#  $\tau$-tilting theory and silting theory of skew group algebra extensions

Yuta Kimura, Ryotaro Koshio, Yuta Kozakai, Hiroyuki Minamoto, Yuya Mizuno
Yuta Kimura : Department of Mechanical Systems Engineering, Faculty of
Engineering, Hiroshima Institute of Technology, 2-1-1 Miyake, Saeki-ku
Hiroshima 731-5143, Japan<EMAIL_ADDRESS>Ryotaro Koshio:
Department of Mathematics, Graduate School of Science, Tokyo University of
Science 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan
<EMAIL_ADDRESS>Yuta Kozakai: Department of Mathematics, Tokyo University
of Science 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan
<EMAIL_ADDRESS>Hiroyuki Minamoto: Department of Mathematics, Graduate
School of Science/ Faculty of Science, Osaka Metropolitan University 1-1
Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan<EMAIL_ADDRESS>Yuya
Mizuno: Faculty of Liberal Arts, Sciences and Global Education / Graduate
School of Science, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku,
Sakai, Osaka 599-8531, Japan<EMAIL_ADDRESS>
###### Abstract.

Let $\Lambda$ be a finite dimensional algebra with an action by a finite group
$G$ and $A:=\Lambda*G$ the skew group algebra. One of our main results asserts
that the canonical restriction-induction adjoint pair of the skew group
algebra extension $\Lambda\subset A$ induces a poset isomorphism between the
poset of $G$-stable support $\tau$-tilting modules over $\Lambda$ and that of
($\operatorname{mod}G$)-stable support $\tau$-tilting modules over $A$. We
also establish a similar poset isomorphism of posets of appropriate classes of
silting complexes over $\Lambda$ and $A$. These two results generalize and
unify preceding results by Zhang-Huang, Breaz-Marcus-Modoi and the second and
the third authors. Moreover, we give a practical condition under which
$\tau$-tilting finiteness and silting discreteness of $\Lambda$ are inherited
to those of $A$. As applications we study $\tau$-tilting theory and silting
theory of the (generalized) preprojective algebras and the folded mesh
algebras. Among other things, we determine the posets of support
$\tau$-tilting modules and of silting complexes over preprojective algebra
$\Pi({\mathbb{L}}_{n})$ of type ${\mathbb{L}}_{n}$.

###### 2020 Mathematics Subject Classification:

Primary: 16G10, Secondary: 16B50, 16E35, 16S35

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries about skew group algebra extensions
  3. 3 $\tau$-tilting theory of skew group algebra extensions
  4. 4 Silting theory of skew group algebra extensions
  5. 5 Application to preprojective algebras and folded mesh algebras
  6. A Reinterpretation to $G$-graded algebras
  7. B A formalism by theory of Hopf algebras and tensor categories

##  1\. Introduction

A decade has been passed since the introduction of $\tau$-tilting theory [1]
and silting theory [4]. During these years, those theories have been
extensively studied in many researchers and now become basic and fundamental
subjects of representation theory of algebras. These theories study hidden
symmetry of the module category and the derived category of algebras by using
specific classes of objects, $\tau$-tilting modules and silting complexes.
Thus, these theories have connections to various areas of mathematics. Among
other things, we point out that since silting theory is strongly related to
$t$-structures of the derived category, it is effectively used to study of
Bridgeland’s stability conditions [18].

The central aim of the present paper is to study $\tau$-tilting theory and
silting theory of the skew group algebra extension $\Lambda\subset\Lambda*G$.
Namely, given an algebra $\Lambda$ with an action $G\curvearrowright\Lambda$
by a finite group $G$, we study relationships between $\tau$-tilting modules
and silting complexes over $\Lambda$ and the skew group algebra $A:=\Lambda*G$
via the induction and the restriction functor. A detailed explanation is given
in the next section, in which the main results are summarized as Theorem 1.1.
Another important result proves that under a practical condition,
$\tau$-tilting finiteness and silting discreteness of $\Lambda$ are inherited.
It is summarized in Theorem 1.2. In Section 1.2, we review recent related
results which study a triangulated category ${\mathcal{T}}$ with an action
$G\curvearrowright{\mathcal{T}}$ by a finite group $G$, and then we explain
relationships between these results and our results.

As applications we study $\tau$-tilting theory and silting theory of the
(generalized) preprojective algebras and the folded mesh algebras in Section
5. Among other things, we determine the posets of support $\tau$-tilting
modules and of silting complexes of preprojective algebra
$\Pi({\mathbb{L}}_{n})$ of type ${\mathbb{L}}_{n}$.

###  1.1. $\tau$-tilting theory and silting theory of skew group algebra
extensions

Let $\Lambda$ be a finite dimensional algebra, $G$ a finite group acting on
$\Lambda$ and $A:=\Lambda*G$ the skew group algebra. The canonical inclusion
$\Lambda\subset A$ induces the adjoint pair
$(\operatorname{Ind}_{\Lambda}^{A},\operatorname{Res}_{\Lambda}^{A})$ of the
induction functor $\operatorname{Ind}_{\Lambda}^{A}=A\otimes_{\Lambda}-$ and
the restriction functor $\operatorname{Res}_{\Lambda}^{A}$ of the categories
of finite dimensional (left) modules.

$\operatorname{Ind}^{A}_{\Lambda}:\operatorname{mod}\Lambda\rightleftarrows\operatorname{mod}A:\operatorname{Res}_{\Lambda}^{A}.$

The one of the main aims of this paper is to compare support $\tau$-tilting
modules $M$ over $\Lambda$ and those $N$ of over $A$, via the adjoint pair
$(\operatorname{Ind}_{\Lambda}^{A},\operatorname{Res}_{\Lambda}^{A})$. Namely
we ask which support $\tau$-tilting modules over $\Lambda$ (resp. $A$) are
sent to those over $A$ (resp. $\Lambda$) by the functor
$\operatorname{Ind}_{\Lambda}^{A}$ (resp. $\operatorname{Res}_{\Lambda}^{A}$).

Since it is a natural question, there are several previous research of this
problem (or of equivalent one).

####  1.1.1. Preceding results

Zhang-Huang [51] studied a skew group algebra extension $\Lambda\subset A$
where $A:=\Lambda*G$ under the assumption that
$\operatorname{char}{\mathbf{k}}\nmid|G|$. Their main result proved that the
induction functor $\operatorname{Ind}_{\Lambda}^{A}$ sends $G$-stable support
$\tau$-tilting modules $M$ over $\Lambda$ to support $\tau$-tilting modules
$\operatorname{Ind}_{\Lambda}^{A}(M)$ over $A$ and that this map is injective
(for the notion $G$-stability see Definition 2.1). They also gave a
restriction on the image of the map induced by
$\operatorname{Ind}_{\Lambda}^{A}$. But they could not identify the image of
this map.

Breaz-Marcus-Modoi [17] dealt with the extension $\Lambda\subset A$ of a
strongly $G$-graded algebra $A=\bigoplus_{g\in G}A_{g}$ and its degree
$e_{G}$-part $\Lambda:=A_{e_{G}}$ and compared support $\tau$-tilting modules
of $\Lambda$ and $A$ via the canonical adjoint pair. (We note that as is
explained in Appendix A, this problem is equivalent to that of skew group
algebra extension by a finite group $G$, via Cohen-Montgomery duality.) They
worked with the assumption that $\Lambda$ is self-injective. They showed that
the induction functor $\operatorname{Ind}_{\Lambda}^{A}$ sends $G$-stable
support $\tau$-tilting modules $M$ over $\Lambda$ to support $\tau$-tilting
modules $\operatorname{Ind}_{\Lambda}^{A}(M)$ over $A$. Moreover, under the
additional assumption $\operatorname{char}{\mathbf{k}}\nmid|G|$, they also
find a condition for a support $\tau$-tilting module $N$ over $A$ that ensures
that $\operatorname{Res}_{\Lambda}^{A}(N)$ is a support $\tau$-tilting modules
over $\Lambda$.

Finally, the second author and the third author of this paper [43] dealt with
a group extension $N\triangleleft H$. They compared support $\tau$-tilting
modules of $\Lambda:={\mathbf{k}}N$ and $A:={\mathbf{k}}H$ of group algebras
via the canonical adjoint pair induced from the extension
$\Lambda={\mathbf{k}}N\subset{\mathbf{k}}H=A$. It is noteworthy that they did
not put any hypothesis on the characteristic of the base field ${\mathbf{k}}$.
(We note that $A={\mathbf{k}}H$ has a canonical structure of a strongly
$G:=H/N$-graded algebras such that $A_{e_{G}}={\mathbf{k}}N$, see Appendix A.)
They proved that the functor $\operatorname{Ind}_{\Lambda}^{A}$ induces an
injective map from the poset of $G$-stable support $\tau$-tilting modules over
$\Lambda$ to the poset of those over $A$. This is analogous to the
aforementioned result by Zhang-Huang. However, they additionally succeeded to
identify the image of the map induced from the functor
$\operatorname{Ind}_{\Lambda}^{A}$.

####  1.1.2. Poset isomorphisms

In the paper we deal with the skew group algebra extension
$\Lambda\subset\Lambda*G$ without any hypothesis on the characteristic of
${\mathbf{k}}$, and obtain results that unifies and generalizes previous
results recalled above. One of the novelties of the paper is the notion of _(
$\operatorname{mod}G$)-stability_ of a module $N$ over $A$ (Definition 2.9).
This notion precisely identifies the image of the map induced by
$\operatorname{Ind}_{\Lambda}^{A}$. Namely we prove that the functor
$\operatorname{Ind}_{\Lambda}^{A}$ induces a poset isomorphism between the
poset $(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}$ of $G$-stable
support $\tau$-tilting modules and the poset
$(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}$ of
($\operatorname{mod}G$)-stable support $\tau$-tilting modules, and that the
functor $\operatorname{Res}_{\Lambda}^{A}$ induces the inverse map between
these posets.

In the paper, we also compare silting complexes, $2$-term silting complexes
and tilting complexes via the canonical adjoint pair and obtain similar poset
isomorphisms between the posets of these complexes.

The following theorem summarizes our results about the skew group algebra
extension.

######  Theorem 1.1.

Let $\Lambda$ be a finite dimensional algebra with an action of a finite group
$G$ and $A:=\Lambda*G$ the skew group algebra. Then the adjoint pair
$(\operatorname{Ind}^{A}_{\Lambda},\operatorname{Res}_{\Lambda}^{A})$ induces
the following commutative diagram of the poset morphisms

$\textstyle{(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Res}^{A}_{\Lambda}}$$\textstyle{(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ind}_{\Lambda}^{A}}$$\textstyle{(\operatorname{2-silt}\Lambda)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Res}^{A}_{\Lambda}}$$\scriptstyle{{\textup{H}}^{0}}$$\textstyle{(\operatorname{2-silt}A)^{\operatorname{mod}G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ind}_{\Lambda}^{A}}$$\scriptstyle{{\textup{H}}^{0}}$$\textstyle{(\operatorname{silt}\Lambda)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Res}^{A}_{\Lambda}}$$\textstyle{(\operatorname{silt}A)^{\operatorname{mod}G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ind}_{\Lambda}^{A}}$$\textstyle{(\operatorname{tilt}\Lambda)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Res}^{A}_{\Lambda}}$$\textstyle{(\operatorname{tilt}A)^{\operatorname{mod}G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ind}_{\Lambda}^{A}}$

where the horizontal arrows are poset isomorphisms inverse to each other. The
vertical arrows ${\textup{H}}^{0}$ from the middle to the top are induce from
the poset isomorphism ${\textup{H}}^{0}$ between $2$-term silting complexes
and support $\tau$-tilting complexes.

The top horizontal isomorphisms are established in Theorem 3.3. The second,
the third and the fourth horizontal isomorphisms, as well as the commutativity
of the second and third squares are established in Theorem 4.3. Proposition
4.4 proves that two vertical arrows ${\textup{H}}^{0}$ are poset isomorphisms.
The commutativity of the first square is easy to check and its verification is
left to the readers.

In a subsequent work, we establish similar correspondences of torsion
theories, semi-bricks and wide subcategories over $\Lambda$ and $A$.

####  1.1.3. $\tau$-tilting finiteness and silting discreteness

In general ($\operatorname{mod}G$)-stability is difficult to check. However,
next theorem tells that under certain conditions, $\tau$-tilting finiteness
(resp. silting discreteness) of $\Lambda$ is inherited to $A=\Lambda*G$ and
moreover that all support $\tau$-tilting modules (resp. silting complexes)
over $A$ are ($\operatorname{mod}G$)-stable.

The basic setup of the theorem is that, as is often with many applications, an
algebra $\Lambda={\mathbf{k}}Q/I$ is given by a quiver $Q$ with relations $I$
and an action $G\curvearrowright\Lambda$ is induced from an action
$G\curvearrowright Q$ on the quiver $Q$. For a vertex $e\in Q_{0}$, we denote
by $\operatorname{stab}(e)<G$ the stabilizer group.

######  Theorem 1.2.

In the above setup, we further assume that for each vertex $e\in Q_{0}$, the
group algebra ${\mathbf{k}}\operatorname{stab}(e)$ is local. If $\Lambda$ is a
$\tau$-tilting finite (resp. silting discrete), then so is $A$ and moreover
all support $\tau$-tilting modules (resp. silting complexes) are
($\operatorname{mod}G$)-stable. Consequently the adjoint pair
$(\operatorname{Ind}^{A}_{\Lambda},\operatorname{Res}_{\Lambda}^{A})$ induces
the following poset isomorphisms

$\begin{split}\operatorname{Ind}_{\Lambda}^{A}:(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}\
\ &\longleftrightarrow\ \
\textup{s}\tau\textup{-}\operatorname{tilt}A:\operatorname{Res}_{\Lambda}^{A}\\\
\textup{(resp.
}\operatorname{Ind}_{\Lambda}^{A}:(\operatorname{silt}\Lambda)^{G}\ \
&\longleftrightarrow\ \
\operatorname{silt}A:\operatorname{Res}_{\Lambda}^{A}\textup{)}\\\
\end{split}$

Theorem 1.2 about support $\tau$-tilting modules (resp. silting complexes) is
proved as Theorem 3.11 (resp. Theorem 4.7) under a slightly more general
condition.

We note that if the action of $G$ on the vertex set $Q_{0}$ is free, the
assumption of Theorem 1.2 is satisfied. For the convenience of the readers, we
recall the following well-known fact in modular representation theory of a
finite group.

######  Lemma 1.3 (See for example [50, Corollary 3.2.4]).

Assume that ${\mathbf{k}}$ is an algebraically closed field of characteristic
$p=\operatorname{char}{\mathbf{k}}>0$. The group algebra ${\mathbf{k}}G$ is
local algebra if and only if $G$ is a $p$-group.

###  1.2. Related results

In this section, we review recent related results which study a triangulated
category ${\mathcal{T}}$ with an action $G\curvearrowright{\mathcal{T}}$ by a
finite group $G$ and then we explain relationships between these results and
our results. We assume that a triangulated category ${\mathcal{T}}$ is
$\operatorname{Hom}$-finite and idempotent complete in the sequel. We denote
by ${\mathcal{T}}^{G}$ the category of $G$-equivariant objects of
${\mathcal{T}}$, which is triangulated, in nice situations, by [14, 19, 27].

Chen-Chen-Ruan [22] deals with a triangulated category ${\mathcal{T}}$
equipped with an action $G\curvearrowright{\mathcal{T}}$ by a finite abelian
group $G$. The point here is that the category ${\mathcal{T}}^{G}$ of
$G$-equivariant object comes equipped with an action
$\hat{G}\curvearrowright{\mathcal{T}}^{G}$ by the character group $\hat{G}$.
Then as one of their main results [22, Theorem A], they proved that $G$-stable
tilting objects of ${\mathcal{T}}$ is bijectively corresponds to
$\hat{G}$-stable tilting objects of ${\mathcal{T}}^{G}$ under the assumption
that the group algebra ${\mathbf{k}}G$ is a direct product of ${\mathbf{k}}$.

Dell-Heng-Licata [25], first study stability conditions of a triangulated
category ${\mathcal{T}}$ equipped with an action
${\mathcal{C}}\curvearrowright{\mathcal{T}}$ of a fusion category
${\mathcal{C}}$ (for fusion categories we refer [29]). The point here is that
the category ${\mathcal{T}}^{G}$ of $G$-equivariant object comes equipped with
an action $\operatorname{mod}G\curvearrowright{\mathcal{T}}^{G}$ by the tensor
category $\operatorname{mod}G$. We note that they need to assume that
$\operatorname{char}{\mathbf{k}}\nmid|G|$ so that the tensor category
$\operatorname{mod}{\mathbf{k}}G$ is a fusion category. One of their main
result [25, Theorem B] proves that the space
$\textup{stab}_{G}({\mathcal{T}})$ of $G$-equivariant stability conditions of
${\mathcal{T}}$ is homeomorphic to the space
$\textup{stab}_{\operatorname{mod}G}({\mathcal{T}}^{G})$ of
($\operatorname{mod}G$)-equivariant stability conditions of ${\mathcal{T}}$.
We note that in [25, Theorem B], the action $G\curvearrowright{\mathcal{T}}$
is reinterpreted to the action $\mathsf{vec}(G)\curvearrowright{\mathcal{T}}$
by the category $\mathsf{vec}(G)$ of $G$-graded vector spaces. (The category
$\mathsf{vec}(G)$ is isomorphic to the category
${\mathbf{k}}G^{*}\operatorname{mod}$ of ${\mathbf{k}}G^{*}$-modules where
${\mathbf{k}}G^{*}=\operatorname{Hom}_{{\mathbf{k}}}({\mathbf{k}}G,{\mathbf{k}})$
denotes the dual Hopf algebra of ${\mathbf{k}}G$. The above reinterpretation
is explained in Appendix B.)

