(3). Since
$\Omega_{R}$ commutes with $\phi_{*}$, we obtain
$\Omega_{R}^{2m+6}\big{(}X\big{)}\simeq X$. Now, it remains to show that
${{}_{R}}X$ is $(2m+4)$-rigid.

Since $R$ is symmetric, the stable module category
$R\mbox{{\rm-{\text@underline{mod}}}}$ of $R$ is a triangulated category with
the shift functor $[1]=\Omega^{-}_{R}:R\mbox{{\rm-{\text@underline{mod}}}}\to
R\mbox{{\rm-{\text@underline{mod}}}}$, where $\Omega^{-}_{R}$ is the cosyzygy
functor on $R\mbox{{\rm-{\text@underline{mod}}}}$. Clearly, ${\rm
Ext}_{R}^{n}(X_{1},X_{2})\simeq{\rm\underline{Hom}}_{R}(X_{1},X_{2}[n])$ for
all $n\geq 1$ and $X_{1},X_{2}\in R\mbox{{\rm-mod}}$, where
${\rm\underline{Hom}}_{R}(X,Y)$ denotes the morphism set from $X$ to $Y$ in
$R$-mod. Since the Auslander-Reiten (AR) translation on $R\mbox{{\rm-mod}}$
coincides with $\Omega^{2}_{R}$, it follows from the AR-formula that there is
a natural isomorphism
$D{\rm\underline{Hom}}_{R}(X_{1},X_{2})\simeq{\rm\underline{Hom}}_{R}(X_{2},X_{1}[-1])$.
Consequently, for each $i\in\mathbb{N}$, there are isomorphisms

${\rm
Ext}_{R}^{m+3+i}(X,X)\simeq{\rm\underline{Hom}}_{R}(\Omega_{R}^{m+3}(X),X[i])\simeq{\rm\underline{Hom}}_{R}(\phi_{*}(X),X[i])\simeq
D{\rm\underline{Hom}}_{R}(X[i],\phi_{*}(X)[-1]).$

Recall that $\phi$ is an algebra isomorphism with $\phi^{2}={\rm Id}_{R}$ by
Lemma 3.3(3). Then

${\rm\underline{Hom}}_{R}(X[i],\phi_{*}(X)[-1])\simeq{\rm\underline{Hom}}_{R}(\phi_{*}(X)[i],X[-1])\simeq{\rm\underline{Hom}}_{R}(\Omega_{R}^{m+3}(X),X[-1-i])\simeq{\rm
Ext}_{R}^{m+2-i}(X,X)$

for $0\leq i\leq m+1$. This implies ${\rm Ext}_{R}^{m+3+i}(X,X)\simeq D{\rm
Ext}_{R}^{m+2-i}(X,X)$ for $0\leq i\leq m+1$. Since $X$ is $(m+2)$-rigid by
$(1)$, it is actually $(2m+4)$-rigid. $\square$

######  Proposition 4.5.

Suppose that $\Lambda$ is a symmetric algebra and $N$ is a basic
$\Lambda$-module without nonzero projective direct summands. Let $A:={\rm
End}_{\Lambda}(\Lambda\oplus N)$, $e$ an idempotent of $A$ corresponding to
the direct summand $\Lambda$ of $\Lambda\oplus N$, and $R:=R(A,e)$.

$(1)$ If ${{}_{\Lambda}}\Lambda\oplus N$ is $m$-rigid, then ${{}_{R}}R\oplus
A(1-e)$ is $(m+2)$-rigid.

$(2)$ If ${{}_{\Lambda}}\Lambda\oplus N$ is $m$-ortho-symmetric, then
${{}_{R}}R\oplus A(1-e)$ is $(2m+4)$-ortho-symmetric.

$(3)$ If ${{}_{\Lambda}}\Lambda\oplus N$ is maximal $m$-orthogonal, then
${{}_{R}}R\oplus A(1-e)$ is maximal $(2m+4)$-orthogonal.

Proof. $(1)$ Since $R$ is a symmetric algebra by Proposition 4.2, (1) follows
from Lemma 4.4(1).

$(2)$ By assumption, ${{}_{\Lambda}}N$ is basic and contains no nonzero
projective direct summands. This implies that ${{}_{A}}A(1-e)$ is basic and
contains no nonzero projective-injective direct summands. We claim that
${{}_{R}}A(1-e)$ contains no nonzero projective direct summands. In fact, by
the proof of Lemma 4.4(1), ${{}_{R}}R(1-e)$ is a projective cover of
${{}_{R}}A(1-e)$. If ${{}_{R}}A(1-e)$ contains an indecomposable projective
direct summand $Y$, then $Y$ is a direct summand of $R(1-e)$. Since $R$ is
symmetric, ${{}_{R}}Y$ must be projective-injective. However, since
$A\mbox{{\rm-mod}}\subseteq R\mbox{{\rm-mod}}$ is a Serre subcategory,
${{}_{A}}Y$ is also a nonzero projective-injective direct summand of
${{}_{A}}A(1-e)$. This is a contradiction and shows that the above claim
holds. Now $(2)$ follows from Lemmas 4.4 and 2.13.

$(3)$ Recall that maximal orthogonal modules over an algebra $B$ are exactly
ortho-symmetric $B$-modules such that their endomorphism algebras have finite
global dimension. Let $A_{1}:={\rm End}_{R}(R\oplus A(1-e))$. By $(2)$, to
show $(3)$, it suffices to show that ${\rm gl.dim}(A_{1})<\infty$ if ${\rm
gl.dim}(A)<\infty$

Let $B_{1}:={\rm End}_{R}(R\oplus A)$. Since ${{}_{R}}A\simeq
R(e-\overline{e})\oplus A(1-e)$ by the proof of Lemma 4.4(1), $A_{1}$ and
$B_{1}$ are Morita equivalent, and therefore ${\rm gl.dim}(A_{1})={\rm
gl.dim}(B_{1})$. Since $eA_{A}$ is faithful, it follows from Proposition
3.5(3) that if ${\rm gl.dim}(A)<\infty$ then ${\rm gl.dim}(B_{1})={\rm
gl.dim}(B_{1}^{\rm op})<\infty$. Hence ${\rm gl.dim}(A_{1})<\infty$. $\square$

Proof of Theorem 1.1. The statement $(1)$ follows from Proposition 4.2. Let
$R:=R(A)$ and $S:=S(A,e)$. Then $R$ and $S$ are symmetric by $(1)$ and
Proposition 4.3(1). Let $A_{2}:=\mathcal{A}(A,e)$ and
$B_{2}:=\mathcal{B}(A,e)$. Then $A_{2}$ and $B_{2}$ are gendo-symmetric.

Next, we show that $(2)$ and $(3)$ hold for $A_{2}$. In fact, since $A$ is
gendo-symmetric, we can identify $A$ with ${\rm End}_{\Lambda}(\Lambda\oplus
X)$, where $\Lambda:=eAe$ is symmetric and $X=eA(1-e)$. As global, dominant
and injective dimensions are invariant under Morita equivalences, the classes
of minimal Auslander-Gorenstein algebras and of higher Auslander algebras are
closed under Morita equivalences. Moreover, for a self-injective algebra
$\Gamma$ and $M\in\Gamma\mbox{{\rm-mod}}$, it follows from [22, Lemma 3] that
${\rm dom.dim}({\rm End}_{\Gamma}(\Gamma\oplus M))$ equals the maximal natural
number $n\geq 2$ or $\infty$ such that $M$ is $(n-2)$-rigid. So, for a basic
module $X$ that has no nonzero projective direct summands, the inequality
${\rm dom.dim}(A_{2})\geq{\rm dom.dim}(A)+2$ and the statement $(3)$ follow
immediately from Proposition 4.5. Further, for an arbitrary module $X$, the
consideration can be reduced by a series of Morita equivalences, as shown
below.