Now we explain relationships between above results and our results. Thus as in
the previous section, let $\Lambda$ be a finite dimensional algebra with an
action $G\curvearrowright\Lambda$ by a finite group and $A:=\Lambda*G$. The
category $\operatorname{mod}\Lambda$ of $\Lambda$-modules is equipped with the
induced action $G\curvearrowright\operatorname{mod}\Lambda$. It is well-know
that $(\operatorname{mod}\Lambda)^{G}\simeq\operatorname{mod}A$. Now we assume
that $\operatorname{char}{\mathbf{k}}\nmid|G|$. Then by[19, Proposition 4.5],
we have an equivalence
${\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}\Lambda)^{G}\simeq{\mathsf{D}}^{\mathrm{b}}((\operatorname{mod}\Lambda)^{G})\simeq{\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A)$
of triangulated categories. As we mentioned before, silting complexes and
stablity conditions are closely related. Therefore, since our poset
isomorphism
$(\operatorname{silt}\Lambda)^{G}\cong(\operatorname{silt}A)^{\operatorname{mod}G}$
of silting complexes is verified without the assumption
$\operatorname{char}{\mathbf{k}}\nmid|G|$, we conjecture that there exists a
homeomorphism
$\textup{stab}_{G}({\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}\Lambda))\cong\textup{stab}_{\operatorname{mod}G}({\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A))$
without any assumption on $\operatorname{char}{\mathbf{k}}$. This conjecture
and its variant for algebraic varieties are left to our future study.

If we further assume that the group algebra ${\mathbf{k}}G$ is a direct
product of ${\mathbf{k}}$ as in [22], then ($\operatorname{mod}G$)-stable
objects of ${\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A)$ are precisely
$\hat{G}$-stable objects of ${\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A)$
(see for the discussion after Definition 2.10). Thus our poset isomorphism
$(\operatorname{tilt}\Lambda)^{G}\cong(\operatorname{tilt}A)^{\operatorname{mod}G}$
established for a finite group $G$ which is not necessarily abelian, can be
looked as a partial generalization of the result given in [22] in the
situation that
${\mathcal{T}}={\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}\Lambda)$ and an
action $G\curvearrowright{\mathcal{T}}$ is induced from an action
$G\curvearrowright\Lambda$ on an algebra $\Lambda$.

#### 1.2.1.

Homma [37] deals with the situation that a (not necessarily finite) group $G$
acting on a $\operatorname{Hom}$-finite triangulated category ${\mathcal{T}}$.
Compare to the setup of our paper, a $\operatorname{Hom}$-finite
${\mathbf{k}}$-linear category ${\mathcal{T}}$ plays the role of the derived
category ${\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}\Lambda)$ with the
induced $G$-action. Using covering theory, he studied the poset of $G$-stable
silting objects of ${\mathcal{T}}$. One of his main results shows that if an
action of $G$ on $\mathop{\mathrm{ind}}\nolimits{\mathcal{T}}$ is free, then
the poset of $G$-stable silting objects of ${\mathcal{T}}$ is isomorphic as
posets to the silting poset of ${\mathcal{T}}^{\prime}$ where
${\mathcal{T}}^{\prime}$ is a $G$-precovering of ${\mathcal{T}}$ such that the
thick-hull of the image of the induction functor coincide with
${\mathcal{T}}^{\prime}$. In the case
${\mathcal{T}}={\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}\Lambda)$, the
consequence of this result may coincide with the silting part of Theorem 1.2.
However, the assumption of freeness for the action
$G\curvearrowright\mathop{\mathrm{ind}}\nolimits{\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}\Lambda)$
is stronger than our assumption.

###  1.3. Organization of the paper

Section 2 gives preliminaries about a skew group algebra extension
$\Lambda\subset A$ where $A:=\Lambda*G$. We recall well-known facts about a
skew group algebra extension. In the case
$\operatorname{char}{\mathbf{k}}\nmid|G|$, it is well-known that the
Auslander-Reiten translations $\tau$ (of $\Lambda$ and of $A$) commute with
the functors $\operatorname{Ind}_{\Lambda}^{A},\
\operatorname{Res}_{\Lambda}^{A}$. As one of the technical heart of the paper,
we discuss this commutativity without the assumption on the characteristic of
the base field. Other important notions introduced in this section are
($\operatorname{mod}G$)-action on modules $N$ over $A$ and stability with
respect to this action which is called ($\operatorname{mod}G$)-stability.

In Section 3 and Section 4, we study $\tau$-tilting theory and silting theory
of a skew group algebra extension, respectively and prove the corresponding
part of Theorem 1.1 and Theorem 1.2.

In Section 5, applying results obtained in the previous sections, we study
silting theory and $\tau$-tilting theory of preprojective algebras and folded
mesh algebras.

In Appendix A, we explain that our results about a skew group algebra
extensions can be reinterpreted to that about a $G$-graded algebras
$A=\bigoplus_{g\in G}A_{g}$ and the canonical adjoint pair
$V:\operatorname{grmod}A\rightleftarrows\operatorname{mod}A:U$ between the
categories of graded $A$-modules and that of ungraded $A$-modules, via Cohen-
Montgomery duality. We also explain that specializing the results about a
$G$-graded algebra, we obtain results about an extension $A_{e_{G}}\subset A$
of a strongly $G$-graded algebra by its degree $e_{G}$-part $A_{e_{G}}$. By
this reinterpretation, our results gives generalizations of previous papers
[17, 43].

In Appendix B, we discuss a formal aspect from theory of tensor categories and
Hopf algebras. We provide a formalism of our result that allows us to give a
possible generalization of our result to an algebra $\Lambda$ with an action
of a Hopf algebra $H$.

###  1.4. Conventions

In the paper ${\mathbf{k}}$ denotes a (not necessarily algebraically closed)
field of arbitrary characteristic. Undecorated tensor product $\otimes$ always
denotes the tensor product $\otimes_{{\mathbf{k}}}$ over ${\mathbf{k}}$.

Groups are assumed to be finite groups. Algebras are assumed to be a finite
dimensional ${\mathbf{k}}$-algebra. Modules over an algebra $\Lambda$ are
always finite dimensional left $\Lambda$-modules, unless otherwise is
mentioned. An action $G\curvearrowright\Lambda$ of a group $G$ on an algebra
$\Lambda$ is assumed to preserve the algebra structure of $\Lambda$.

Let $M$ be a $\Lambda$-module. We denote its basic part by
$M^{\mathsf{basic}}$, i.e., a basic module such that
$\operatorname{add}M^{\mathsf{basic}}=\operatorname{add}M$. We note that
$M^{\mathsf{basic}}$ is unique up to isomorphisms. We denote by $|M|$ the
number of the indecomposable direct summands of $M^{\mathsf{basic}}$.

In the same way, we define the basic part $M^{\mathsf{basic}}$ of an object
$M$ of the derived category
${\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}\Lambda)$.

### Acknowledgment

The authors thank H. Asashiba for answering a question about relationship
between skew group algebras and orbit categories, and suggesting his paper
[10]. We also appreciate his comments and suggestions to the first draft of
this paper. The authors thank T. Adachi for discussions about the paper [2] by
him with R. Kase. The authors thank T. Homma for explaining his results give
in [37]. The authors thank N. Hiramae for pointing out an error in the first
draft of this paper. The authors would like to express their gratitude to R.
Wang for bringing the papers [20, 21] to their attention.

The first author is partly supported by Grant-in-Aid for JSPS Fellows
JP22J01155. The fourth author is partly supported by Grant-in-Aid for
Scientific Research (C) Grant Number JP21K03210. The fifth author is partly
supported by Grant-in-Aid for Scientific Research (C) Grant Number JP20K03539.

##  2\. Preliminaries about skew group algebra extensions

Let $\Lambda$ be an algebra with an action of a group $G$. Recall that the
_skew group algebra_ $A:=\Lambda\ast G$ is defines in the following way: The
underlying ${\mathbf{k}}$-vector space of $A$ is defined to be
$\Lambda\otimes{\mathbf{k}}G$ where ${\mathbf{k}}G$ is the group algebra of
$G$. For $r\in\Lambda$ and $g\in G$, we set $r*g:=r\otimes
g\in\Lambda\otimes{\mathbf{k}}G$. Then the multiplication of $A$ is given by

$(r*g)(s*h):=rg(s)*gh\ \ \ (r,s\in\Lambda,\ g,h\in G).$

###  2.1. $G$-action on $\Lambda$-modules and ($\operatorname{mod}G$)-action
on $A$-modules

####  2.1.1. $G$-action and $G$-stability

For $g\in G$ and a $\Lambda$-module $M$, we denote by ${}_{g}M$ the
$\Lambda$-module whose underlying vector space is that of $M$ and the action
of $\Lambda$ is twisted by $g$, i.e.

$r\cdot_{{}_{g}M}m:=g(r)\cdot_{M}m$

for $r\in\Lambda,g\in G$ and $m\in M$. We remark that
${}_{h}({}_{g}M)\cong{}_{gh}M$. Note that the map
${}_{g}\Lambda\to\Lambda,r\mapsto g^{-1}(r)$ is an isomorphism of
$\Lambda$-modules.

######  Definition 2.1.

A $\Lambda$-module $M$ is called $G$-_stable_ if ${}_{g}M\cong M$ for all
$g\in G$.

####  2.1.2. $\operatorname{mod}G$ action and
($\operatorname{mod}G$)-stability

######  Definition 2.2.

Let $N$ be an $A=\Lambda*G$-module and $X$ a $G$-module. We define an
$A$-module $X\otimes^{G}_{{\mathbf{k}}}N$ in the following way. As a
${\mathbf{k}}$-vector space $X\otimes^{G}_{{\mathbf{k}}}N$ is defined to be
$X\otimes_{{\mathbf{k}}}N$. We define an action of an element $r*g\in A$ to
$x\otimes n\in X\otimes_{{\mathbf{k}}}N$ by the following formula:

$(r*g)\cdot(x\otimes n):=g(x)\otimes(r*g)n.$

We note that this construction is already used in representation theory of
skew group algebras (see e.g., [44, Definition 5.17]).

######  Remark 2.3.

The construction $(X,M)\mapsto X\otimes_{{\mathbf{k}}}^{G}M$ given in
Definition 2.2 is natural in $X\in\operatorname{mod}G$ and
$M\in\operatorname{mod}\Lambda$ respectively and hence induces a functor
$\otimes_{{\mathbf{k}}}^{G}:(\operatorname{mod}G)\times(\operatorname{mod}A)\to\operatorname{mod}A$.
It is straightforward to check that this functor together with the
isomorphisms given in Lemma 2.5 below endows the category
$\operatorname{mod}A$ with a structure of a left module category over the
tensor category $\operatorname{mod}G$ in the sense of [29, Definition 7.1.2].
(It should be noted that familiarity with tensor categories is not necessary
for understanding this paper, with the exception of Appendix B.)

In other words, the construction of Definition 2.2 gives an action
$\operatorname{mod}G\curvearrowright\operatorname{mod}A$. We note that as is
explained in Section 4, this action
$\operatorname{mod}G\curvearrowright\operatorname{mod}A$ on
$\operatorname{mod}A$ extends to an action
$\operatorname{mod}G\curvearrowright{\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A)$
on the derived categories ${\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A)$,
which coincides with the ($\operatorname{mod}G$)-action used in [25].

######  Example 2.4.

If $\Lambda={\mathbf{k}}$ with the trivial $G$-action, then the skew group
algebra $A$ is the group algebra ${\mathbf{k}}G$ and the $A$-module
$X\otimes_{{\mathbf{k}}}^{G}N$ is the ${\mathbf{k}}G$-modules
$X\otimes_{{\mathbf{k}}}N$ with the diagonal action.

We leave the verification of the following lemma to the readers.

######  Lemma 2.5.

For an $A$-module $N$ and $G$-modules $X,Y$, the following assertions hold.

  1. (1)

${\mathbf{k}}_{G}\otimes_{{\mathbf{k}}}^{G}N\cong N$.

  2. (2)

$(X\otimes_{{\mathbf{k}}}^{G}Y)\otimes_{{\mathbf{k}}}^{G}N\cong
X\otimes_{{\mathbf{k}}}^{G}(Y\otimes_{{\mathbf{k}}}^{G}N)$.

Thanks to the second statement of above lemma, we may write both of
$(X\otimes_{{\mathbf{k}}}^{G}Y)\otimes_{{\mathbf{k}}}^{G}N$ and
$X\otimes_{{\mathbf{k}}}^{G}(Y\otimes_{{\mathbf{k}}}^{G}N)$ by
$X\otimes_{{\mathbf{k}}}^{G}Y\otimes_{{\mathbf{k}}}^{G}N$.

######  Example 2.6.

Observe that $\Lambda$ has a canonical $A$-module structure. For a $G$-module
$X$, we often write $\Lambda*X=X\otimes_{{\mathbf{k}}}^{G}\Lambda$ and in that
case an element $x\otimes r$ is written as $r*x$. The $A$-module
$\Lambda*{\mathbf{k}}G={\mathbf{k}}G\otimes_{{\mathbf{k}}}^{G}\Lambda$ is
canonically isomorphic to the $A$-module $A=\Lambda*G$.

Let ${\mathbf{k}}G=\bigoplus_{i=1}^{n}P_{i}$ be a decomposition as a
$G$-module. Then we have a decomposition $A=\bigoplus_{i=1}^{n}\Lambda*P_{i}$
as an $A$-module.

######  Lemma 2.7.

Let $X$ be a $G$-module. Then $X\otimes_{{\mathbf{k}}}^{G}A$ is isomorphic to
$X\otimes_{{\mathbf{k}}}A\cong A^{\oplus\dim X}$ as $A$-$\Lambda$-bimodules.

###### Proof.

It is straightforward to check that the following map is an isomorphism of
$A$-$\Lambda$-bimodules

$\phi:X\otimes_{{\mathbf{k}}}^{G}A\to
X\otimes_{{\mathbf{k}}}A,\phi(x\otimes(r*g)):=g^{-1}(x)\otimes(r*g).$

∎

######  Corollary 2.8.

Let $Q$ be a projective $A$-module and $X$ a finite dimensional $G$-module.
Then $X\otimes_{{\mathbf{k}}}^{G}Q$ is a projective $A$-module.

Now we introduce the notion of ($\operatorname{mod}G$)-stability.

######  Definition 2.9.

An $A$-module $N$ is called ($\operatorname{mod}G$)-_stable_ if
$X\otimes_{{\mathbf{k}}}^{G}N\in\operatorname{add}N$ for any
$X\in\operatorname{mod}G$.

We introduce ${\mathbb{X}}$-stability.

######  Definition 2.10 (The $\mathbb{X}$-stability(cf.[51, p.1070])).

An $A$-module $N$ is called ${\mathbb{X}}$-_stable_ if
$S\otimes_{{\mathbf{k}}}^{G}N\in\operatorname{add}N$ for any $1$-dimensional
simple $G$-module $S$.

We explain that the above definition of ${\mathbb{X}}$-stability coincides
with the original definition given by Zhang-Huang [51]. Let
$\chi:G\to{\mathbf{k}}^{\times}$ be a group homomorphism and
${\mathbf{k}}_{\chi}$ the induced $G$-module. For a $1$-dimensional $G$-module
$S$, there exists a unique $\chi:G\to{\mathbf{k}}^{\times}$ such that
$S\cong{\mathbf{k}}_{\chi}$. Then
$S\otimes_{{\mathbf{k}}}^{G}N={\mathbf{k}}_{\chi}\otimes^{G}_{{\mathbf{k}}}N$
is ${}^{\chi^{-1}}N$ in the notation of [51]. Observe that in case of $N$ is
basic, then we have $S\otimes_{{\mathbf{k}}}^{G}N\in\operatorname{add}N$ if
and only if ${\mathbf{k}}_{\chi}\otimes_{{\mathbf{k}}}^{G}N\cong N$. Thus, we
see that for a basic $A$-module $N$, the two ${\mathbb{X}}$-stability are the
same notion.

We note that an $A$-module $N$ is ($\operatorname{mod}G$)-stable, then it is
$\mathbb{X}$-stable. In Example 3.6, we give an example of support
$\tau$-tilting $A$-modules which is $\mathbb{X}$-stable, but not
($\operatorname{mod}G$)-stable.

#### 2.1.3.

The adjoint property of ($\operatorname{mod}G$)-action given in the following
proposition plays a key role in the proof of Theorem 3.11.

######  Proposition 2.11.

Let $X$ be a finite dimensional $G$-module and
${\textup{D}}(X)=\operatorname{Hom}_{{\mathbf{k}}}(X,{\mathbf{k}})$ the dual
$G$-module of $X$. Then the pair
$(X\otimes_{{\mathbf{k}}}^{G}-,{\textup{D}}(X)\otimes_{{\mathbf{k}}}^{G}-)$ of
endofunctors of $\operatorname{mod}A$ is an adjoint pair.

This proposition follows from a general result in the theory of module
categories over a tensor category [29, Proposition 7.1.6]. But for the
convenience of the readers, we provide a direct proof.

###### Proof.

Let $\\{x_{i}\\}_{i=1}^{n}$ be a basis of $X$ and $\\{x^{i}\\}_{i=1}^{n}$ the
dual basis of ${\textup{D}}(X)$. For an $A$-module $N$, we define maps
$\epsilon_{N}:N\to{\textup{D}}(X)\otimes_{{\mathbf{k}}}^{G}X\otimes_{{\mathbf{k}}}^{G}N$
and
$\eta_{N}:X\otimes_{{\mathbf{k}}}^{G}{\textup{D}}(X)\otimes_{{\mathbf{k}}}^{G}N\to
N$ to be

$\begin{split}\epsilon_{N}&:N\to{\textup{D}}(X)\otimes_{{\mathbf{k}}}^{G}X\otimes_{{\mathbf{k}}}^{G}N,\
\epsilon_{N}(n):=\sum_{i=1}^{n}x^{i}\otimes x_{i}\otimes n\\\
\eta_{N}&:X\otimes_{{\mathbf{k}}}^{G}{\textup{D}}(X)\otimes_{{\mathbf{k}}}^{G}N\to
N,\ \eta_{N}(x\otimes\xi\otimes n):=\xi(x)n.\end{split}$

Then, it is straightforward to check that these maps are homomorphisms of
$A$-modules natural in $N$ that satisfy the triangle identities
$(\eta_{X\otimes^{G}N})\circ(X\otimes^{G}\epsilon_{N})=\operatorname{id}_{X\otimes^{G}N}$,
$({\textup{D}}(X)\otimes^{G}\eta_{N})\circ(\epsilon_{{\textup{D}}(X)\otimes^{G}N})=\operatorname{id}_{{\textup{D}}(X)\otimes^{G}N}$.
∎

###  2.2. Induction-Reduction adjunction

Recall that the extension (i.e., the canonical inclusion) $\Lambda\to A$ is a
Frobenius extension, that means that, $A$ is a free as left $\Lambda$-module
and that we have an isomorphism of $A$-$\Lambda$-bimodules

$\phi:A\to\operatorname{Hom}_{\Lambda}(A,\Lambda),$

which is given by $\phi(r*g)(s*h):=sh(r)\delta_{g,h^{-1}}$. Consequently the
functors $\operatorname{Ind}_{\Lambda}^{A}=A\otimes_{\Lambda}-,\
\operatorname{Hom}_{\Lambda}(A,-):\operatorname{mod}\Lambda\to\operatorname{mod}A$
are naturally isomorphic. Hence the induction functor
$F:=\operatorname{Ind}_{\Lambda}^{A}$ is left and right adjoint of the
restriction functor $R:=\operatorname{Res}_{\Lambda}^{A}$.

$\textstyle{\operatorname{mod}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{\operatorname{mod}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{F}$
(2-1)

####  2.2.1. Basic isomorphisms

We collect isomorphisms of $\Lambda$-modules and $A$-modules, that play
important roles in the sequel. Although these are well-known and can be
(partly) found in the standard references (e.g., [49]), we provides proofs for
the convenience of the readers.

######  Lemma 2.12.

  1. (1)

For a $\Lambda$-module $M$, an element $g\in G$ and $X\in\operatorname{mod}G$,
we have the following isomorphisms

$\begin{split}&\textup{(a)}\ F({}_{g}M)\cong F(M),\ \textup{(b)}\
RF(M)\cong\bigoplus_{g\in G}{}_{g}M,\\\ &\textup{(c)}\ FRF(M)\cong
F(M)^{\oplus|G|},\ \textup{(d)}\
X\otimes_{{\mathbf{k}}}^{G}FM\cong(FM)^{\oplus\dim X}.\end{split}$

  2. (2)

For an $A$-module $N$, an element $g\in G$ and $X\in\operatorname{mod}G$, we
have the following isomorphisms.

$\begin{split}&\textup{(a)}\ R(N)\cong{}_{g}R(N),\ \textup{(b)}\ FR(N)\cong
A\otimes_{\Lambda}N\cong{\mathbf{k}}G\otimes_{{\mathbf{k}}}^{G}N,\\\
&\textup{(c)}\ RFR(N)\cong R(N)^{\oplus|G|},\ \textup{(d)}\
R(X\otimes^{G}_{{\mathbf{k}}}N)\cong X\otimes_{{\mathbf{k}}}N\cong
N^{\oplus\dim X}.\end{split}$

###### Proof.

(1) The isomorphism (a) is given by

$\phi:A\otimes_{\Lambda}{}_{g}M\to A\otimes_{\Lambda}M,\ \phi((r*h)\otimes
m):=(r*hg^{-1})\otimes m.$

The isomorphism (b) is given by

$\phi:\bigoplus_{g\in G}{}_{g}M\to A\otimes_{\Lambda}M,\phi((m_{g})_{g\in
G}):=\sum_{g\in G}(1*g^{-1})\otimes m_{g}.$

The isomorphism (c) follows from (a) and (b). The isomorphism (d) is a
consequence of Lemma 2.7.

(2) It is straightforward to check that the map $\phi_{N}:R(N)\to{}_{g}R(N),\
\phi(n):=(1*g)n$ gives the desired isomorphism (a).

It follows from the definition of $F,R$ that $RF(N)\cong A\otimes_{\Lambda}N$.
It is straightforward to prove that the map below is an isomorphism of
$A$-modules.

$\phi:{\mathbf{k}}G\otimes_{{\mathbf{k}}}^{G}N\to A\otimes_{\Lambda}N=FR(N),\
\phi(g\otimes n):=(1*g)\otimes(1*g^{-1})n$

The isomorphism (c) follows from (a) and (1-b). It is straightforward to check
that the map $\phi:R(X\otimes_{{\mathbf{k}}}^{G}N)\to
X\otimes_{{\mathbf{k}}}R(N),\ \phi(x\otimes n):=x\otimes n$ is an isomorphism
of $\Lambda$-modules. ∎

We point out the following immediate consequence of Lemma 2.12.

######  Corollary 2.13.

  1. (1)

For a $\Lambda$-module $M$, the $A$-modules $F(M)$ is
($\operatorname{mod}G$)-stable. Moreover we have
$\operatorname{add}FRF(M)=\operatorname{add}F(M)$.

  2. (2)

For an $A$-module $N$, the $\Lambda$-modules $R(M)$ is $G$-stable. Moreover we
have $\operatorname{add}RFR(N)=\operatorname{add}R(N)$.

####  2.2.2. A criterion of $G$-stability

As a consequence of Lemma 2.12, we have the following criterion of
$G$-stability

######  Corollary 2.14.

A $\Lambda$-module $M$ is $G$-stable if and only if we have
$\operatorname{add}M=\operatorname{add}RF(M)$.

####  2.2.3. A criterion of ($\operatorname{mod}G$)-stability

Recall that an $A$-module $N$ is called _rigid_ if
$\operatorname{Ext}_{A}^{1}(N,N)=0$. To obtain a criterion of
($\operatorname{mod}G$)-stability similar to Corollary 2.14, we need to assume
that an $A$-module $N$ to be rigid.

######  Proposition 2.15 (cf. [43, Lemma 3.7]).

For a rigid $A$-module $N$, the following conditions are equivalent.

  1. (1)

$N$ is ($\operatorname{mod}G$)-stable.

  2. (2)

$S\otimes_{{\mathbf{k}}}^{G}N\in\operatorname{add}N$ for any simple
$G$-modules $S$.

  3. (3)

$\operatorname{add}FR(N)=\operatorname{add}N$.

Moreover, if we further assume one of the following conditions: (i)
$\operatorname{char}{\mathbf{k}}\nmid|G|$, (ii) ${\mathbf{k}}G$ is local, then
the above conditions are equivalent to the following condition (4).

(4) $FR(N)\in\operatorname{add}N$.

We leave the verification of the following lemma to the readers, since it can
be prove in the same way of [43, Lemma 3.6].

######  Lemma 2.16 (cf. [43, Lemma 3.6]).

Let $N$ be an $A$-module and $X$ a $G$-module. Assume that $N$ is rigid and
that for any composition factors $S$ of $X$, we have
$S\otimes^{G}_{{\mathbf{k}}}N\in\operatorname{add}N$. Then we have
$X\otimes^{G}_{{\mathbf{k}}}N=\bigoplus S\otimes_{{\mathbf{k}}}^{G}N$ where
$S$ runs over all composition factor of $X$.

###### Proof of Proposition 2.15.

(3)$\Rightarrow$(1) follows from Lemma 2.12 (1-d). (1)$\Rightarrow$(2) is
trivial. We prove the implication (2)$\Rightarrow$(3). By Lemma 2.12(2-b) and
Lemma 2.16, we have

$A\otimes_{\Lambda}N\cong{\mathbf{k}}G\otimes_{{\mathbf{k}}}^{G}N\cong\bigoplus
S\otimes_{{\mathbf{k}}}^{G}N$

where $S$ runs all composition factors of ${\mathbf{k}}G$. Since we are
assuming (2) holds, it follows that
$A\otimes_{\Lambda}N\in\operatorname{add}N$. Since the trivial $G$-module
${\mathbf{k}}_{G}$ is a composition factor of ${\mathbf{k}}G$, we see that
$N\cong{\mathbf{k}}_{G}\otimes_{{\mathbf{k}}}^{G}N$ belongs to
$\operatorname{add}(A\otimes_{\Lambda}N)$.

It is clear (3) always implies (4). By Lemma 2.17 below, we see that (4)
together with one of (i), (ii) implies (3). ∎

######  Lemma 2.17.

Let $N$ be an $A$-module. If one of the following conditions holds, then we
have $N\in\operatorname{add}FR(N)$.

  1. (i)

$\operatorname{char}{\mathbf{k}}\nmid|G|$.

  2. (ii)

$N$ is rigid and ${\mathbf{k}}G$ is local.

###### Proof.

It is enough to show that the canonical map $f:FR(N)\to N$ splits. We note
that we have
$f=\epsilon\otimes^{G}_{{\mathbf{k}}}N:{\mathbf{k}}G\otimes_{{\mathbf{k}}}^{G}N\to{\mathbf{k}}_{G}\otimes_{{\mathbf{k}}}^{G}N$
where $\epsilon:{\mathbf{k}}G\to{\mathbf{k}}_{G}$ denotes the augmentation
map.

If the condition (i) holds. Then the canonical map $g$ splits and hence so
does $f$.

If the condition (ii) holds. Then every $G$-module $X$ has a surjective
homomorphism $X\to{\mathbf{k}}_{G}$ of $G$-modules. By induction of length of
$X$ and utilizing the rigidity of $N$, we can show that the induced map
$X\otimes_{{\mathbf{k}}}^{G}N\to{\mathbf{k}}_{G}\otimes_{{\mathbf{k}}}^{G}N$
splits and $X\otimes^{G}_{{\mathbf{k}}}N\in\operatorname{add}N$. ∎

######  Remark 2.18.

Even if an $A$-module $N$ is not rigid, the implications (3) $\Rightarrow$ (1)
$\Rightarrow$ (2) hold true, but converses do not hold true in general.

We provide an example that (2) $\Rightarrow$ (1) $\Rightarrow$ (3) does not
hold true in general. Assume that $\operatorname{char}{\mathbf{k}}=2$. Let
$\Lambda={\mathbf{k}}$ be the base field with the trivial action by a cyclic
group $G=C_{2}$ of order $2$. Then $A={\mathbf{k}}G$ is a usual group algebra.
Since there is only one simple $G$-module ${\mathbf{k}}_{G}$, the trivial
$G$-module, any $A$-module $N$ satisfies the condition (2). But the $A$-module
$N={\mathbf{k}}_{G}$ is not ($\operatorname{mod}G$)-stable, since
${\mathbf{k}}G\otimes_{{\mathbf{k}}}^{G}{\mathbf{k}}_{G}={\mathbf{k}}G\notin\operatorname{add}{\mathbf{k}}_{G}$.
It is easy to see that $N^{\prime}:={\mathbf{k}}_{G}\oplus{\mathbf{k}}G$ is
($\operatorname{mod}G$)-stable but
$\operatorname{add}FR(N^{\prime})\subsetneq\operatorname{add}N^{\prime}$.

####  2.2.4. The local case

Assume that the group algebra ${\mathbf{k}}G$ is local. Then the the unique
simple module is the trivial representation ${\mathbf{k}}_{G}$ of $G$. In this
case, the condition (2) of Proposition 2.15 is always satisfied. Thus as a
consequence, we have the following corollary.

######  Corollary 2.19.

If $G$ is a finite group such that ${\mathbf{k}}G$ is local, then every rigid
module $N$ over $A=\Lambda*G$ is ($\operatorname{mod}G$)-stable as well as it
satisfies $\operatorname{add}N=\operatorname{add}FR(N)$.

####  2.2.5. A lemma

The following lemma is used to prove Lemma 2.24.

######  Lemma 2.20.

Let $e,e^{\prime}\in\Lambda$ be idempotent elements, $r\in e\Lambda
e^{\prime}$ and $g\in G$. Then the following assertions hold.

  1. (1)

We have an isomorphism ${}_{g}\Lambda e\cong\Lambda g^{-1}(e)$ of
$\Lambda$-modules.

  2. (2)

We have an isomorphism $RF(\Lambda e)\cong\bigoplus_{g\in G}\Lambda g(e)$ of
$\Lambda$-modules.

  3. (3)

We denote the right multiplication map $r:\Lambda e\to\Lambda e^{\prime}$ by
the same symbol. Then under the above isomorphism, we have
$RF(r)=\bigoplus_{g\in G}g(r)$.

$RF(r):RF(\Lambda e)\cong\bigoplus_{g\in G}\Lambda g(e)\xrightarrow{\bigoplus
g(r)}\bigoplus_{g\in G}\Lambda g(e^{\prime})=RF(\Lambda e^{\prime})$

###### Proof.

(1) The desired isomorphism is given by

$\phi:{}_{g}\Lambda e\to\Lambda g^{-1}(e),\ \phi(re):=g^{-1}(r)g^{-1}(e).$

(2)(3) follow from (1) and Lemma 2.12(1-b). ∎

###  2.3. Compatibility of the Auslander-Reiten translation $\tau$ with $F$
and $R$

####  2.3.1. Recollection of the radical morphisms

For $M,M^{\prime}\in\operatorname{mod}\Lambda$, the _radical_
$\operatorname{rad}(M^{\prime},M)$ is defined to be a subspace of
$\operatorname{Hom}_{\Lambda}(M^{\prime},M)$ consisting of such elements
$f:M^{\prime}\to M$ that satisfy the following property: for any
$L\in\mathop{\mathrm{ind}}\nolimits\Lambda$ and any morphisms $s:L\to
M^{\prime},t:M\to L$, the composition $tfs:L\to L$ is not an isomorphism. We
note that $\operatorname{rad}(-,+)$ is a ${\mathbf{k}}$-linear sub-bifunctor
of $\operatorname{Hom}_{\Lambda}(-,+)$. We also note that
$\operatorname{rad}(M,M)$ is the Jacobson radical of
$\operatorname{End}_{\Lambda}(M)$ [11, Proposition A.3.5].

Recall that a homomorphism $g:M\to M^{\prime\prime}$ is called _right minimal_
if any endomorphism $h:M\to M$ such that $g\circ h=g$ is an isomorphism.

We need the following two lemmas about minimal morphisms and radical
morphisms.

######  Lemma 2.21.

Let $0\to M^{\prime}\xrightarrow{f}M\xrightarrow{g}M^{\prime\prime}$ be an
exact sequence in $\operatorname{mod}\Lambda$. Then $g$ is right minimal if
and only if $f\in\operatorname{rad}(M^{\prime},M)$.

###### Proof.

“If” part. Assume that $f\in\operatorname{rad}(M^{\prime},M)$. Let $h:M\to M$
be such that $g\circ h=g$. Then, $g\circ(h-\operatorname{id}_{M})=0$. It
follows that there exists $k:M\to M^{\prime}$ such that
$h-\operatorname{id}_{M}=f\circ k$. Since $h-\operatorname{id}_{M}=f\circ k$
belongs to $\operatorname{rad}(M,M)$, it is nilpotent. Therefore,
$h=\operatorname{id}_{M}+(h-\operatorname{id}_{M})$ is an automorphism of $M$.

“Only if” part. We prove that if $f$ does not belongs to
$\operatorname{rad}(M^{\prime},M)$, then $g$ is not a right minimal. Let $L$
be an indecmoposable $\Lambda$-module and $s:L\to M^{\prime},t:M\to L$
homomorphisms of $\Lambda$-modules such that $t\circ f\circ s$ is an
automorphism of $L$. We set $l:=t\circ f\circ s$ and
$h:=\operatorname{id}_{M}-f\circ s\circ l^{-1}\circ t$. Then, since $g\circ
f=0$, we have $g\circ h=g$. On the other hand, $t\circ h=t-t\circ f\circ
s\circ l^{-1}\circ t=t-t=0$. Since $t\neq 0$, we conclude that $h$ is not an
automorphism. ∎

######  Lemma 2.22.

Let $M,M^{\prime}$ be $\Lambda$-modules, $P$ projective $\Lambda$-modules and
$f:M\to P$, $g:M^{\prime}\to M$ homomorphisms of $\Lambda$-modules. Assume
that $g:M^{\prime}\to M$ is surjective. Then $f$ belongs to
$\operatorname{rad}(M,P)$ if and only if $f\circ g$ belongs to
$\operatorname{rad}(M^{\prime},P)$.

###### Proof.

“Only if” part is clear. We prove “if” part by contraposition. Assume that $f$
does not belong to $\operatorname{rad}(M,P)$. Then there exists an
indecomposable $\Lambda$-module $L$ and homomorphisms $s:L\to M,\ t:P\to L$
such that $l:=t\circ f\circ s$ is an automorphism of $L$. It follows that
$f\circ s\circ l^{-1}$ is a section of $t$ (i.e., $t\circ(f\circ s\circ
l^{-1})=\operatorname{id}_{L}$) and hence the $\Lambda$-module $L$ is a
projective. Since $g:M^{\prime}\to M$ is surjective, there exists a
homomorphism $r:L\to M^{\prime}$ such that $g\circ r=s$. It follows that
$t\circ(f\circ g)\circ r=t\circ f\circ s$ is an automorphism. This shows that
$f\circ g$ does not belong to $\operatorname{rad}(M^{\prime},P)$.

---
$\textstyle{M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\scriptstyle{s}$

∎

####  2.3.2. The Jacobson radicals and radical morphisms

The following lemma about the Jacobson radicals seems to be well-known. But
for the convenience of the readers we provide a proof.

######  Lemma 2.23.

Let $J_{\Lambda}$ and $J_{A}$ be the Jacobson radicals of $\Lambda$ and $A$.
Then the following assertions hold.

  1. (1)

For an element $g\in G$, we have $g(J_{\Lambda})=J_{\Lambda}$.