We take a direct summand $N$ of $X$ such that $N$ is basic, has no nonzero
projective direct summands and satisfies ${\rm add}(\Lambda\oplus N)={\rm
add}(\Lambda\oplus X)$. Let $B:={\rm End}_{\Lambda}(\Lambda\oplus N)$ and
$f^{2}=f\in A$ correspond to the direct summand $\Lambda\oplus N$ of
$\Lambda\oplus X$. Then ${{}_{A}}Af$ is a progenerator (that is, a projective
generator), and therefore $B=fAf$ is Morita equivalent to $A$. Since
$ef=e=fe$, we have $R(B)=fAf\oplus fAe\otimes_{\Lambda}eAf=fRf$. Due to
$R\otimes_{A}Af\simeq Rf$, the module ${{}_{R}}Rf$ is a progenerator. Thus $R$
and $R(B)$ are Morita equivalent. Now, let $B_{2}:={\rm End}_{R(B)}(R(B)\oplus
B(f-e))$. If $A$ is $n$-minimal Auslander-Gorenstein (respectively,
$n$-Auslander), then so is $B$, and therefore, so is $B_{2}$ by the above-
proved case. Next, we shall show that $A_{2}$ and $B_{2}$ are Morita
equivalent. Recall that the restriction of $\pi_{1}$ to $A$ is the identity
map of $A$. This implies $A\otimes_{R}Rf=Af$ as $R$-modules, and therefore
${\rm add}(_{R}A)={\rm add}(_{R}Af)$. Let $A_{2}^{\prime}:={\rm
End}_{R}(Rf\oplus A(1-e)f)={\rm End}_{R}(Rf\oplus A(f-e))$. Then $A_{2}$ and
$A_{2}^{\prime}$ are Morita equivalent. Since the functor
$(f\cdot):R\mbox{{\rm-mod}}\to R(B)\mbox{{\rm-mod}}$ is an equivalence and
$f(Rf\oplus A(f-e))=R(B)\oplus B(f-e)$, there is an algebra isomorphism
$A_{2}^{\prime}\simeq B_{2}$. Hence $A_{2}$ and $B_{2}$ are Morita equivalent.
Thus $(2)$ and $(3)$ hold true for $A_{2}$.

It remains to show ${\rm dom.dim}(B_{2})\geq{\rm dom.dim}(A)+2$. Up to Morita
equivalence, we assume $A={\rm End}_{\Lambda}(\Lambda\oplus N)$. If
${{}_{\Lambda}}\Lambda\oplus N$ is $m$-rigid for some $m\in\mathbb{N}$, then
it follows from the first part of the proof of Lemma 4.4(1) that $I$ is an
$(m+2)$-idempotent ideal of $R$. Let $e_{0}:=(1-e)+\overline{e}\in R$. By
Lemma 3.3, we have $\overline{e}e_{0}=\overline{e}=e_{0}\overline{e}$,
$I:=R\overline{e}R=S\overline{e}S$ and $S/I\simeq(1-e)A(1-e)$ as algebras.
Thanks to Corollary 2.4(1), $I$ is an $(m+2)$-idempotent ideal of $S$.
Further, by Lemma 2.3(2), ${{}_{S}}S/I$ is $(m+2)$-rigid, and therefore
${{}_{S}}S\oplus S/I$ is $(m+2)$-rigid since $S$ is symmetric by Proposition
4.3(1). Thus ${\rm dom.dim}(B_{2})\geq{\rm dom.dim}(A)+2$, due to [22, Lemma
3]. $\square$

###  4.3 Recollements of mirror-reflective algebras and Tachikawa’s second
conjecture

In this subsection, we study the iterated process of constructing (reduced)
mirror symmetric algebras from gendo-symmetric algebras and prove Theorems 1.2
and 1.3.

Throughout this section, let $(A,e)$ be a gendo-symmetric algebra. We define
inductively for $n\geq 1$

$A_{1}=B_{1}:=A,\quad R_{1}:=R(A_{1},e_{1}),\quad\quad S_{1}:=S(A_{1},f_{1}),$
$\quad\quad\;\;A_{n+1}:={\rm End}_{R_{n}}\big{(}R_{n}\oplus
A_{n}(1_{A_{n}}-e_{n})\big{)},\quad R_{n+1}:=R(A_{n+1},e_{n+1}),$
$\quad\quad\quad\quad\quad\quad B_{n+1}:={\rm
End}_{S_{n}}\big{(}S_{n}\oplus(1_{B_{n}}-f_{n})B_{n}(1_{B_{n}}-f_{n})\big{)},\quad
S_{n+1}:=S(B_{n+1},f_{n+1}),$

where $e_{1}=f_{1}:=e$, and for $n\geq 1$, $e_{n+1}\in A_{n+1}$ is the
idempotent corresponding to the direct summand $R_{n}$ of the $R_{n}$-module
$R_{n}\oplus A_{n}(1_{A_{n}}-e_{n})$, and $f_{n+1}\in B_{n+1}$ is the
idempotent corresponding to the direct summand $S_{n}$ of the $S_{n}$-module
$S_{n}\oplus(1_{B_{n}}-f_{n})B_{n}(1_{B_{n}}-f_{n})$. In other words,

$A_{n+1}=\mathcal{A}(A_{n},e_{n}),\quad
B_{n+1}=\mathcal{B}(B_{n},f_{n})\,\mbox{ for }n\geq 1.$