  2. (2)

$J_{\Lambda}\subset J_{A}$.

###### Proof.

(1) Let ${\mathfrak{m}}$ be a maximal left ideal of $\Lambda$. Then the image
$g({\mathfrak{m}})$ by the action of $g\in G$ is again a maximal left ideal of
$\Lambda$. It follows that $g(J_{\Lambda})=J_{\Lambda}$.

(2) Let $j\in J_{\Lambda}$. To prove $j\in J_{A}$, we show that for any
$\alpha,\beta\in A$, the element $1-\alpha j\beta$ of $A$ is invertible. We
set $\gamma:=\alpha j\beta$. If we write $\alpha=\sum_{g\in G}\alpha_{g}*g,\
\beta=\sum_{g\in G}\beta_{g}*g$, then
$\gamma=\sum_{g,h}\alpha_{g}g(j)g(\beta_{h})*gh$. It follows from (1) and the
fact that $J_{\Lambda}$ is a two-sided ideal that $\gamma$ is of the form
$\sum_{g}\gamma_{g}*g$ for some $\gamma_{g}\in J_{\Lambda}$. Since $\Lambda$
is finite dimensional, the Jacobson radical $J_{\Lambda}$ is nilpotent. Hence
$\gamma$ is nilpotent. It follows that $1-\gamma=1-\alpha j\beta$ is
invertible. ∎

We fix a complete set $\\{e_{i}\\}_{i\in I}$ of primitive orthogonal
idempotent element of $\Lambda$. Let $P,P^{\prime}$ be projective
$\Lambda$-modules. We take decomposition $P=\bigoplus_{s=1}^{m}P_{s},\
P^{\prime}=\bigoplus_{t=1}^{n}P^{\prime}_{t}$ into indecomposable projective
$\Lambda$-modules $P_{s}\cong\Lambda e_{i(s)},P^{\prime}_{t}\cong\Lambda
e_{j(t)}$. Recall that a homomorphism $f:P\to P^{\prime}$ of $\Lambda$-modules
belongs to $\operatorname{rad}(P,P^{\prime})$ if and only if each component
$f_{ts}:P_{s}\to P^{\prime}_{t}$ belongs to the radical
$e_{i(s)}J_{\Lambda}e_{j(t)}$ under the identification

$\operatorname{Hom}_{\Lambda}(P_{s},P^{\prime}_{t})\cong\operatorname{Hom}_{\Lambda}(\Lambda
e_{i(s)},\Lambda e_{j(t)})\cong e_{i(s)}\Lambda e_{j(t)}.$

By Lemma 2.23 and Lemma 2.20, we see that $F$ and $RF$ preserve radical
morphisms between projective modules. Thus we obtain the following lemma.

######  Lemma 2.24.

Let $P,P^{\prime}$ be projective $\Lambda$-modules. Then for a radical
morphism $f:P\to P^{\prime}$, the morphism $F(f):F(P)\to F(P^{\prime})$ (resp.
$RF(f):RF(P)\to RF(P^{\prime})$) is a radical morphism between projective $A$
(resp. $\Lambda$)-modules.

Let $M$ be a $\Lambda$-module and $P$ a projective $\Lambda$-module. Recall
from [8, Proposition I.4.1] that a surjective homomorphism $p:P\to M$ is
projective cover if and only if it is right minimal.

The following lemma shows that the functors $F,RF$ preserve projective covers.

######  Lemma 2.25.

Let $p:P\to M$ be a projective cover of a $\Lambda$-module $M$. Then

  1. (1)

$F(p):F(P)\to F(M)$ is a projective cover.

  2. (2)

$RF(p):RF(P)\to RF(M)$ is a projective cover.

###### Proof.

(1) Let $K=\mathop{\mathrm{Ker}}\nolimits p$ be the kernel of $p$ and $i:K\to
P$ the canonical inclusion. By Lemma 2.21, $i\in\operatorname{rad}(K,P)$. By
the same lemma, we only have to show that
$F(i)\in\operatorname{rad}(F(K),F(P))$, since $F$ is exact and preserves
projective modules.

Let $q:P^{\prime}\to K$ be a projective cover. Then $i\circ
q\in\operatorname{rad}(P^{\prime},P)$. By Lemma 2.24, we have $F(i)\circ
F(q)=F(i\circ q)\in\operatorname{rad}(F(P^{\prime}),F(P))$. Since $F$ is
exact, $F(q):F(P^{\prime})\to F(K)$ is a surjective homomorphism. Thus we can
apply Lemma 2.22 and conclude that $F(i)$ belongs to
$\operatorname{rad}(F(K),F(P))$.

(2) Since $RF$ is exact and preserves projective modules and radical morphisms
between projective modules, the same proof with (1) works for $RF$. ∎

Let $M$ be a $\Lambda$-module. Recall that a projective presentation
$P_{1}\xrightarrow{f}P_{0}\xrightarrow{p}M\to 0$ is called _minimal_ if the
homomorphisms $p:P_{0}\to M$ and $f:P_{1}\to\mathop{\mathrm{Ker}}\nolimits p$
are projective covers.

As a consequence of Lemma 2.25, we obtain the following corollary that shows
that the functors $F,RF$ preserve minimal projective presentations.

######  Corollary 2.26.

Let $M$ be a $\Lambda$-module and $P_{1}\xrightarrow{f}P_{0}\to M\to 0$ a
minimal projective presentation. Then, the following assertions hold.

  1. (1)

The projective presentation $F(P_{1})\xrightarrow{F(f)}F(P_{0})\to F(M)\to 0$
of the $A$-module $F(M)$ is minimal.

  2. (2)

The projective presentation $RF(P_{1})\xrightarrow{RF(f)}RF(P_{0})\to RF(M)\to
0$ of the $\Lambda$-module $RF(M)$ is minimal.

####  2.3.3. Compatibility of the AR-translation $\tau$ with $F$

It was proved that the Auslander-Reiten translations $\tau$ (of $\Lambda$ and
of $A$) commutes with the induction functor $F$ in the case that $|G|$ is
invertible in ${\mathbf{k}}$ by Reiten-Riedtmann [49, Proof of Lemma 4.2].
(Breaz-Marcus-Modoi [17, Proposition 3.5] proved the same commutativity for
group graded algebras satisfying some conditions.) In the next proposition, we
show that the commutativity holds true in general.

######  Proposition 2.27 ([20, Lemma 7.3]).

Let $M$ be a $\Lambda$-module. Then, $\tau F(M)=F\tau(M)$.

This proposition has been demonstrated in [20, Lemma 7.3] using the same
method as our proof. However, for the reader’s convenience, we include a proof
here.

We need the following compatibility of $F=A\otimes_{\Lambda}-$ with the
${\mathbf{k}}$-duality
${\textup{D}}(-)=\operatorname{Hom}_{{\mathbf{k}}}(-,{\mathbf{k}})$.

######  Lemma 2.28.

For a right $\Lambda$-module $M^{\prime}$, we have the following isomorphism
of left $A$-modules which is natural in $M^{\prime}$

$A\otimes_{\Lambda}{\textup{D}}(M^{\prime})\cong{\textup{D}}(M^{\prime}\otimes_{\Lambda}A)$

###### Proof.

The desired isomorphism is obtained from the following string of isomorphisms:

$\begin{split}A\otimes_{\Lambda}{\textup{D}}(M^{\prime})&\cong\operatorname{Hom}_{\Lambda}(A,\Lambda)\otimes_{\Lambda}{\textup{D}}(M^{\prime})\cong\operatorname{Hom}_{\Lambda}(A,{\textup{D}}(M^{\prime}))\\\
&\cong\operatorname{Hom}_{{\mathbf{k}}}(M^{\prime}\otimes_{\Lambda}A,{\mathbf{k}})={\textup{D}}(M^{\prime}\otimes_{\Lambda}A).\end{split}$

∎

For simplicity we use the same symbol $(-)^{*}$ to denote the dualities
$(-)^{*}:=\operatorname{Hom}_{\Lambda}(-,\Lambda)$ and
$(-)^{*}:=\operatorname{Hom}_{A}(-,A)$

###### Proof of Proposition 2.27.

Let $P_{1}\xrightarrow{f}P_{0}\to M\to 0$ be a minimal projective presentation
of $M$. Then we have
$\tau(M)\cong\mathop{\mathrm{Ker}}\nolimits[D(f^{*}):{\textup{D}}(P_{1}^{*})\to{\textup{D}}(P_{0}^{*})]$
and by Lemma 2.28

$\begin{split}A\otimes_{\Lambda}\tau(M)&\cong\mathop{\mathrm{Ker}}\nolimits[A\otimes
D(f^{*}):A\otimes_{\Lambda}{\textup{D}}(P_{1}^{*})\to
A\otimes_{\Lambda}{\textup{D}}(P_{0}^{*})]\\\
&\cong\mathop{\mathrm{Ker}}\nolimits[D(f^{*}\otimes
A):{\textup{D}}(P_{1}^{*}\otimes_{\Lambda}A)\to{\textup{D}}(P_{0}^{*}\otimes_{\Lambda}A)]\\\
&\cong\mathop{\mathrm{Ker}}\nolimits[D((A\otimes
f)^{*}):{\textup{D}}((A\otimes_{\Lambda}P_{1})^{*})\to{\textup{D}}((A\otimes_{\Lambda}P_{0})^{*})]\\\
\end{split}$

where for the last isomorphism we use the isomorphism
$\operatorname{Hom}_{\Lambda}(P,\Lambda)\otimes_{\Lambda}A\cong\operatorname{Hom}_{A}(A\otimes_{\Lambda}P,A)$
for a $\Lambda$-module $P$.

By Lemma 2.26(1), the projective presentation
$A\otimes_{\Lambda}P_{1}\xrightarrow{A\otimes f}A\otimes_{\Lambda}P_{0}\to
A\otimes_{\Lambda}M\to 0$ of $A\otimes_{\Lambda}M$ is minimal. It follows that
$\tau(A\otimes_{\Lambda}M)=\mathop{\mathrm{Ker}}\nolimits D((A\otimes
f)^{*})\cong A\otimes_{\Lambda}\tau(M)$. ∎

####  2.3.4. Compatibility of the AR-translation $\tau$ with $R$

It is well-known that if $\operatorname{char}{\mathbf{k}}\nmid|G|$, the
Auslander-Reiten translations $\tau$ (of $\Lambda$ and of $A$) commutes with
the restriction functor $R$. (Dade [24, Corollary 5.7] (see also [17,
Proposition 3.7]) proved the commutativity for group graded algebras
satisfying certain condition.)

We remark that the commutativity does not hold true in general.

######  Example 2.29.

For example, let ${\mathbf{k}}={\mathbb{F}}_{2},G={\mathbb{Z}}/2{\mathbb{Z}}$
and $\Lambda={\mathbf{k}}$ with the trivial $G$-action. Then
$A={\mathbf{k}}G\cong{\mathbf{k}}[x]/(x^{2})$ and for a trivial $G$-module
$N:={\mathbf{k}}$ we have $R\tau(N)=R(N)$ and $\tau R(N)=0$. Thus $\tau
R(N)\neq R\tau(N)$.

In the next proposition, we show that if we put a mild condition (irrelevant
to $|G|$ nor $\operatorname{char}{\mathbf{k}}$) on an $A$-module $N$, then we
have $\tau R(N)\cong R\tau(N)$.

######  Proposition 2.30.

Let $N$ be an $A$-module and $Q_{1}\xrightarrow{g}Q_{0}\to N\to 0$ be a
minimal projective presentation. Assume that $N\in\operatorname{add}FR(N)$.
Then the following assertions hold.

  1. (1)

The morphism $R(g):R(Q_{1})\to R(Q_{0})$ gives a minimal projective
presentation of $R(N)$.

  2. (2)

The morphism $FR(g):FR(Q_{1})\to FR(Q_{0})$ gives a minimal projective
presentation of $FR(N)$.

  3. (3)

We have $\tau R(N)\cong R\tau(N)$.

We note that by Lemma 2.17, in the case
$\operatorname{char}{\mathbf{k}}\nmid|G|$, we have
$N\in\operatorname{add}FR(N)$ for any $A$-module $N$. Hence we can recover the
commutativity $\tau R(N)\cong R(\tau N)$ in the classical case
$\operatorname{char}{\mathbf{k}}\nmid|G|$ from the above proposition. We also
note that thanks to Lemma 2.15, a ($\operatorname{mod}G$)-stable rigid
$A$-module $N$ satisfies the condition $N\in\operatorname{add}FR(N)$.

To prove the proposition, we need the following compatibility of $R$ with
dualities $(-)^{*}=\operatorname{Hom}_{\Lambda}(-,\Lambda)$ and
$(-)^{*}=\operatorname{Hom}_{A}(-,A)$

######  Lemma 2.31.

For an $A$-module $N$, there is an isomorphism $R(N^{*})\cong R(N)^{*}$ which
is natural in $N$.

###### Proof.

Since $F$ is a right adjoint of $R$, we have the following isomorphism of
${\mathbf{k}}$-vector spaces

$\operatorname{Hom}_{A}(N,A)\cong\operatorname{Hom}_{A}(N,F(\Lambda))\cong\operatorname{Hom}_{\Lambda}(R(N),\Lambda).$

It is straightforward to check that this map is compatible with the action of
$\Lambda$. ∎

###### Proof of Proposition 2.30.

(1) We set $M:=R(N)$. Let $P_{1}\xrightarrow{f}P_{0}\to M\to 0$ be a minimal
projective presentation of $M$. It follows from the assumption
$N\in\operatorname{add}F(M)$ that the complex $[Q_{1}\to Q_{0}]$ is a direct
summand of a direct sum of the complex $[F(P_{1})\to F(P_{0})]$. Hence the
complex $[R(Q_{1})\to R(Q_{0})]$ is a direct summand of a direct sum of the
complex $[RF(P_{1})\to RF(P_{0})]$. On the other hand, by Corollary 2.26, the
projective presentation $RF(P_{1})\xrightarrow{RF(f)}RF(P_{0})\to RF(M)\to 0$
is minimal. It is easy to see that if a projective presentation is a direct
summand of a minimal projective presentation, then it is a minimal projective
presentation. Thus we conclude that $R(g)$ gives a minimal projective
presentation of $R(N)$.

(2) follows from Corollary 2.26.

(3) We have the following isomorphisms by Lemma 2.31

$\begin{split}R\tau(N)&\cong\mathop{\mathrm{Ker}}\nolimits[R{\textup{D}}(g^{*}):R{\textup{D}}(Q_{1}^{*})\to
R{\textup{D}}(Q_{0}^{*})]\\\
&\cong\mathop{\mathrm{Ker}}\nolimits[{\textup{D}}(R(g^{*})):{\textup{D}}(R(Q_{1}^{*}))\to{\textup{D}}(R(Q_{0}^{*}))]\\\
&\cong\mathop{\mathrm{Ker}}\nolimits[{\textup{D}}((Rg)^{*}):{\textup{D}}(R(Q_{1})^{*})\to{\textup{D}}(R(Q_{0})^{*})].\end{split}$

By (1), $R(Q_{1})\xrightarrow{R(g)}R(Q_{0})\to R(N)\to 0$ is a minimal
projective presentation of $R(N)$. It follows that
$R\tau(N)\cong\mathop{\mathrm{Ker}}\nolimits{\textup{D}}((Rg)^{*})=\tau R(N)$.
∎

##  3\. $\tau$-tilting theory of skew group algebra extensions

In this section we discuss $\tau$-tilting theory of a skew group algebra
extension $\Lambda\subset\Lambda*G$. For the definition of support
$\tau$-tilting modules and other basic notions and results, we refer [1].

###  3.1. The bijection

The next theorem is the first step to prove Theorem 3.3, which is the
$\tau$-tilting part of Main Theorem 1.1.

######  Theorem 3.1.

  1. (1)

If $M$ is a $G$-stable support $\tau$-tilting $\Lambda$-module, then $F(M)$ is
a support $\tau$-tilting $A$-module.

  2. (2)

If $N$ is a ($\operatorname{mod}G$)-stable support $\tau$-tilting $A$-module,
then $R(N)$ is a support $\tau$-tilting $\Lambda$-module.

###### A sketch of the first proof of Theorem 3.1..

(1) is a generalization of [41, Theorem 3.2] in our setting as well as a
generalization of [51, Theorem 4.2]. With the preparations given before mainly
the commutativity $\tau F(M)\cong F(\tau M)$ (Proposition 2.27), the same
proof with [51] works.

(2) is a generalization of [43, Theorem 3.4] in our setting. With the
preparations given before, mainly the commutativity $\tau R(N)\cong R(\tau N)$
for ($\operatorname{mod}G$)-stable $N$ (Proposition 2.30), the same proof with
[43] works. ∎

######  Remark 3.2.

In Section 4.3 we give an alternative proof of Theorem 3.1 by using $2$-term
silting complexes. In the proof we do not use the commutativity $\tau
F(M)\cong F(\tau M),\ \tau R(N)\cong R(\tau N)$ that played crucial roles in
the first proof.

Let $\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda,\
\textup{s}\tau\textup{-}\operatorname{tilt}A$ be the posets of (isomorphism
classes of) basic support $\tau$-tilting modules over $\Lambda$ and $A$
respectively. We denote by
$(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}$ the subposet of
$G$-stable basic support $\tau$-tilting $\Lambda$-modules, and by
$(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}$ the
poset of ($\operatorname{mod}G$)-stable basic support $\tau$-tilting
$A$-modules.

We remark that since we are not assuming that our algebras $\Lambda,A$ are
basic, strictly speaking the $\tau$-tilting module $\Lambda$ (resp. $A$) does
not necessarily belong to $\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda$
(resp. $\textup{s}\tau\textup{-}\operatorname{tilt}A$). However, to simplify
the notation, we often denote $\Lambda$ and $A$ to denote the basic
$\tau$-tilting modules $\Lambda^{\mathsf{basic}}$ and $A^{\mathsf{basic}}$.

######  Theorem 3.3 (cf. [43, Corollary 3.9]).

The adjoint pair
$(F=\operatorname{Ind}_{\Lambda}^{A},R=\operatorname{Res}_{\Lambda}^{A})$
induces an isomorphism of posets

$F=\operatorname{Ind}_{\Lambda}^{A}:(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}\longleftrightarrow(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}:\operatorname{Res}_{\Lambda}^{A}=R.$

###### Proof.

Thanks to Theorem 3.1, the maps
$F:(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}\to(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G},\
$ $M\mapsto F(M)^{\mathsf{basic}}$ and
$R:(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}\to(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G},\
N\mapsto R(N)^{\mathsf{basic}}$ are well-defined. Moreover, it follows from
Corollary 2.14 and Proposition 2.15 that these maps are the inverse maps to
each other.

Recall that for two support $\tau$-tilting modules $M_{1},M_{2}$, the
inequality $M_{1}\geq M_{2}$ is defined to be existence of a surjection
$M^{\prime}_{1}\twoheadrightarrow M_{2}$ from some
$M^{\prime}_{1}\in\operatorname{add}M_{1}$. Since the functors $F,R$ preserve
surjections, it is straightforward to check that the above maps are poset
morphisms. ∎

Recall that a finite abelian group $G$ is said to split over ${\mathbf{k}}$ if
all its simple modules are $1$-dimensional. It follows that the case that a
finite group $G$ is abelian and splits over ${\mathbf{k}}$,
($\operatorname{mod}G$)-stability of $A$-modules coincides with
${\mathbb{X}}$-stability of $A$-modules. Thus we obtain the following
corollary in which we denote the poset of ${\mathbb{X}}$-stable support
$\tau$-tilting $A$-modules by
$(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{{\mathbb{X}}}$.

######  Corollary 3.4.

Assume that $G$ is a finite abelian group which splits over ${\mathbf{k}}$.
Then the adjoint pair
$(\operatorname{Ind}_{\Lambda}^{A},\operatorname{Res}_{\Lambda}^{A})$ induces
an isomorphism of posets

$\operatorname{Ind}_{\Lambda}^{A}:(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}\longleftrightarrow(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{{\mathbb{X}}}:\operatorname{Res}_{\Lambda}^{A}.$

######  Remark 3.5.

We note that if ${\mathbf{k}}$ is algebraically closed, then every finite
abelian group $G$ splits over ${\mathbf{k}}$. In the case that ${\mathbf{k}}$
is algebraically closed, the statement of Corollary 3.4 for an abelian group
$G$ such that $\operatorname{char}{\mathbf{k}}\nmid|G|$ was claimed in [51,
Corollary 4.7], which was obtained as a consequence of [51, Theorem 4.6] that
claimed the same statement holds true even in the case that $G$ is solvable
such that $\operatorname{char}{\mathbf{k}}\nmid|G|$. However, as is shown in
Example 3.6, there exists an example of an algebra $\Lambda$ with an action
$G\curvearrowright\Lambda$ of a solvable group $G$ such that
$\operatorname{char}{\mathbf{k}}\nmid|G|$ that satisfies
$(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}\subsetneq(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{{\mathbb{X}}}$
where we set $A:=\Lambda*G$. Consequently, the map
$\operatorname{Ind}_{\Lambda}^{A}:(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}\to(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{{\mathbb{X}}}$
is not a bijection.

####  3.1.1. An example

######  Example 3.6.

Assume that ${\mathbf{k}}$ is algebraically closed.

Let $G={\mathfrak{S}}_{3}$ be the symmetric group of degree $3$ and
$\Lambda:={\mathbf{k}}[x]/(x^{2})$. We equip an action of $G$ on $\Lambda$ to
be $\sigma\cdot x:=\operatorname{sgn}(\sigma)x$ for $\sigma\in S_{3}$. We set
$A:=\Lambda*G$.

Since the algebra $\Lambda$ is local, we have
$\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda=\\{0<\Lambda\\}$. It is
clear that both of $\Lambda,0$ are $G$-stable. Hence we have
$(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}=\\{0<\Lambda\\}$. It
follows from Theorem 3.3 that
$(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}=\\{0<A\\}$.

From now, we directly compute the Hasse quiver of
$\textup{s}\tau\textup{-}\operatorname{tilt}A$, determine the actions
$S\otimes_{{\mathbf{k}}}^{G}-$ of simple $G$-module on
$\textup{s}\tau\textup{-}\operatorname{tilt}A$ and finally check that
$(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}=\\{0<A\\}$
actually holds. We see that the $\tau$-tilting theory of $A$ is heavily
depends on the characteristic of the base fields.

(I) The case $\operatorname{char}{\mathbf{k}}\neq 2,3$.

To describe the skew group algebra $A$, we set $Q$ to be the quiver below and
$A^{\prime}_{12}:={\mathbf{k}}Q/(x^{\prime}x^{\prime\prime},x^{\prime\prime}x^{\prime})$.

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x^{\prime}}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x^{\prime\prime}}$

Then using the general method given in [15, Section 3] (or direct
computation), we can check that $A$ is isomorphic to $A^{\prime}_{12}\times
M_{2}(\Lambda)$ and hence Morita equivalent to
$A^{\prime}:=A^{\prime}_{12}\times\Lambda$. We set $e_{3}:=(0,1_{\Lambda})\in
A^{\prime}$.

The Hasse diagram of the poset
$\textup{s}\tau\textup{-}\operatorname{tilt}A=\textup{s}\tau\textup{-}\operatorname{tilt}A^{\prime}$
is given below:

---
$\textstyle{Q_{1}\oplus Q_{2}\oplus
Q_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{1}\oplus
Q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{2}\oplus
Q_{2}\oplus
Q_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{1}\oplus
S_{1}\oplus
Q_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{2}\oplus
Q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{1}\oplus
S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{2}\oplus
Q_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{1}\oplus
Q_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{2}}$$\textstyle{S_{1}}$$\textstyle{Q_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $Q_{1},Q_{2},Q_{3}$ denote the indecomposable projective
$A^{\prime}$-modules corresponding to the vertices $1,2,3$ and $S_{1},S_{2}$
denote the simple $A^{\prime}$-modules corresponding to the vertices $1,2$.

We study ($\operatorname{mod}G$)-stability and ${\mathbb{X}}$-stability. Let
${\mathbf{k}}_{G}$ be the trivial representation,
${\mathbf{k}}_{\operatorname{sgn}}$ the sign representation and $V$ the unique
irreducible $2$-dimensional representation of $G$. Using Example 2.6 we can
check that

$\begin{split}{\mathbf{k}}_{\operatorname{sgn}}\otimes_{{\mathbf{k}}}^{G}Q_{1}\cong
Q_{2},\ {\mathbf{k}}_{\operatorname{sgn}}\otimes_{{\mathbf{k}}}^{G}Q_{2}\cong
Q_{1},\ {\mathbf{k}}_{\operatorname{sgn}}\otimes_{{\mathbf{k}}}^{G}Q_{3}\cong
Q_{3},\\\
{\mathbf{k}}_{\operatorname{sgn}}\otimes_{{\mathbf{k}}}^{G}S_{1}\cong S_{2},\
{\mathbf{k}}_{\operatorname{sgn}}\otimes_{{\mathbf{k}}}^{G}S_{2}\cong S_{1},\
{\mathbf{k}}_{\operatorname{sgn}}\otimes_{{\mathbf{k}}}^{G}S_{3}\cong
S_{3},\\\ V\otimes_{{\mathbf{k}}}^{G}Q_{1}\cong Q_{3},\
V\otimes_{{\mathbf{k}}}^{G}Q_{2}\cong Q_{3},\
V\otimes_{{\mathbf{k}}}^{G}Q_{3}\cong Q_{1}\oplus Q_{2}\oplus
Q_{3}.\end{split}$ (3-2)

Thus, ($\operatorname{mod}G$)-stable support $\tau$-tilting modules are
precisely $Q_{1}\oplus Q_{2}\oplus Q_{3}$ and $0$. These are
${\mathbb{X}}$-stable. But there are two more ${\mathbb{X}}$-stable support
$\tau$-tilting modules $Q_{3}$ and $Q_{1}\oplus Q_{2}$.

######  Remark 3.7.

The above isomorphisms (3-2) are that in the category
$A^{\prime}\operatorname{mod}$. Thus, in particular, we regard
${\mathbf{k}}_{\operatorname{sgn}}\otimes_{{\mathbf{k}}}^{G}-,\
V\otimes_{{\mathbf{k}}}^{G}-$ are endofunctors of
$A^{\prime}\operatorname{mod}$ via the equivalence
$(A^{\prime}\times\Lambda)\operatorname{mod}\cong\operatorname{mod}A$. The
same remark is applied to the isomorphisms (3-3).

(II) The case $\operatorname{char}{\mathbf{k}}=2$. In this case, the action
$G\curvearrowright\Lambda$ is trivial. It follows that the skew group algebra
$A$ is the ordinary group algebra $A=\Lambda\otimes{\mathbf{k}}G$. It is well-
known that ${\mathbf{k}}G\cong{\mathbf{k}}[y]/(y^{2})\times
M_{2}({\mathbf{k}})$. Hence we see that $A\cong\Lambda[y]/(y^{2})\times
M_{2}(\Lambda)$ and that $A$ is Morita equivalent to
$A^{\prime\prime}:=\Lambda[y]/(y^{2})\times\Lambda={\mathbf{k}}[x,y]/(x^{2},y^{2})\times{\mathbf{k}}[x]/(x^{2})$.

The Hasse diagram of the poset
$\textup{s}\tau\textup{-}\operatorname{tilt}A=\textup{s}\tau\textup{-}\operatorname{tilt}A^{\prime\prime}$
is given below:

---
$\textstyle{Q_{1}\oplus
Q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $Q_{1}$ and $Q_{2}$ denote the indecomposable projective
$A^{\prime\prime}$-modules corresponding to the primitive idempotent elements
$e_{1}:=(1,0),e_{2}:=(0,1)$ of $A^{\prime\prime}$.

Since we are assuming $\operatorname{char}{\mathbf{k}}=2$, the symmetric group
$G={\mathfrak{S}}_{3}$ has two simple modules, the trivial module
${\mathbf{k}}_{G}$ and the unique $2$-dimensional irreducible representation
$V$. It follows that every $A$-module is ${\mathbb{X}}$-stable. We can check
that

$\begin{split}V\otimes_{{\mathbf{k}}}^{G}Q_{1}\cong Q_{2}^{\oplus 2},\
V\otimes_{{\mathbf{k}}}^{G}Q_{2}\cong Q_{1}\oplus Q_{2}.\end{split}$ (3-3)

Thus, ($\operatorname{mod}G$)-stable support $\tau$-tilting modules are
$Q_{1}\oplus Q_{2}$ and $0$.

(III) The case $\operatorname{char}{\mathbf{k}}=3$.

Let $\eta,\zeta\in G$ be elements of order $3$ and $2$ respectively. Then we
have $G\cong\langle\eta\rangle\rtimes\langle\zeta\rangle$. Since $\eta$
trivially acts on $\Lambda$, we have
$A=\Lambda*G\cong(\Lambda*\langle\eta\rangle)*\langle\zeta\rangle\cong(\Lambda\otimes{\mathbf{k}}[y]/(y^{3}))*\langle\zeta\rangle\cong({\mathbf{k}}[x,y]*\langle\zeta\rangle)/(x^{2},y^{3})$
where in the right most term, we take the action
$\langle\zeta\rangle\curvearrowright{\mathbf{k}}[x,y]$ given by $\zeta\cdot
x:=-x,\ \zeta\cdot y:=-y$. It is well-known that the algebra
${\mathbf{k}}[x,y]*\langle\zeta\rangle$ is isomorphic to the preprojective
algebra $\Pi(\tilde{{\mathbb{A}}}_{1})$ of the extended Dynkin quiver
$\tilde{{\mathbb{A}}}_{1}$. It follows that the skew group algebra $A$ is
isomorphic to the following path algebra ${\mathbf{k}}Q^{\prime\prime}/I$ with
the relations

$Q^{\prime\prime}:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
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59.50044pt\raise 13.71976pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
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0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
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59.50044pt\raise 7.9758pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 59.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
27.64412pt\raise-25.0625pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.0625pt\hbox{$\scriptstyle{x^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
5.42169pt\raise-12.99985pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
27.80292pt\raise-7.25694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.38196pt\hbox{$\scriptstyle{y^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
5.49463pt\raise-7.42926pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces$

and the relations are
$xx^{\prime}=0,x^{\prime}x=0,yy^{\prime}y=0,y^{\prime}yy^{\prime}=0,x^{\prime}y=y^{\prime}x,xy^{\prime}=yx^{\prime}$.

The Hasse quiver of the poset $\textup{s}\tau\textup{-}\operatorname{tilt}A$
has two connected components both of which are infinite:

---
$\textstyle{Q_{1}\oplus Q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{(2)}\oplus Q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{1}\oplus N^{\prime}_{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{(2)}\oplus N_{(3)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{\prime}_{(3)}\oplus N^{\prime}_{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{(4)}\oplus N_{(3)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{\prime}_{(3)}\oplus N^{\prime}_{(4)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots}$$\textstyle{\vdots}$ |
---|---
$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{(3)}\oplus
L_{(4)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{\prime}_{(3)}\oplus
L^{\prime}_{(4)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{(3)}\oplus
L_{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{\prime}_{(3)}\oplus
L^{\prime}_{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{1}\oplus
L_{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{2}\oplus
L^{\prime}_{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(3-4)

where

$\begin{split}&N_{(2)}:=\begin{smallmatrix}&2&&2&&\\\ 1&&1&&1&\\\
&&&2&&2\end{smallmatrix},N_{(3)}:=\begin{smallmatrix}&2&&2&&2&&\\\
1&&1&&1&&1&\\\
&&&&&2&&2\end{smallmatrix},N_{(4)}:=\begin{smallmatrix}&2&&2&&2&&2&&\\\
1&&1&&1&&1&&1&\\\ &&&&&&&2&&2\end{smallmatrix},\dots\\\
&L_{(2)}:=\begin{smallmatrix}2&&2\\\
&1\end{smallmatrix},L_{(3)}:=\begin{smallmatrix}2&&2&&2\\\
&1&&1\end{smallmatrix},L_{(4)}:=\begin{smallmatrix}2&&2&&2&&2\\\
&1&&1&&1\end{smallmatrix},\dots\end{split}$

and the composition series of $N^{\prime}_{(i)},L^{\prime}_{(i)}$ for
$i=2,3,\dots$ are obtained from that of $N_{(i)},L_{(i)}$ by replacing the
composition factors $1,2$ with $2,1$.

We can show by the same method of the proof of [26, Theorem 6.17] that the
Hasse quiver of $\textup{s}\tau\textup{-}\operatorname{tilt}A$ has precisely
two components given in (3-4). (We have the following alternative proof. The
element $\eta:=yy^{\prime}+y^{\prime}y$ of $A$ is central and belongs to the
Jacobson radical. It follows from [30, Theorem 4.1] that the canonical map
$\textup{s}\tau\textup{-}\operatorname{tilt}A\to\textup{s}\tau\textup{-}\operatorname{tilt}A/(\eta)$
is bijective. On the other hand, the algebra $A/(\eta)$ is precisely the
algebra dealt in [26, Theorem 6.17]. Hence we can obtain the Hasse quiver of
the poset
$\textup{s}\tau\textup{-}\operatorname{tilt}A/(\eta)\cong\operatorname{2-silt}A/(\eta)$.)

Since we are assuming $\operatorname{char}{\mathbf{k}}=3$, the symmetric group
$G={\mathfrak{S}}_{3}$ has two simple modules, the trivial module
${\mathbf{k}}_{G}$ and the sign representation
${\mathbf{k}}_{\operatorname{sgn}}$. It follows in particular that
($\operatorname{mod}G$)-stability coincides with ${\mathbb{X}}$-stability. We
can check that

$\begin{split}{\mathbf{k}}_{\operatorname{sgn}}\otimes_{{\mathbf{k}}}^{G}Q_{1}\cong
Q_{2},\ {\mathbf{k}}_{\operatorname{sgn}}\otimes_{{\mathbf{k}}}^{G}Q_{2}\cong
Q_{1},{\mathbf{k}}_{\operatorname{sgn}}\otimes_{{\mathbf{k}}}^{G}S_{1}\cong
S_{2},\ {\mathbf{k}}_{\operatorname{sgn}}\otimes_{{\mathbf{k}}}^{G}S_{2}\cong
S_{1}.\end{split}$

It follows that the functor
${\mathbf{k}}_{\operatorname{sgn}}\otimes^{G}_{{\mathbf{k}}}-$ induces the
reflection with respect to the vertical dotted lines of the Hasse quiver of
the poset $\textup{s}\tau\textup{-}\operatorname{tilt}A$.

Thus, we check that ($\operatorname{mod}G$)-stable support $\tau$-tilting
modules are $Q_{1}\oplus Q_{2}$ and $0$.

###  3.2. $\tau$-tilting finiteness

Recall that an algebra $\Lambda$ is said to be _$\tau$ -tilting finite_ if
$|\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda|<\infty$. From Example
3.6(III) we observe that $\tau$-tilting finiteness is not necessarily
preserved by a skew group algebra extension. The aim of this section is to
prove Theorem 3.11 that tells that finiteness of support $\tau$-tilting
modules of $\Lambda$ is inhered to $A=\Lambda*G$ under certain condition that
can be easily checked in many situations.

####  3.2.1. Preparations

We need preparations. In many situation of applications, an algebra
$\Lambda={\mathbf{k}}Q/(r_{1},\dots,r_{n})$ is often given by a quiver $Q$
with the relations $r_{1},\dots,r_{n}$ and an action
$G\curvearrowright\Lambda$ is often induced from an action $G\curvearrowright
Q$ on the quiver $Q$ that preserves the ideal $(r_{1},\dots,r_{n})$. The
following definition is an abstraction of this situation.

######  Definition 3.8.

Let $\Lambda$ be an algebra with a $G$-action.

We say that the action $G\curvearrowright\Lambda$ _preserves idempotent
elements_ if the following conditions (1)(2) hold. We say that the action
$G\curvearrowright\Lambda$ _preserves idempotent elements and has local
stabilizers_ if the following conditions (1)(2)(3) hold.

  1. (1)

A finite dimensional algebra $\Lambda$ is basic and elementary with a complete
set ${\mathsf{E}}\subset\Lambda$ of primitive orthogonal idempotent elements.
Namely, for $e,e^{\prime}\in{\mathsf{E}},\ e\neq e^{\prime}$, we have
$e(\Lambda/J_{\Lambda})e^{\prime}=0$ and
$e(\Lambda/J_{\Lambda})e={\mathbf{k}}$.

  2. (2)

The action of $G$ preserves ${\mathsf{E}}$. Namely, for any $e\in{\mathsf{E}}$
and $g\in G$, we have $g(e)\in{\mathsf{E}}$

  3. (3)

For any $e\in{\mathsf{E}}$, the group algebra
${\mathbf{k}}\operatorname{stab}(e)$ of the stabilizer group
$\operatorname{stab}(e)$ is local.

As is mentioned above, the following is an elementary and basic example of an
action $G\curvearrowright\Lambda$ that preserves idempotent elements.

######  Example 3.9.

Let $Q$ be a finite quiver and $I$ a two-sided admissible ideal of
${\mathbf{k}}Q$. If a finite group $G$ acts on $Q$ in such a way that the
extended action on ${\mathbf{k}}Q$ preserves $I$. Then the induced action of
$G$ on $\Lambda:={\mathbf{k}}Q/I$ preserves the canonical idempotent elements
${\mathsf{E}}:=\\{e_{i}\mid i\in Q_{0}\\}$ that correspond to the vertices of
$Q$.

The following lemma provides basic properties of indecomposable projective
modules over $A=\Lambda*G$ in the case that an action
$G\curvearrowright\Lambda$ preserves idempotent elements.

######  Lemma 3.10.

Let $\Lambda$ be an algebra with a $G$-action that preserves idempotent
elements and ${\mathsf{E}}$ the fixed set in Definition 3.8. Then the
following assertions hold.

  1. (1)

Let $e,e^{\prime}\in{\mathsf{E}}$. Then the idempotent elements $e,e^{\prime}$
are equivalent as idempotent elements of $A$ if and only if these elements are
in the same $G$-orbit.

  2. (2)

An element $e\in{\mathsf{E}}$ (regarded as an element of $A$ via the canonical
inclusion $\Lambda\subset A$) is primitive in $A$ if and only if the group
algebra ${\mathbf{k}}\operatorname{stab}(e)$ is local where
$\operatorname{stab}(e)$ is the stabilizer subgroup of $e$.

  3. (3)

The projective module $Ae$ is ($\operatorname{mod}G$)-stable for each
$e\in{\mathsf{E}}$.

###### Proof.

(1) “If” part. We assume that there exists $g\in G$ such that
$e^{\prime}=g(e)$. Then the elements $a:=e*g^{-1},\ b:=g(e)*g$ satisfy the
equations

$ea=ae^{\prime}=a,\ e^{\prime}b=be=b,\ ab=e,\ ba=e^{\prime}.$ (3-5)

“Only if” part. We prove only if part by contradiction. Assume that $e$ and
$e^{\prime}$ belong to different $G$-orbit. Let $a,b\in A$ that satisfy the
equations (3-5). We write $a=\sum_{g\in G}a_{g}*g,\ b=\sum_{g\in G}b_{g}*g$
with $a_{g},b_{g}\in\Lambda$. It follows from the first and second equations
of (3-5) that $a_{g}\in e\Lambda g(e^{\prime})=eJ_{\Lambda}g(e^{\prime}),\
b_{g}\in e^{\prime}\Lambda g(e)=e^{\prime}J_{\Lambda}g(e)$. Therefore,
$e=ab,e^{\prime}=ba$ belong to $J_{\Lambda}*G$. In particular, we come to the
contradiction that these elements are nilpotent.

(2) The two-sided ideal $\langle J_{\Lambda}\rangle:=AJ_{\Lambda}A$ of $A$
generated by $J_{\Lambda}\subset\Lambda\subset A$ is the linear span of the
elements of the forms $j*g\ (j\in J_{\Lambda},g\in G)$. It follows that
$A/\langle J_{\Lambda}\rangle\cong\Lambda/J_{\Lambda}*G$. Observe that for
$r\in\Lambda,g\in G$, we have $e(\overline{r}*g)e=(e\overline{r}g(e))*g$ where
$\overline{r}:=r+J_{\Lambda}$. It follows from the condition (1) of Definition
3.8 that $e(A/\langle
J_{\Lambda}\rangle)e\cong{\mathbf{k}}\operatorname{stab}(e)$.

Since $\langle J_{\Lambda}\rangle\subset J_{A}$ by Lemma 2.23, the algebra
$e(A/J_{A})e$ is the quotient algebra of $e(A/\langle J_{\Lambda}\rangle)e$ by
its Jacobson radical $e(J_{A}/\langle J_{\Lambda}\rangle)e$. It follows that
${\mathbf{k}}\operatorname{stab}(e)\cong e(A/\langle J_{\Lambda}\rangle)e$ is
local if and only if $e(A/J_{A})e$ is a field if and only if $e$ is primitive
as an element of $A$.

(3) We fix a complete set $R$ of the representatives of
$G/\operatorname{stab}(e)$ that contains the unit $e_{G}$ of $G$. Then
$\bigoplus_{g\in R}\Lambda g(e)$ is a direct summand of the $A$-module
$\Lambda$ and $\bigoplus_{g\in R}Ag(e)=(\bigoplus_{g\in R}\Lambda g(e))*G$.
Hence $\bigoplus_{g\in R}Ag(e)$ is ($\operatorname{mod}G$)-stable. By (1), we
have $\operatorname{add}Ae=\operatorname{add}(\bigoplus_{g\in R}Ag(e))$.
Therefore, we conclude that $Ae$ is ($\operatorname{mod}G$)-stable. ∎

####  3.2.2. Finiteness theorem

######  Theorem 3.11.

Let $G$ be a finite group and $\Lambda$ an algebra with $G$-action that
preserves idempotent elements and has local stabilizers. Moreover, we assume
$|(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}|<\infty$. Then the
following assertions hold.

  1. (1)

$A=\Lambda*G$ is $\tau$-tilting finite.

  2. (2)

All supprot $\tau$-tilting $A$-modules are ($\operatorname{mod}G$)-stable.

  3. (3)

The adjoint pair
$(\operatorname{Ind}_{\Lambda}^{A},\operatorname{Res}_{\Lambda}^{A})$ induces
a bijection

$\operatorname{Ind}_{\Lambda}^{A}:(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}\longleftrightarrow\textup{s}\tau\textup{-}\operatorname{tilt}A:\operatorname{Res}_{\Lambda}^{A}.$

To prove this theorem, we introduce the notion of
ind-($\operatorname{mod}G$)-stability.

######  Definition 3.12.

A support $\tau$-tilting $A$-module $N$ is said to be
_ind-($\operatorname{mod}G$)-stable_ if each indecomposable direct summand is
($\operatorname{mod}G$)-stable. Namely, if $N=\bigoplus_{i=1}^{n}N_{i}$ is a
decomposition into indecomposable modules $N_{i}$, then each $N_{i}$ is
($\operatorname{mod}G$)-stable.

We note the condition is equivalent to $X\otimes_{{\mathbf{k}}}^{G}N_{i}\cong
N_{i}^{\oplus\dim X}$ for any $X\in\operatorname{mod}G$.

A key step is the following lemma that states that
ind-($\operatorname{mod}G$)-stability of support $\tau$-tilting modules is
preserved by irreducible left mutations.

######  Lemma 3.13.

If $N=\bigoplus_{i=1}^{n}N_{i}$ is a basic ind-($\operatorname{mod}G$)-stable
support $\tau$-tilting $A$-module with decompostion into indecmoposable direct
summand $N_{i}$. Then for any $i=1,\dots,n$, the irreducible left mutation
$\mu_{i}^{-}(N)$ is also ind-($\operatorname{mod}G$)-stable.

###### Proof.

The case $n=1$ is clear. Thus we may assume that $n\geq 2$. We may also assume
that $i=1$. We set $N^{\prime}:=\bigoplus_{i=2}^{n}N_{i}$. Let $f:N_{1}\to L$
be a left minimal $\operatorname{add}N^{\prime}$-approximation. We only have
to prove that the cokernel $C:=\operatorname{Cok}f$ is
($\operatorname{mod}G$)-stable.

For this purpose, we take a finite dimensional $G$-module $X$ and show that
$X\otimes_{{\mathbf{k}}}^{G}C\cong C^{\oplus\dim X}$. First we claim that
$X\otimes_{{\mathbf{k}}}^{G}f$ is a left
$\operatorname{add}N^{\prime}$-approximation. Indeed, let
$g:X\otimes_{{\mathbf{k}}}^{G}N_{1}\to K$ be a homomorphism of $A$-modules
with $K\in\operatorname{add}N^{\prime}$. Recall that the pair
$(X\otimes_{{\mathbf{k}}}^{G}-,{\textup{D}}(X)\otimes_{{\mathbf{k}}}^{G}-)$ is
an adjoint pair by Proposition 2.11. Let
$g^{\circ}:N_{1}\to{\textup{D}}(X)\otimes_{{\mathbf{k}}}^{G}K$ be the
homomorphism corresponding to $g$ via the adjoint isomorphism. Since
$N^{\prime}$ is ($\operatorname{mod}G$)-stable, we have
${\textup{D}}(X)\otimes_{{\mathbf{k}}}^{G}K\in\operatorname{add}N^{\prime}$.
It follows that there exists a homomorphism
$h:L\to{\textup{D}}(X)\otimes_{{\mathbf{k}}}^{G}K$ such that $h\circ
f=g^{\circ}$. Let ${}^{\circ}h:X\otimes_{{\mathbf{k}}}^{G}L\to K$ be the
homomorphism corresponding to $h$ via the adjoint isomorphism. Then by the
naturality of adjoint isomorphisms, we have
${}^{\circ}h\circ(X\otimes^{G}_{{\mathbf{k}}}f)=g$. This shows that
$X\otimes^{G}_{{\mathbf{k}}}f$ is a left
$\operatorname{add}N^{\prime}$-approximation as desired.

$\textstyle{X\otimes_{{\mathbf{k}}}^{G}N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X\otimes^{G}_{{\mathbf{k}}}f}$$\scriptstyle{g}$$\textstyle{X\otimes_{{\mathbf{k}}}^{G}L,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{\circ}h}$$\textstyle{K}$
$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g^{\circ}}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{{\textup{D}}(X)\otimes_{{\mathbf{k}}}^{G}K}$

For simplicity, we set $d=\dim X$. We fix an isomorphism
$X\otimes_{{\mathbf{k}}}^{G}N_{1}\cong N_{1}^{\oplus d}$. By the above claim
and the assumption that $f$ is a minimal left
$\operatorname{add}N^{\prime}$-approximation, we have homomorphism
$s:L^{\oplus d}\to X\otimes_{{\mathbf{k}}}^{G}L,\
t:X\otimes_{{\mathbf{k}}}^{G}L\to L^{\oplus d}$ that compatible with
$f^{\oplus d}$ and $X\otimes_{{\mathbf{k}}}^{G}f$.

$\textstyle{N_{1}^{\oplus
d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{f^{\oplus
d}}$$\textstyle{L^{\oplus
d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{X\otimes_{{\mathbf{k}}}^{G}N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X\otimes_{{\mathbf{k}}}^{G}f}$$\textstyle{X\otimes_{{\mathbf{k}}}^{G}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$

Since $f^{\oplus d}$ is left minimal, the composition $t\circ s$ is an
isomorphism. Moreover, as $X\otimes_{{\mathbf{k}}}^{G}L$ and $L^{\oplus d}$
are (non-canonically) isomorphic to each other and in particular have the same
dimension, it follows that the morphisms $s$ and $t$ are isomorphisms. Now we
conclude that
$X\otimes_{{\mathbf{k}}}^{G}C=\operatorname{Cok}(X\otimes_{{\mathbf{k}}}^{G}f)\cong\operatorname{Cok}(f^{\oplus
d})\cong C^{\oplus d}$ as desired. ∎

###### Proof of Theorem 3.11.

Let ${\mathsf{F}}\subset{\mathsf{E}}$ be a complete set of representatives of
${\mathsf{E}}/G$. It follows from Lemma 3.10 that
$A^{\mathsf{basic}}\cong\bigoplus_{e\in{\mathsf{F}}}Ae$ and that it is
ind-($\operatorname{mod}G$)-stable.

Let $\mu^{-{\mathbb{N}}}_{\bullet}(A)$ be the set of support $\tau$-tilting
modules that are obtained from $A^{\mathsf{basic}}$ by iterated left
mutations. Applying Lemma 3.13 repeatedly, we see that
$\mu^{-{\mathbb{N}}}_{\bullet}(A)\subset(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}$.
Since we are assuming
$|(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}|<\infty$, the set
$(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}$ is a
finite set by Theorem 3.3. It follows that the set
$\mu^{-{\mathbb{N}}}_{\bullet}(A)$ is also finite. Thus, by [1, Corollary
2.38], we have
$\mu^{-{\mathbb{N}}}_{\bullet}(A)=\textup{s}\tau\textup{-}\operatorname{tilt}A$.
Consequently we have
$(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}=\textup{s}\tau\textup{-}\operatorname{tilt}A$.
∎

###  3.3. Complements

#### 3.3.1.

The aim of Section 3.3.1 is to show that if $(M,P)$ is a support
$\tau$-tilting pair over $\Lambda$ such that $M$ is $G$-stable, then the pair
$(F(M),F(P))$ is a support $\tau$-tilting pair over $A$ such that $F(M)$ is
($\operatorname{mod}G$)-stable. Moreover, we study a converse problem. As is
shown in Example 3.17 below, even if a support $\tau$-tilting $\Lambda$-module
$M$ is not $G$-stable, the $A$-module $F(M)$ possibly becomes a support
$\tau$-tilting $A$-module. However, the next proposition shows that a support
$\tau$-tilting $\Lambda$-modules with the complement $P$ is $G$-stable if and
only if the pair $(F(M),F(P))$ is a support $\tau$-tilting pair over $A$.

######  Proposition 3.14.

Let $(M,P)$ be a support $\tau$-tilting pair for $\Lambda$. Then the following
conditions are equivalent.

  1. (a)

$M$ is $G$-stable.

  2. (b)

The pair $(F(M),F(P))$ is a support $\tau$-tilting pair for $A$.

Via the reinterpretation given in Appendix A, this proposition and the proof
of Proposition 4.4 gives a generalization of [17, Theorem 4.3].

We need the following lemma.

######  Lemma 3.15.

For a $\Lambda$-module $M$, the following conditions are equivalent.

  1. (1)

$F(M)$ is a support $\tau$-tilting $A$-module.

  2. (2)

$RF(M)=\bigoplus_{g\in G}{}_{g}M$ is a support $\tau$-tilting
$\Lambda$-module.

###### Proof.

(1)$\Rightarrow$(2). By Corollary 2.13, $F(M)$ is
($\operatorname{mod}G$)-stable. The implication follows from Theorem 3.1.

(2)$\Rightarrow$(1). By Lemma 2.12, $RF(M)$ is $G$-stable. It follows from
Theorem 3.1 that $FRF(M)$ is support $\tau$-tilting $A$-module. Since
$\operatorname{add}FRF(M)=\operatorname{add}F(M)$ by Lemma 2.12, we conclude
that $F(M)$ is support $\tau$-tilting $A$-module. ∎

###### Proof of Proposition 3.14.

(a)$\Rightarrow$(b) By Theorem 3.1, $F(M)$ is support $\tau$-tilting
$A$-module. Let $Q$ be a projective $A$-modules such that $(F(M),Q)$ is a
support $\tau$-tilting pair.

It follows from the adjoint isomorphism
$\operatorname{Hom}_{\Lambda}(R(Q),M)\cong\operatorname{Hom}_{A}(Q,F(M))=0$
that $R(Q)\in\operatorname{add}P$ and hence $FR(Q)\in\operatorname{add}F(P)$.
Since $Q$ is projective, the canonical surjection
$FR(Q)=A\otimes_{\Lambda}Q\to Q$ splits. Thus we have
$Q\in\operatorname{add}F(P)$.

We recall the adjoint isomorphism

$\operatorname{Hom}_{A}(F(P),F(M))\cong\operatorname{Hom}_{\Lambda}(P,RF(M))\cong\operatorname{Hom}_{\Lambda}(P,\bigoplus_{g\in
G}{}_{g}M).$ (3-6)

Now by the assumption that $M$ is $G$-stable, we have
$\operatorname{add}(\bigoplus_{g\in G}{}_{g}M)=\operatorname{add}M$. It
follows from (3-6) that $\operatorname{Hom}_{A}(F(P),F(M))=0$ and
$F(P)\in\operatorname{add}Q$. Thus we conclude that
$\operatorname{add}Q=\operatorname{add}F(P)$.

(b)$\Rightarrow$(a). The module $\bigoplus_{g\in G}{}_{g}M$ is support
$\tau$-tilting $\Lambda$-module by Lemma 3.15. In particular it is
$\tau$-rigid. It follows that for any $g\in G$, we have

$\operatorname{Hom}_{\Lambda}(M,\tau({}_{g}M))=0,\
\operatorname{Hom}_{\Lambda}({}_{g}M,\tau M)=0.$

Moreover, since $\operatorname{Hom}_{A}(F(P),F(M))=0$, it follows from (3-6)
that $\operatorname{Hom}_{\Lambda}(P,{}_{g}M)=0$ for any $g\in G$. Thanks to
[1, Corollary 2.13] we see that ${}_{g}M\in\operatorname{add}M$. Now it is
easy to deduce that ${}_{g}M\cong M$. ∎

We provide an example of $\Lambda$-module $M$ which is not a support
$\tau$-tilting module, but $F(M)$ is a support $\tau$-tilting $A$-module.

######  Example 3.16.

Let $\Lambda:={\mathbf{k}}Q/(ab,ba)$ where $Q$ is the following quiver

$Q:\ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.34439pt\raise 6.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
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Let $G=\langle g\rangle$ be a cyclic group of order $2$ with a generator $g$.
We give an action $G\curvearrowright\Lambda$ by $g:\ 1\leftrightarrow 2,\ \
a\leftrightarrow b.$ Then there is an isomorphism
$A:=\Lambda*G\xrightarrow{\cong}M_{2}({\mathbf{k}}[x]/(x^{2}))$ of algebras
under which the element $e_{1}\in A$ corresponds to the matrix unit
$\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$.

The projective $\Lambda$-module $P_{1}=\Lambda e_{1}$ is not a support
$\tau$-tilting $\Lambda$-module, but $F(P_{1})=Ae_{1}$ is a support
$\tau$-tilting $A$-module.

We provide an example of a support $\tau$-tilting $\Lambda$-module $M$ which
is not $G$-stable, but $F(M)$ is a support $\tau$-tilting $A$-module.

######  Example 3.17.

Let $\Lambda:={\mathbf{k}}\times{\mathbf{k}}$. We set
$e_{1}:=(1_{{\mathbf{k}}},0),\ e_{2}:=(0,1_{{\mathbf{k}}})$. Let $G:=\langle
g\rangle$ be a cyclic group of order $2$ with a generator $g$. We give an
action $G\curvearrowright\Lambda$ by setting $g(e_{1}):=e_{2},\
g(e_{2}):=e_{1}$. The skew group algebra $A:=\Lambda*G$ is isomorphic to the
matrix algebra $M_{2}({\mathbf{k}})$ of order $2$.

It is easy to check that the simple $\Lambda$-module $S_{1}=\Lambda e_{1}$
corresponding to the vertex $1$ is a support $\tau$-tilting $\Lambda$ module
which is not $G$-stable, but $F(S_{1})=A(e_{1}*e_{G})$ is a support
$\tau$-tilting $A$-module.

#### 3.3.2.

In this section, we show that if $(N,Q)$ is a support $\tau$-tilting pair over
$A$ such that $N$ is ($\operatorname{mod}G$)-stable, then the pair
$(R(N),R(Q))$ is a support $\tau$-tilting pair over $\Lambda$ such that $R(N)$
is $G$-stable. We also study related problems as in the previous section.

First we establish the following lemma which is an
$R=\operatorname{Res}_{\Lambda}^{A}$-version of Lemma 3.15.

######  Lemma 3.18.

Let $N$ be an $A$-module. The following conditions are equivalent

  1. (1)

$R(N)$ is a support $\tau$-tilting $\Lambda$-module.

  2. (2)

$FR(N)={\mathbf{k}}G\otimes_{{\mathbf{k}}}^{G}N$ is a support $\tau$-tilting
$A$-module.

###### Proof.

(1) $\Rightarrow$ (2). By Corollary 2.13, $R(N)$ is $G$-stable. Thanks to
Theorem 3.1, $FR(N)={\mathbf{k}}G\otimes_{{\mathbf{k}}}^{G}N$ is a support
$\tau$-tilting $A$-module.

(2) $\Rightarrow$ (1). By Corollary 2.13, $FR(N)$ is
($\operatorname{mod}G$)-stable. It follows from Theorem 3.1 that $RFR(N)$ is a
support $\tau$-tilting $\Lambda$-module. By Corollary 2.13, we have
$\operatorname{add}RFR(N)=\operatorname{add}R(N)$. Thus, we conclude that
$R(N)$ is a support $\tau$-tilting $A$-module. ∎

The following is an $R=\operatorname{Res}^{A}_{\Lambda}$-version of
Proposition 3.14. However, the two conditions (a) and (b) are not equivalent
in this situation.

######  Proposition 3.19.

Let $(N,Q)$ be a support $\tau$-tilting pair over $A$. We consider the
following conditions.

  1. (a)

$N$ is ($\operatorname{mod}G$)-stable.

  2. (b)

$(R(N),R(Q))$ is a support $\tau$-tilting pair over $\Lambda$.

Then the following assertions hold.

  1. (1)

The condition (a) implies (b).

  2. (2)

If one of the following conditions hold, then the condition (b) implies (a).

    1. (i)

$\operatorname{char}{\mathbf{k}}$ does not divide $|G|$.

    2. (ii)

The group algebra ${\mathbf{k}}G$ is local.

Via the reinterpretation given in Appendix A, this proposition and the proof
of Proposition 4.4 gives a generalization of [17, Theorem 4.5].

###### Proof.

(1) Assume that $N$ is ($\operatorname{mod}G$)-stable. By Theorem 3.1, $R(N)$
is a support $\tau$-tilting $\Lambda$-module. Let $P$ be a projective
$\Lambda$-module such that the pair $(R(N),P)$ is a support $\tau$-tilting
pair for $\Lambda$. Then we have
$\operatorname{Hom}_{A}(F(P),N)\cong\operatorname{Hom}_{\Lambda}(P,R(N))=0$.
It follows that $F(P)\in\operatorname{add}Q$ and hence
$RF(P)\in\operatorname{add}R(Q)$. By Lemma 2.12, $P$ is a direct summand of
$RF(P)$, we have $P\in\operatorname{add}R(Q)$.

Since $N$ is ($\operatorname{mod}G$)-stable, we have
$FR(N)\in\operatorname{add}N$. By the adjoint isomorphism, we have
$\operatorname{Hom}_{\Lambda}(R(Q),R(N))\cong\operatorname{Hom}_{A}(Q,FR(N))=0$.
It follows that $R(Q)\in\operatorname{add}P$. Thus we conclude that
$\operatorname{add}P=\operatorname{add}R(Q)$.

(2) Before assuming one of (i) or (ii) holds, we only assume that the
condition (b) holds. Namely, we assume that $(R(N),R(Q))$ is a support
$\tau$-tilting pair for $\Lambda$. Since $R(N)$ is $G$-stable, the pair
$(FR(N),FR(Q))$ is a support $\tau$-tilting pair for $A$ by Proposition 3.14.
We claim that $\operatorname{add}FR(Q)=\operatorname{add}Q$ i.e., $Q$ is
($\operatorname{mod}G$)-stable. Indeed, since $Q$ is projective, the canonical
surjection $FR(Q)={\mathbf{k}}G\otimes_{{\mathbf{k}}}^{G}Q\to Q$ splits. Hence
we have $Q\in\operatorname{add}FR(Q)$. On the other hand, we have
$\operatorname{Hom}_{A}(FR(Q),N)\cong\operatorname{Hom}_{\Lambda}(R(Q),R(N))=0$.
It follows that $FR(Q)\in\operatorname{add}Q$. Thus we conclude that
$\operatorname{add}FR(Q)=\operatorname{add}Q$.

By the claim we have $|Q|=|FR(Q)|$. On the other hand, by a basic property of
support $\tau$-tilting modules, we have $|A|=|N|+|Q|=|FR(N)|+|FR(Q)|$ . It
follows that $|N|=|FR(N)|$.

If we additionally assume that one of (i) or (ii) holds, then we have
$\operatorname{add}N\subset\operatorname{add}FR(N)$ by Lemma 2.17. Thus, we
conclude $\operatorname{add}N=\operatorname{add}FR(N)$ as desired. ∎

We provide an example that shows that the condition (b) does not always imply
the condition (a).

######  Example 3.20.

Assume that $\operatorname{char}{\mathbf{k}}=3$. Let $G={\mathfrak{S}}_{3}$ be
the symmetric group of degree $3$. We set $\Lambda:={\mathbf{k}}$ with the
trivial $G$ action. Then $A=\Lambda*G$ is the group algebra ${\mathbf{k}}G$
and isomorphic to ${\mathbf{k}}Q/(aba,bab)$ where $Q$ is a quiver given below

$Q:\ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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Note that we may identify the simple modules $S_{1},S_{2}$ corresponding to
the vertices $1,2$ with the trivial representation ${\mathbf{k}}_{G}$ and the
sign representation ${\mathbf{k}}_{\operatorname{sgn}}$.

Since ${\mathbf{k}}G\otimes_{{\mathbf{k}}}^{G}S_{2}\cong A$ as $A$-modules,
the module $N:=S_{2}\oplus P_{2}$ is not ($\operatorname{mod}G$)-stable. But
the pair $(N,0)$ is a supprot $\tau$-tilting pair for $A$ such that the pair
$(R(N),R(0))$ is a support $\tau$-tilting pair for ${\mathbf{k}}$.

##  4\. Silting theory of skew group algebra extensions

In this section we discuss silting theory of a skew group algebra extension
$\Lambda\subset\Lambda*G$. For the definition of silting complexes and other
basic notions and results, we refer [4].

###  4.1. Preliminaries

Recall that the induction functor
$F=\operatorname{Ind}_{\Lambda}^{A}:\operatorname{mod}\Lambda\to\operatorname{mod}A$
and the restriction functor
$R=\operatorname{Res}_{\Lambda}^{A}:\operatorname{mod}A\to\operatorname{mod}\Lambda$
are exact and preserve projective modules. Therefore, these functors $F,R$
induce exact functors
$F=\operatorname{Ind}_{\Lambda}^{A}:{\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}\Lambda)\to{\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A)$
and
$R=\operatorname{Res}_{\Lambda}^{A}:{\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A)\to{\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}\Lambda)$
of derived categories, which are denoted by the same symbols, and the functors
$F$ and $R$ on the derived categories restrict to exact functors
$F=\operatorname{Ind}_{\Lambda}^{A}:{\mathsf{K}}^{\mathrm{b}}(\Lambda\operatorname{proj})\to{\mathsf{K}}^{\mathrm{b}}(A\operatorname{proj})$
and
$R=\operatorname{Res}_{\Lambda}^{A}:{\mathsf{K}}^{\mathrm{b}}(A\operatorname{proj})\to{\mathsf{K}}^{\mathrm{b}}(\Lambda\operatorname{proj})$
of homotopy categories, which are also denoted by the same symbols.

Let $g\in G$. Then the endofunctor
${}_{g}(-):\operatorname{mod}\Lambda\to\operatorname{mod}\Lambda,\
M\mapsto{}_{g}M$ is exact and preserves finitely generated projective modules.
Thus it gives rise to an exact endofunctor ${}_{g}(-)$ of the derived category
${\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}\Lambda)$ and the homotopy
category ${\mathsf{K}}^{\mathrm{b}}(\Lambda\operatorname{proj})$. We define
the $G$-stability of an object
$M\in{\mathsf{K}}^{\mathrm{b}}(\Lambda\operatorname{proj})$ in a similar way
of Definition 2.1.

Let $X$ be a finite dimensional $G$-module. Then the endofunctor
$X\otimes_{{\mathbf{k}}}^{G}-:\operatorname{mod}A\to\operatorname{mod}A$ is
exact and preserves finitely generated projective modules by Corollary 2.8.
Thus it gives rise to an exact endofunctor $X\otimes_{{\mathbf{k}}}^{G}-$ of
the derived category ${\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A)$ and the
homotopy category ${\mathsf{K}}^{\mathrm{b}}(A\operatorname{proj})$. We define
the ($\operatorname{mod}G$)-stability of an object
$N\in{\mathsf{K}}^{\mathrm{b}}(A\operatorname{proj})$ in a similar way of
Definition 2.9.

Moreover, all isomorphisms established in Section 2.1 and Section 2.2 are
natural in relevant modules over $\Lambda$ or $A$, it follows that all the
results given in these sections have their derived category version. Among
other things we point out the following derived category version of
Proposition 2.15 for later quotations. To state this, we recall that an object
$N\in{\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A)$ is called _rigid_ if
$\operatorname{Hom}_{{\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A)}(N,N[1])=0$.

######  Proposition 4.1.

For a rigid object $N\in{\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}A)$, the
following conditions are equivalent.

  1. (1)

$N$ is ($\operatorname{mod}G$)-stable.

  2. (2)

$S\otimes_{{\mathbf{k}}}^{G}N\in\operatorname{add}N$ for any simple
$G$-modules $S$.

  3. (3)

$\operatorname{add}FR(N)=\operatorname{add}N$.

Moreover, if we further assume one of the following conditions: (i)
$\operatorname{char}{\mathbf{k}}\nmid|G|$, (ii) ${\mathbf{k}}G$ is local, then
the above conditions are equivalent to the following condition (4).

(4) $FR(N)\in\operatorname{add}N$.

###  4.2. The bijection

The aim of this section is to prove that the induction-restriction adjoint
pair $(F,R)$ induces a bijection between $G$-stable silting objects and
($\operatorname{mod}G$)-stable silting complexes, whose restriction maps give
bijections of the sets of $2$-term silting complexes having appropriate
stabilities, and the set of tilting objects having appropriate stabilities.
For this purpose, first we prove the following lemma.

######  Lemma 4.2.

  1. (1)

If $S\in{\mathsf{K}}^{\mathrm{b}}(\Lambda\operatorname{proj})$ is a $G$-stable
tilting (resp. silting, $2$-term silting) object of
${\mathsf{K}}^{\mathrm{b}}(\Lambda\operatorname{proj})$, then
$F(S)\in{\mathsf{K}}^{\mathrm{b}}(A\operatorname{proj})$ is a tilting (resp.
silting, $2$-term silting) object of
${\mathsf{K}}^{\mathrm{b}}(A\operatorname{proj})$.

  2. (2)

If $T\in{\mathsf{K}}^{\mathrm{b}}(A\operatorname{proj})$ is a
($\operatorname{mod}G$)-stable tilting (resp. silting, $2$-term silting)
object of ${\mathsf{K}}^{\mathrm{b}}(A\operatorname{proj})$, then
$R(T)\in{\mathsf{K}}^{\mathrm{b}}(\Lambda\operatorname{proj})$ is a tilting
(resp. silting, $2$-term silting) object of
${\mathsf{K}}^{\mathrm{b}}(\Lambda\operatorname{proj})$.

###### Proof.

(1) Since
$\Lambda\in{\mathsf{K}}^{\mathrm{b}}(\Lambda\operatorname{proj})=\mathop{\mathsf{thick}}\nolimits
S$, we have $A=F(\Lambda)\in\mathop{\mathsf{thick}}\nolimits F(S)$. It follows
that $\mathop{\mathsf{thick}}\nolimits
F(S)={\mathsf{K}}^{\mathrm{b}}(A\operatorname{proj})$.

For $n\in{\mathbb{Z}}$, we have

$\operatorname{Hom}_{{\mathsf{K}}^{\mathrm{b}}(A\operatorname{proj})}(F(S),F(S)[n])\cong\operatorname{Hom}_{{\mathsf{K}}^{\mathrm{b}}(\Lambda\operatorname{proj})}(S,RF(S)[n]).$
(4-7)

Since $S$ is $G$-stable, we have $RF(S)\in\operatorname{add}S$ by (a derived
category version of) Corollary 2.14. It follows from the assumption that $S$
is tilting (resp. silting) that
$\operatorname{Hom}_{{\mathsf{K}}^{\mathrm{b}}(A\operatorname{proj})}(F(S),F(S)[n])=0$
for $n\neq 0$ (resp. $n>0$). Now we have shown that if $S$ is a tilting (resp.
silting) object, then so is $F(S)$.

Finally, since the functor $F$ preserves projective modules, it follows that
if $S$ is a $2$-term silting complex, then so is $F(S)$.

(2) can be proved in a similar argument by using Proposition 4.1 and the fact
that $R(A)$ contains $\Lambda$ as a direct summand. ∎

Let $\operatorname{silt}\Lambda$ be the poset (of isomorphism classes) of
basic silting complexes over $\Lambda$. We denote by
$(\operatorname{silt}\Lambda)^{G}$ the subposet of $G$-stable basic silting
complexes. We also introduce $\operatorname{2-silt}\Lambda$ and
$(\operatorname{2-silt}\Lambda)^{G}$ (resp. $\operatorname{tilt}\Lambda$ and
$(\operatorname{tilt}\Lambda)^{G}$) the poset of $2$-term silting complexes
(resp. tilting complexes) and the subposet of $G$-stable ones.

In a similar way, we define
$(\operatorname{silt}A)^{\operatorname{mod}G},(\operatorname{2-silt}A)^{\operatorname{mod}G},(\operatorname{tilt}A)^{\operatorname{mod}G}$
the poset of ($\operatorname{mod}G$)-stable silting complexes, $2$-term
silting complexes and tilting complexes.

We remark that since we are not assuming that our algebras $\Lambda,A$ are
basic, strictly speaking the silting complex $\Lambda$ (resp. $A$) does not
necessarily belong to $\operatorname{silt}\Lambda$ (resp.
$\operatorname{silt}A$). However, to simplify the notation, we often denote
$\Lambda$ and $A$ to denote the basic silting complexes
$\Lambda^{\mathsf{basic}}$ and $A^{\mathsf{basic}}$.

######  Theorem 4.3.

The adjoint pair
$(F=\operatorname{Ind}_{\Lambda}^{A},R=\operatorname{Res}_{\Lambda}^{A})$
induces isomorphisms of posets

$F=\operatorname{Ind}_{\Lambda}^{A}:(\operatorname{silt}\Lambda)^{G}\longleftrightarrow(\operatorname{silt}A)^{\operatorname{mod}G}:\operatorname{Res}_{\Lambda}^{A}=R.$

Moreover, the restrictions of these maps give isomorphisms
$F:(\operatorname{2-silt}\Lambda)^{G}\leftrightarrow(\operatorname{2-silt}A)^{\operatorname{mod}G}:R$
and
$F:(\operatorname{tilt}\Lambda)^{G}\leftrightarrow(\operatorname{tilt}A)^{\operatorname{mod}G}:R$.

###### Proof.

Thanks to Lemma 4.2 the map
$F:(\operatorname{silt}\Lambda)^{G}\to(\operatorname{silt}A)^{\operatorname{mod}G},S\mapsto
F(S)^{\mathsf{basic}}$ and
$R:(\operatorname{silt}A)^{G}\to(\operatorname{silt}\Lambda)^{G},T\mapsto
R(T)^{\mathsf{basic}}$ are well defined. By (a derived category version of)
Corollary 2.14 and Proposition 4.1, we see that these maps are inverse to each
other. It also follows from Lemma 4.2 that restrictions of these maps give
bijections
$F:(\operatorname{2-silt}\Lambda)^{G}\leftrightarrow(\operatorname{2-silt}A)^{\operatorname{mod}G}:R$
and
$F:(\operatorname{tilt}\Lambda)^{G}\leftrightarrow(\operatorname{tilt}A)^{\operatorname{mod}G}:R$.

Recall that for two silting complexes $S_{1},S_{2}$, the inequality $S_{1}\geq
S_{2}$ is defined by the equality $\operatorname{Hom}(S_{1},S_{2}[n])=0\
(\forall n>0)$. Using adjoint isomorphisms as in (4-7) and (a derived category
version of) Corollary 2.14 and Proposition 4.1, we can check that the above
maps are poset morphisms. ∎

###  4.3. Correspondence between $2$-term silting complexes and support
$\tau$-tilting modules

Recall that taking the $0$-th chomology group ${\textup{H}}^{0}(S)$ of a
$2$-term silting complex $S$ gives a bijection
${\textup{H}}^{0}:\operatorname{2-silt}\Lambda\to\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda$.

######  Proposition 4.4.

  1. (1)

The bijection
${\textup{H}}^{0}:\operatorname{2-silt}\Lambda\to\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda$
restricts to a bijection
${\textup{H}}^{0}:(\operatorname{2-silt}\Lambda)^{G}\to(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}$.

  2. (2)

The bijection
${\textup{H}}^{0}:\operatorname{2-silt}A\to\textup{s}\tau\textup{-}\operatorname{tilt}A$
restricts to a bijection
${\textup{H}}^{0}:(\operatorname{2-silt}A)^{\operatorname{mod}G}\to(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}$.

###### Proof.

(1) We denote the inverse map of
${\textup{H}}^{0}:\operatorname{2-silt}\Lambda\to\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda$
by
${\textup{K}}:\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda\to\operatorname{2-silt}\Lambda$.
It is clear that if $S$ is a $G$-stable $2$-term silting complex, then
${\textup{H}}^{0}(S)$ is $G$-stable. Thus, we only have to show that if $M$ is
a $G$-stable support $\tau$-tilting module, then the $2$-term silting complex
${\textup{K}}(M)$ is $G$-stable.

Recall ${\textup{K}}(M)$ is constructed in the following way. Let $P$ be a
complement projective module of $M$ and $P^{-1}\to P^{0}\to M\to 0$ a minimal
projective presentation. For simplicity we set $P_{M}:=[P^{-1}\to P^{0}]$.
Then ${\textup{K}}(M):=P_{M}\oplus P[1]$.

By Corollary 2.26, the complex $RF(P_{M})$ gives a minimal projective
presentation of $RF(M)$. Since $\operatorname{add}RF(M)=\operatorname{add}M$,
we have $\operatorname{add}RF(P_{M})=\operatorname{add}P_{M}$ in
${\mathsf{K}}^{\mathrm{b}}(\Lambda\operatorname{proj})$. Next note that
$RF(P)$ is projective $\Lambda$-module. Since the pairs $(R,F)$ and $(F,R)$
are adjoint pairs, we have
$\operatorname{Hom}_{\Lambda}(RF(P),M)\cong\operatorname{Hom}_{\Lambda}(P,RF(M))=0$.
It follows $RF(P)\in\operatorname{add}P$. On the other hand, we have
$P\in\operatorname{add}RF(P)$ by Lemma 2.12. It follows that
$\operatorname{add}RF(P)=\operatorname{add}P$. Combining above two
observations we conclude that
$\operatorname{add}RF{\textup{K}}(M)=\operatorname{add}{\textup{K}}(M)$.

(2) is proved in a similar way by using Proposition 2.30(2) and the fact that
the canonical morphism $RF(Q)\to Q$ is surjective. ∎

######  Remark 4.5.

The equality $\operatorname{add}RF(P)=\operatorname{add}P$ in the above proof
immediately follows from Propositions 3.14 and 3.19. However, these two
propositions rely on Theorem 3.1. Therefore, if we use these two propositions
in the proof, we can not use Proposition 4.4 to the second proof of the
theorem given below.

###### The second proof of Theorem 3.1.

(1) Let $M$ be a $G$-stable support $\tau$-tilting $\Lambda$-module. Then
${\textup{H}}^{0}F{\textup{K}}(M)$ is a ($\operatorname{mod}G$)-stable support
$\tau$-tilting $A$-module by Theorem 4.3 and Proposition 4.4.

$(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}\xrightarrow{\
{\textup{K}}\ }(\operatorname{2-silt}\Lambda)^{G}\xrightarrow{\ F\
}(\operatorname{2-silt}A)^{\operatorname{mod}G}\xrightarrow{\
{\textup{H}}^{0}\
}(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}.$

On the other hand, it is straightforward to check that
${\textup{H}}^{0}F{\textup{K}}(M)=F(M)$. This finishes the proof of (1). The
second statement is proved in a similar way. ∎

###  4.4. Silting discreteness

We introduce the notion of $G$-stable silting discreteness.

######  Definition 4.6.

An algebra $\Lambda$ with a $G$-action is called _$G$ -stable silting
discrete_ if for any $n\geq 0$, there are only a finite number of $G$-stable
silting complexes $S$ over $\Lambda$ such that $\Lambda[n]\leq S\leq\Lambda$.
In other words, we have
$|(\operatorname{silt}\Lambda)^{G}\cap[\Lambda[n],\Lambda]|<\infty$ where

$[\Lambda[n],\Lambda]:=\\{S\in\operatorname{silt}\Lambda\mid\Lambda[n]\leq
S\leq\Lambda\\}.$

We provide a silting version of Theorem 3.11.

######  Theorem 4.7.

Let $G$ be a finite group and $\Lambda$ an algebra with $G$-action that
preserves idempotent elements and has local stabilizers. Assume that $\Lambda$
is $G$-stable silting discrete. Then the following assertions hold.

  1. (1)

$A=\Lambda*G$ is silting discrete.

  2. (2)

All silting complexes over $A$ are ($\operatorname{mod}G$)-stable.

  3. (3)

The adjoint pair
$(\operatorname{Ind}_{\Lambda}^{A},\operatorname{Res}_{\Lambda}^{A})$ induces
a bijection

$\operatorname{Ind}_{\Lambda}^{A}:(\operatorname{silt}\Lambda)^{G}\longleftrightarrow\operatorname{silt}A:\operatorname{Res}_{\Lambda}^{A}.$

###### Proof.

We can introduce the notion of ind-($\operatorname{mod}G$)-stability of
objects of ${\mathsf{D}}^{\mathrm{b}}(\operatorname{mod}\Lambda)$ as in the
same way of Definition 3.12. Then we can prove by the same method of Lemma
3.13 that every irreducible left or irreducible right silting mutation
$\mu^{\operatorname{silt},\pm}_{i}(S)$ of ind-($\operatorname{mod}G$)-stable
silting complexes over $A$ is again ind-($\operatorname{mod}G$)-stable. On the
other hand, as is mentioned in the proof of Theorem 3.11, the $A$-module
$A^{\mathsf{basic}}$ is ind-($\operatorname{mod}G$)-stable. Therefore, if we
denote by $\mu^{\operatorname{silt},{\mathbb{Z}}}_{\bullet}(A)$ the set of all
silting complexes over $A$ obtained by silting mutations from
$A^{\mathsf{basic}}$, then
$\mu^{\operatorname{silt},{\mathbb{Z}}}_{\bullet}(A)\subset(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}$.

Thus we only have to show that every silting complex $T$ over $A$ belongs to
$\mu_{\bullet}^{\operatorname{silt},{\mathbb{Z}}}(A)$. Shifting the
cohomological grading, we may assume that $T\in[A[n],A]$ for some $n\geq 0$.
Assume that $T\notin\mu_{\bullet}^{\operatorname{silt},{\mathbb{Z}}}(A)$. Then
there exists an infinite sequence of irreducible left silting mutations
starting from $A$ that is bounded by $T$ and hence by $A[n]$.

$A^{\mathsf{basic}}\xrightarrow{\ \mu_{i_{1}}^{\operatorname{silt},-}\
}T_{1}\xrightarrow{\ \mu_{i_{2}}^{\operatorname{silt},-}\ }T_{2}\xrightarrow{\
\mu_{i_{3}}^{\operatorname{silt},-}\ }\cdots>T>A^{\mathsf{basic}}[n].$

Thus the set $(\operatorname{silt}A)^{\operatorname{mod}G}\cap[A[n],A]$
contains an infinite set $\\{A,T_{1},T_{2},\dots\\}$. It follows from Theorem
4.3 that $|(\operatorname{silt}\Lambda)^{G}\cap[\Lambda[n],\Lambda]|=\infty$,
a contradiction. ∎

###  4.5. The case that $\Lambda$ is Frobenius

In this section we assume that $\Lambda$ is (finite dimensional) Frobenius.

####  4.5.1. Preliminary

We always equip a Frobenius algebra $\Lambda$ with a Nakayama automorphism
$\nu_{\Lambda}$ and a Nakayama form
$\langle-,+\rangle_{\Lambda}:\Lambda\times\Lambda\to{\mathbf{k}}$, which now
we recall. A Nakayama automorphism $\nu_{\Lambda}:\Lambda\to\Lambda$ is an
algebra automorphism and a Nakayama form
$\langle-,+\rangle_{\Lambda}:\Lambda\times\Lambda\to{\mathbf{k}}$ is a non-
degenerate bilinear form that satisfies the equations $\langle
rs,t\rangle_{\Lambda}=\langle r,st\rangle_{\Lambda}=\langle
s,t\nu_{\Lambda}(r)\rangle$ for any $r,s,t\in\Lambda$. We note that a Nakayama
automorphism is unique up to inner automorphisms. We also note that the map
$\phi:\Lambda\to{\textup{D}}(\Lambda),\ \phi_{r}(s):=\langle
r,s\rangle_{\Lambda}$ gives an isomorphism
$\phi:\Lambda\to{}_{\nu}{\textup{D}}(\Lambda)$ of
$\Lambda$-$\Lambda$-bimodules.

Following [6, Definition 11], we say that a group action
$G\curvearrowright\Lambda$ on a Frobenius algebra $\Lambda$ _preserves the
Nakayama form_ if we have $\langle g(r),g(s)\rangle_{\Lambda}=\langle
r,s\rangle_{\Lambda}$ for all $r,s\in\Lambda$ and $g\in G$.

######  Proposition 4.8 (cf. [6, Proposition 4.4, Corollary 4.5]).

Let $\Lambda$ be a Frobenius algebra with an action of $G$ that preserves the
Nakayama form. Then the skew group algebra $A=\Lambda*G$ is a Frobenius
algebra with a Nakayama automorphism $\nu_{A}$ given by

$\nu_{A}(r*g):=\nu_{\Lambda}(r)*g\ \ (r\in\Lambda,\ g\in G).$

###### Proof.

For simplicity we write $\nu=\nu_{\Lambda},\
\langle-,+\rangle=\langle-,+\rangle_{\Lambda}$. First we claim that $\nu
g(r)=g\nu(r)$ for all $r\in\Lambda$. Indeed we can verify, as below, that the
equation $\langle\nu^{-1}g^{-1}\nu g(r),s\rangle=\langle r,s\rangle$ holds for
any $s\in\Lambda$:

$\begin{split}\langle\nu^{-1}g^{-1}\nu g(r),s\rangle=\langle s,g^{-1}\nu
g(r)\rangle=\langle g(s),\nu g(r)\rangle=\langle g(r),g(s)\rangle=\langle
r,s\rangle\end{split}$

Now it is straightforward to check that the bilinear form
$\langle-,+\rangle_{A}:A\times A\to{\mathbf{k}}$ given by

$\langle r*g,s*h\rangle_{A}:=\delta_{g,h^{-1}}\langle r,g(s)\rangle\ \
(r,s\in\Lambda,\ g,h\in G)$

is a Nakayama form whose Nakayama automorphism is $\nu_{A}$. ∎

####  4.5.2. Twist by Nakayama automorphism of finite order

Let $\Lambda$ be a Frobenius algebra. Now we assume that $\Lambda$ has a
Nakayama automorphism $\nu$ of order $N<\infty$ in the group
$\operatorname{Aut}\Lambda$ of algebra automorphisms of $\Lambda$. Let
$\langle\nu\rangle\cong{\mathbb{Z}}/N{\mathbb{Z}}$ be the subgroup of
$\operatorname{Aut}\Lambda$ generated by $\nu$. We consider the canonical
action $\langle\nu\rangle\curvearrowright\Lambda$ to be $\nu^{n}\cdot
r:=\nu^{n}(r)$ for $r\in\Lambda$ and $n\in{\mathbb{Z}}/N{\mathbb{Z}}$.

We denote the posets of $\langle\nu\rangle$-stable objects
$(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{\nu}:=(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{\langle\nu\rangle},$
$(\operatorname{silt}\Lambda)^{\nu}:=(\operatorname{silt}\Lambda)^{\langle\nu\rangle}$
etc.

We point out the lemma below that follows from [3, Theorem A.4.], [2,
Propositions 2.5, 2.6].

######  Lemma 4.9.

We have the following equalities and an isomorphism of the posets

$\operatorname{2-tilt}\Lambda=(\operatorname{2-silt}\Lambda)^{\nu}\cong(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{\nu},\
\operatorname{tilt}\Lambda=(\operatorname{silt}\Lambda)^{\nu}.$

The following corollary plays a key role in applications given in the next
section.

######  Corollary 4.10.

In the above setup, the skew group algebra $A:=\Lambda*\langle\nu\rangle$ is
symmetric and we have the following equalities and an isomorphism of posets

$\operatorname{2-tilt}A=\operatorname{2-silt}A\cong\textup{s}\tau\textup{-}\operatorname{tilt}A,\
\operatorname{tilt}A=\operatorname{silt}A.$

###### Proof.

By Proposition 4.8, the automorphism $\nu_{A}$ of $A$ given by
$\nu_{A}(r*\nu^{m}):=\nu_{\Lambda}(r)*\nu^{m}$ is a Nakayama automorphism of
$A$. It is straightforward to check that $\nu_{A}$ is the inner automorphism
$\operatorname{ad}_{\epsilon}=\epsilon\cdot-\cdot\epsilon^{-1}$ associated to
the element $\epsilon:=1_{\Lambda}*\nu$. It follows that $A$ is symmetric.
Applying Lemma 4.9 to the algebra $A$, we obtain the desired equalities of
posets. ∎

Applying Theorem 3.3, Theorem 4.3 and Proposition 4.4, we obtain the following
result.

######  Theorem 4.11.

The adjoint pair
$(\operatorname{Ind}_{\Lambda}^{A},\operatorname{Res}^{A}_{\Lambda})$ induces
the following isomorphisms of posets

$\begin{split}&\operatorname{2-tilt}\Lambda\cong(\operatorname{2-tilt}A)^{\langle\nu\rangle\operatorname{mod}}\cong(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\langle\nu\rangle\operatorname{mod}},\
\ \
\operatorname{tilt}\Lambda\cong(\operatorname{silt}A)^{\langle\nu\rangle\operatorname{mod}}.\end{split}$

Finally, applying Theorem 3.11 and Theorem 4.7, we obtain the following
result.

######  Theorem 4.12.

Assume that the action $\langle\nu\rangle\curvearrowright\Lambda$ preserves
idempotent elements and has local stabilizers and that $\Lambda$ is tilting
discrete. Then the adjoint pair
$(\operatorname{Ind}_{\Lambda}^{A},\operatorname{Res}^{A}_{\Lambda})$ induces
the following isomorphisms of posets

$\begin{split}&\operatorname{2-tilt}\Lambda\cong\operatorname{2-tilt}A\cong\textup{s}\tau\textup{-}\operatorname{tilt}A,\
\ \ \operatorname{tilt}\Lambda\cong\operatorname{silt}A.\end{split}$

##  5\. Application to preprojective algebras and folded mesh algebras

In this section, applying results obtained in the previous sections, we study
silting theory and $\tau$-tilting theory of preprojective algebras and folded
mesh algebras.

###  5.1. The preprojective algebra $\Pi({\mathbb{L}}_{n})$ of type
${\mathbb{L}}_{n}$

In this section, we determine the posets
$\textup{s}\tau\textup{-}\operatorname{tilt}\Pi({\mathbb{L}}_{n}),\
\operatorname{silt}\Pi({\mathbb{L}}_{n})$ in terms of the Weyl group and the
braid group.

#### 5.1.1.

Recall that the generalized Dynkin diagrams introduced by Happel-Preiser-
Ringel [35] consist of the usual Dynkin diagrams ${\mathbb{A}}_{n}$,
${\mathbb{B}}_{n}$, ${\mathbb{C}}_{n}$, ${\mathbb{D}}_{n}$,
${\mathbb{E}}_{6}$, ${\mathbb{E}}_{7}$, ${\mathbb{E}}_{8}$,
${\mathbb{F}}_{4}$, ${\mathbb{G}}_{2}$ and one additional case
${\mathbb{L}}_{n}$ defined as

${\mathbb{L}}_{n}\ \ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern