Normal subgroups and support τ -tilting modules ∗†Ryotaro KOSHIO

Yuta KOZAKAI

December 26, 2023
Abstract
Let G̃ be a ﬁnite group, G a normal subgroup of G̃ and k an algebraically closed ﬁeld of characteristic p > 0. The ﬁrst main result in this paper is to show that support τ -tilting kG̃-modules
with some properties are support τ -tilting modules as kG-modules, too. As the second main result,
we give equivalent conditions for support τ -tilting kG̃-modules to satisfy the above properties, andshow that the set of the support τ -tilting kG̃-modules with the properties is isomorphic to the setof G̃-invariant support τ -tilting kG-modules as posets. As an application, we show that the set ofG̃-invariant support τ -tilting kG-modules is isomorphic to the set of support τ -tilting kG̃-modules in
the case that the index of G in G̃ is a p-power. As a further application, we give a feature of vertices
of indecomposable τ -rigid kG̃-modules. Finally, we give block versions of the above results.

1

Introduction

Since 2014 when τ -tilting theory was introduced by T. Adachi, O. Iyama, and I. Reiten [1], the theory
continues to develop rapidly. The main theme of the theory is to study of support τ -tilting modules, andmany researchers have already contributed to it. In fact, the support τ -tilting modules are in one-toone correspondences with the various representation-theoretically important objects including two-term
silting complexes [1], functorially ﬁnite torsion classes [1], left-ﬁnite semibricks [3], two-term simpleminded collections [3, 15], and so on. In particular, the theory is expected to be helpful in solving Broué’sabelian defect group conjecture because the theory is useful for the classiﬁcation of two-term tilting
complexes over group algebras or block algebras of ﬁnite groups. Even though the studies on the τ -tilting
theory related to the modular representation theory of ﬁnite groups are very important for these reasons,
there are few such studies. Therefore, it should be quite important to study modular representation
theory in relation to τ -tilting theory. Indeed, the authors started such a study in [16, 18, 17, 19].
However, the results obtained so far are all concerned with the induction functor; they tell under somedistinct situations that the induction functor IndG̃
G from kG-modules to k G̃-modules induces a poset
homomorphisms with some nice property from the support τ -tilting modules over kG to those over k G̃,where G is a normal subgroup of a ﬁnite group G̃ and k an algebraically closed ﬁeld of characteristicp > 0. (Poset means partially ordered set, as usual.) Naturally, there arises the following question.
Question 1.1. When does the restriction functor from k G̃-modules to kG-modules give the maps fromthe support τ -tilting modules over k G̃ to those over kG?
Regarding this question, in [5], S. Breaz, A. Marcus, and G. C. Modoi gave a positive answer in casethat the quotient group G̃/G is a p-prime group (i.e. the prime number p does not divide the order ofthe factor group G̃/G). Therefore, we consider the case that G̃/G is not necessarily a p-prime group, andget the following positive answer to the question.
Theorem 1.2 (Theorem 3.4 and Corollary 3.5). Let G̃ be a ﬁnite group, G a normal subgroup ofG̃, M̃ a relatively G-projective support τ -tilting k G̃-module, and (M̃ , P̃ ) a corresponding support τ G̃G̃tilting pair. If it holds that IndG̃
G ResG M̃ ∈ add M̃ , then ResG M̃ is a support τ -tilting kG-module, andG̃(ResG̃
G M̃ , ResG P̃ ) a corresponding support τ -tilting pair. Moreover, for relatively G-projective supportG̃τ -tilting k G̃-modules M̃1 and M̃2 with the property that IndG̃
G ResG M̃i ∈ add M̃i for i = 1, 2, if M̃1 ≥ M̃2G̃
in sτ -tilt k G̃, then ResG̃
G M̃1 ≥ ResG M̃2 in sτ -tilt kG.
∗ Mathematics Subject Classiﬁcation (2020). 20C20, 16G10.
† Keywords. Support τ -tilting modules, blocks of ﬁnite groups, induction functors, restriction functors

1Let M be a G̃-invariant support τ -tilting kG-module. The ﬁrst author showed that IndG̃
G M is a
support τ -tilting k G̃-module [16, Theorem 3.2]. We are interested in what is the image of the set of G̃invariant support τ -tilting kG-modules under the map induced by the induction functor IndG̃
G . Therefore,
we give equivalent conditions to the assumption of Theorem 1.2, and ﬁnally we clarify what the image ofthe map induced by IndG̃
G is in the following theorem.
Theorem 1.3 (Theorem 3.8 and Corollary 3.9). Let M̃ be a support τ -tilting k G̃-module. Then thefollowing conditions are equivalent:(1) M̃ =add IndG̃
G M for some G̃-invariant support τ -tilting kG-module M .
G̃(2) IndG̃
G ResG M̃ ∈ add M̃ and M̃ is relatively G-projective.

(3) S ⊗k M̃ ∈ add M̃ for each simple k(G̃/G)-module S.
Moreover, denoting by (sτ -tilt kG)G̃ the subset of sτ -tilt kG consisting of G̃-invariant support τ -tilting kGmodules and by (sτ -tilt k G̃)⋆ the subset of sτ -tilt k G̃ consisting of support τ -tilting k G̃-modules satisfying
the above equivalent conditions, the induction functor IndG̃G induces a poset isomorphism(sτ -tilt kG)G̃∼(sτ -tilt k G̃)⋆
IndG̃
G M.

M

The studies on the vertices of indecomposable modules over group algebras have been done by many
researchers for a long time, for example, see [6, 7, 8, 11, 14, 23]. On the other hand, τ -rigid modules overﬁnite-dimensional algebras are important classes and there are many studies on the modules. They havenice properties and are essential objects for the representation theory, for example, see [1, 4, 9, 10, 13].
Therefore, one of our interests is to give a feature of the vertices of indecomposable τ -rigid modules overgroup algebras. As a further application of Theorem 1.3, we give a feature of vertices of indecomposableτ -rigid modules.
Theorem 1.4 (Theorem 3.14). Let G̃ be a ﬁnite group. Then any indecomposable τ -rigid k G̃-module
has a vertex contained in a Sylow p-subgroup of G̃ properly if and only if G̃ has a normal subgroup of
p-power index in G̃.
As a natural question, we wonder if we get a block version of our theorems for group algebras. In
particular, we are interested in how we give block versions of Theorems 1.2 and 1.3. As the results, we
get block versions of the ntheorems. Let G be aonormal subgroup of a ﬁnite group G̃ and B a block of kG.
We denote by IG̃ (B) := g̃ ∈ G̃ g̃Bg̃ −1 = B the inertial group of B in G̃.
Theorem 1.5 (Theorem 5.6). Let G be a normal subgroup of a ﬁnite group G̃, B a block of kG, B̃ ablock of k G̃ covering B, β the block of kIG̃ (B) satisfyingX
x1β x−1 = 1B̃
x∈[G̃/IG̃ (B)]I (B)and M̃ a support τ -tilting B̃-module. If it holds that βIndGG̃

I (B)ResGG̃

G̃
βResG̃
IG̃ (B) M̃ ∈ add βResIG̃ (B) M̃

I (B)

G̃
and βResG̃
βResG̃
IG̃ (B) M̃ is relatively G-projective, then we have that ResG
IG̃ (B) M̃ is a support τ -tilting
B-module. Moreover, if (M̃ , P̃ ) is a support τ -tilting pair for B̃ corresponding to M̃ , then the pair

I (B)(ResGG̃I (B)G̃βResG̃
IG̃ (B) M̃ , ResG

I (B)is a support τ -tilting pair for B corresponding to ResGG̃

βResG̃
IG̃ (B) P̃ )βResG̃
IG̃ (B) M̃ .

Theorem 1.6 (Theorem 5.7 and Corollary 5.8). Let G be a normal subgroup of a ﬁnite group G̃, B a
block of kG, B̃ a block of k G̃ covering B, β the block of kIG̃ (B) satisfyingX
x1β x−1 = 1B̃
x∈[G̃/IG̃ (B)]and M̃ a support τ -tilting B̃-module. Then the following conditions are equivalent:2(1) M̃ =add B̃IndG̃
G M for some IG̃ (B)-invariant support τ -tilting B-module M .
I (B)(2) βIndGG̃

I (B)ResGG̃G̃
G̃
βResG̃
IG̃ (B) M̃ ∈ add βResIG̃ (B) M̃ and βResIG̃ (B) M̃ is relatively G-projective.

G̃(3) β(S ⊗k βResG̃
IG̃ (B) M̃ ) ∈ add βResIG̃ (B) M̃ for each simple k(IG̃ (B)/G)-module S.

Moreover, denoting by (sτ -tilt B)IG̃ (B) the subset of sτ -tilt B consisting of IG̃ (B)-invariant support τ tilting B-modules and by (sτ -tilt B̃)⋆⋆⋆ the subset of sτ -tilt B̃ consisting of support τ -tilting B̃-modules
satisfying the above equivalent conditions, the functor B̃IndG̃G induces a poset isomorphism(sτ -tilt B)IG̃ (B)∼(sτ -tilt B̃)⋆⋆⋆B̃IndG̃G M.

M

Our particular interest is in the case that the index of G in G̃ is a p-power. In fact, under someassumptions, it is expected that tilting complexes over the block B of kG give those over the uniqueblock B̃ of k G̃ covering B (for example, see [12, 20, 26]). In this regard, the authors showed the following
result in [17].
Theorem 1.7 ([17, Theorem 1.2]). Let G be a normal subgroup of a ﬁnite group G̃ of p-power index inG̃, B a block of kG, and B̃ the unique block of k G̃ covering B. Assume that the following two conditions
are satisﬁed:(1) Any indecomposable B-module is IG̃ (B)-invariant.
(2) The set of isomorphism classes of basic support τ -tilting B-modules is a ﬁnite set.
Then the induction functor IndG̃G induces an isomorphism from sτ -tilt B to sτ -tilt B̃ of posets.
This theorem can be applied to the case that the block B has a cyclic defect group, but the two
conditions limit the scope of its use. For example, the theorem cannot be applied to the case that p = 2,
G is the alternating group A4 of degree 4 and that G̃ is the symmetric group S4 of degree 4, because thenontrivial simple kA4 -modules are not S4 -invariant. Indeed, sτ -tilt kA4 is not isomorphic to sτ -tilt kS4
because the number of isomorphism classes of simple kA4 -modules is three and that of kS4 is two.
However, we wonder if the induction functor might give some kinds of good relation between the special
subsets of the two, and ﬁnally, as an application of Theorem 1.6, we could get the following theorem
which can be applied to the case of kA4 and kS4 . The following theorem is a signiﬁcant generalization ofTheorem 1.7 and enables us to explain the phenomenon occurred in [16, Example 3.9] (see Example 3.11).
Theorem 1.8 (Theorem 5.9). Let G be a normal subgroup of a ﬁnite group G̃, B a block of kG and B̃a block of k G̃ covering B. If the quotient group IG̃ (B)/G is a p-group, then the functor IndG̃G inducesIG̃ (B)IG̃ (B)an isomorphism as posets between (sτ -tilt B)
and sτ -tilt B̃, where (sτ -tilt B)
is the subset ofsτ -tilt B consisting of IG̃ (B)-invariant support τ -tilting B-modules.
Throughout this paper, we ﬁx the following notation:
Let k be an algebraically closed ﬁeld of characteristic p > 0. An algebra means a k-algebra. For aﬁnite-dimensional algebra Λ, a Λ-module means a ﬁnite-dimensional left Λ-module. For a Λ-module M ,we denote the Auslander-Reiten translate of M by τ M . In case that Λ is a symmetric algebra, τ M is
isomorphic to Ω2 M . We denote the category of all direct summands of ﬁnite direct sums of copies of
M by add M . For Λ-modules M and N , we write M =add N if add M = add N . This relation is anequivalence relation. We denote by sτ -tilt Λ the set of equivalence classes of support τ -tilting Λ-modules
under the equivalence relation =add .
Let G be a ﬁnite group and H a subgroup of G. We denote the restriction functor from kG-modules
G
to kH-modules by ResG
H and the induction functor kG ⊗kH − from kH-modules to kG-modules by IndH .
We denote the trivial kG-module by kG .
Let G̃ be a ﬁnite group, G a normal subgroup of G̃. We denote a set of coset representatives of G in
G̃ by [G̃/G]. For a kG-module M and g̃ ∈ G̃, we deﬁne a kG-module g̃M consisting of symbols g̃m as aset, where m ∈ M , and its kG-module structure is given by g̃m + g̃m′ := g̃(m + m′ ), g(g̃m) := g̃(g̃ −1 gg̃m)and λ(g̃m) = g̃(λm) for m, m′ ∈ M, g ∈ G and λ ∈ k. For a kG-module M , we say that M is G̃-invariant
if M is isomorphic to g̃M for any g̃ ∈ G̃.
3

2

Preliminaries

In this section, we give elementary facts on the modular representation theory which are helpful to prove
our results.
Proposition 2.1 ([2, Lemma 8.5, Lemma 8.6]). Let G be a ﬁnite group, K a subgroup of G, H a
subgroup of K. For any kG-module U and kH-module V , the following hold:KG∼(1) ResG
H U = ResH ResK U .
GK∼(2) IndG
H V = IndK IndH V .
GG∼(3) IndG
H (V ⊗k ResH U ) = (IndH V ) ⊗k U .
G∼(4) HomkG (U, IndG
H V ) = HomkH (ResH U, V ).
G∼(5) HomkG (IndG
H V, U ) = HomkH (V, ResH U ).
G(6) The functors ResG
H and IndH send free modules (projective modules) to free modules (projective
modules, respectively).

In the modular representation theory of ﬁnite groups, Mackey’s decomposition formula is well-known
and important. We recall Mackey’s decomposition formula for normal subgroups.
Proposition 2.2 ([2, Lemma 8.7]). Let G be a normal subgroup of a ﬁnite group G̃ and M a kG-module.
Then the following isomorphism as kG-modules holds:
M
G̃
∼
xM.
ResG̃
G IndG M =x∈[G̃/G]The following is known as Eckmann-Shapiro Lemma.
Lemma 2.3 ([22, Proposition 2.20.7]). Let H be a ﬁnite group of a ﬁnite group G, M a kH-module andN a kG-module. Then for all n ∈ N, there exists an isomorphism of k-vector spaces:Gn∼ExtnkH (M, ResG
H N ) = ExtkG (IndH M, N )The following lemma is a reﬁnement of [16, Lemma 3.1] which requires the G̃-invariance for the
kG-module.
Lemma 2.4. Let G be a normal subgroup of G̃ and M a kG-module. Then the following hold:G̃∼(1) IndG̃
G (ΩM ) = Ω(IndG M ).
G̃∼(2) IndG̃G (τ M ) = τ (IndG M ).

Proof. It is enough to show that the statement (1) holds since τ ∼= Ω2 for symmetric algebras. There exists
G̃
G̃
G̃
∼
a projective kG-module Q such that IndG (ΩM ) ∼= Ω(IndG M )⊕Q and that IndG̃
G P (M ) = P (IndG M )⊕Q.
Hence, we have that
G̃
G̃
G̃
G̃
∼
ResG̃
G IndG (ΩM ) = ResG Ω(IndG M ) ⊕ ResG Q,
Land the left-hand side is isomorphic to x∈[G̃/G] xΩM by Proposition 2.2. However, each xΩM has no
projective summands and the restricted module ResG̃
G Q is a projective kG-module by Proposition 2.1 (6),G̃∼which implies that Q = 0. Therefore, we conclude that IndG̃
G (ΩM ) = Ω(IndG M ).
Lemma 2.5. Let G be a normal subgroup of a ﬁnite group G̃ and M̃ be a k G̃-module. Then ResG̃G M̃ isa G̃-invariant kG-module.

4

Proof. Take g̃ ∈ G̃ arbitrarily. We consider the map
f : ResG̃
G M̃

g̃ResG̃
G M̃

m

g̃m.

Clearly, this map is linear and bijective. We only show that the map is kG-homomorphism, but for any
g ∈ G and m ∈ ResG̃G M̃ , it holds thatf (gm) = g̃gm = g̃gg̃ −1 g̃m = g · g̃m = g · f (m).

3

Main Theorems

In this section we give theorems stated in Section 1 and their proofs. Throughout this section, G̃ denotes
a ﬁnite group and G a normal subgroup of G̃.
First, we start with a consideration on restricted modules of rigid modules and τ -rigid modules. Let Λ
be a ﬁnite-dimensional algebra. We recall that a Λ-module M is rigid (resp. τ -rigid) if Ext1Λ (M, M ) = 0(resp. HomΛ (M, τ M ) = 0). We remark that τ -rigid modules are rigid modules by the Auslander-Reiten
duality HomΛ (X, Y ) ∼= D Ext1Λ (Y, τ X), where HomΛ (X, Y ) denotes the quotient of HomΛ (X, Y ) by thesubspace of the homomorphisms factoring through projective Λ-modules.
G̃
Lemma 3.1. Let M̃ be a k G̃-module with the property that IndG̃
G ResG M̃ ∈ add M̃ . Then the following
hold:(1) If M̃ is a rigid k G̃-module, then the restricted module ResG̃G M̃ is a rigid kG-module.
(2) If M̃ is a τ -rigid k G̃-module, then the restricted module ResG̃G M̃ is a τ -rigid kG-module.
Proof. (1) Let M̃ be a rigid k G̃-module. Then, by Lemma 2.3, we have that
G̃
G̃
1
G̃
∼
Ext1kG (ResG̃G M̃ , ResG M̃ ) = ExtkG̃ (M̃ , IndG ResG M̃ ).
G̃
By the assumption that IndG̃
G ResG M̃ ∈ add M̃ and the rigidity of M̃ , we have that the right-hand side isG̃
0. Hence, ResG M̃ is a rigid kG-module.
(2) Let M̃ be a τ -rigid k G̃-module. Then we have that
G̃
G̃
G̃
G̃
G̃
∼
∼
HomkG (ResG̃G M̃ , τ ResG M̃ ) = HomkG̃ (M̃ , IndG τ ResG M̃ ) = HomkG̃ (M̃ , τ IndG ResG M̃ ),G̃
where the last isomorphism comes from Lemma 2.4. By the assumption that IndG̃
G ResG M̃ ∈ add M̃ andG̃G̃
the τ -rigidity of M̃ , we have that HomkG̃ (M̃ , τ IndG ResG M̃ ) = 0, which implies that ResG̃G M̃ is a τ -rigid
kG-module.

For a ﬁnite group H and a subgroup K of H, we recall that a kH-module M is relatively K-projectiveH
if M is a direct summand of IndH
K ResK M .
G̃∼Lemma 3.2. Let M̃ be a relatively G-projective k G̃-module. Then ResG̃
G (ΩM̃ ) = Ω(ResG M̃ ). In
G̃
G̃
particular, it holds that τ (ResG M̃ ) ∼= ResG (τ M̃ ).
G̃
∼
Proof. There exists a projective kG-module P such that ResG̃
G (ΩM̃ ) = Ω(ResG M̃ ) ⊕ P . Hence, it isenough to show that P = 0. It is clear in the case that M̃ is a projective k G̃-module.
We may assume that M̃ has no projective summands. Since M̃ is relatively G-projective, ΩM̃ isrelatively G-projective too (for example see [2, Proposition 20.7]). Hence, ΩM̃ is a direct summand
G̃
G̃
G̃
∼
of IndG̃
G ResG ΩM̃ . On the other hand, by the isomorphism ResG ΩM̃ = Ω(ResG M̃ ) ⊕ P , we have that

5

G̃
G̃
G̃
G̃
G̃
∼
IndG̃
G ResG ΩM̃ = IndG (Ω(ResG M̃ )) ⊕ IndG P . Here, since IndG P is a projective k G̃-module by Proposition 2.1 (6) and ΩM̃ has no projective summands by the self-injectivity of k G̃, we have that ΩM̃ is aG̃G̃
direct summand of IndG̃
G (Ω(ResG M̃ )). Therefore, ResG (ΩM̃ ) is a direct summand of
M
G̃
G̃
∼
ResG̃
g̃Ω(ResG̃
G IndG (Ω(ResG M̃ )) =G M̃ )g̃∈[G̃/G]G̃
by Proposition 2.2, which implies that ResG̃
G (ΩM̃ ) is has no projective summands because each g̃Ω(ResG M̃ )G̃
has no projective summands by the self-injectivity of kG. Thus, we conclude that P = 0 and ResG (ΩM̃ ) ∼=G̃Ω(ResG M̃ ).
The later assertion follows from the fact that τ ∼= Ω2 and the relative G-projectivity of ΩM̃ .

The following is important for the proof of Theorem 3.4.
Proposition 3.3 ([1, Corollary 2.13]). Let Λ be a ﬁnite-dimensional algebra. For a τ -rigid pair (M, P )for Λ the following are equivalent:(1) (M, P ) is a support τ -tilting pair for Λ.
(2) If HomΛ (M, τ X) = 0, HomΛ (X, τ M ) = 0 and HomΛ (P, X) = 0, then X ∈ add M .
G̃
Theorem 3.4. Let M̃ be a support τ -tilting k G̃-module. If it holds that IndG̃
G ResG M̃ ∈ add M̃ and M̃ isrelatively G-projective, then we have that ResG̃G M̃ is a support τ -tilting kG-module. Moreover, if (M̃ , P̃ )G̃is a support τ -tilting pair for k G̃ corresponding to M̃ , then (ResG̃G M̃ , ResG P̃ ) is a support τ -tilting pairfor kG corresponding to ResG̃G M̃ .

Proof. Let (M̃ , P̃ ) be a support τ -tilting pair for k G̃ corresponding to the support τ -tilting k G̃-moduleM̃ .
G̃
First, we show that (ResG̃G M̃ , ResG P̃ ) is a τ -rigid pair for kG. Since the k G̃-module M̃ is a support τ tilting module, it is a τ -rigid module. Hence, we have that ResG̃G M̃ is a τ -rigid k G̃-module by Lemma 3.1.
On the other hand, by Proposition 2.1 we have that
G̃
G̃
G̃
∼
HomkG (ResG̃G P̃ , ResG M̃ ) = HomkG̃ (P̃ , IndG ResG M̃ ).
G̃
G̃
G̃
Now, by the assumption that IndG̃
G ResG M̃ ∈ add M̃ , we have that HomkG (ResG P̃ , ResG M̃ ) = 0 becauseG̃G̃(M̃ , P̃ ) is a support τ -tilting pair for k G̃. Therefore, we conclude that (ResG M̃ , ResG P̃ ) is a τ -rigid pairfor kG.
G̃Next, we show that the τ -rigid pair (ResG̃G M̃ , ResG P̃ ) is a support τ -tilting pair for kG. We showthat X ∈ add ResG̃G M̃ under the assumption thatG̃G̃
HomkG (X, τ (ResG̃G M̃ )) = HomkG (ResG M̃ , τ X) = HomkG (ResG P̃ , X) = 0,G̃which implies that the pair (ResG̃G M̃ , ResG P̃ ) is a support τ -tilting pair for kG by Proposition 3.3. Under
these assumptions, we have the following:G̃∼
HomkG̃ (IndG̃
G X, τ M̃ ) = HomkG (X, ResG (τ M̃ ))(Proposition 2.1)∼
= HomkG (X, τ (ResG̃G M̃ ))(Lemma 3.2)

= 0.
G̃
∼
HomkG̃ (M̃ , τ (IndG̃
G X)) = HomkG̃ (M̃ , IndG (τ X))∼= HomkG (ResG̃G M̃ , τ X)
= 0.

6

(Lemma 2.4)
(Proposition 2.1)

G̃∼
HomkG̃ (P̃ , IndG̃
G X) = HomkG̃ (ResG P̃ , X)
= 0.

(Proposition 2.1)

By these three isomorphisms and the fact that (M̃ , P̃ ) is a support τ -tilting pair for k G̃, applyG̃G̃
ing Proposition 3.3, we have that IndG̃G X ∈ add M̃ . Also, X is a direct summand of ResG IndG X byG̃
Proposition 2.2. Therefore, we have that X ∈ add ResG M̃ .
Corollary 3.5. Let M̃1 and M̃2 be relatively G-projective support τ -tilting k G̃-modules such that
G̃
G̃
G̃
IndG̃
G ResG M̃i ∈ add M̃i for i = 1, 2. Then M̃1 ≥ M̃2 in sτ -tilt k G̃ implies that ResG M̃1 ≥ ResG M̃2 insτ -tilt kG.
Proof. The consequence immediately follows from Theorem 3.4 and the exactness of the functor ResG̃
G.
We wish to restate the assumption of Theorem 3.4 with some equivalent conditions. First, we givethe lemmas which can be applied in case of rigid k G̃-modules not only support τ -tilting k G̃-modules.
Lemma 3.6. Let M̃ be a rigid k G̃-module and L a k G̃-module. If it holds that S ⊗k M̃ ∈ add M̃ forany composition factor S of L, then the following isomorphism as k G̃-modules holds:M
L ⊗k M̃ ∼S ⊗k M̃ ,=S

where S runs over all composition factors of L.
Proof. Let L be an arbitrary k G̃-module and M̃ a rigid k G̃-module satisfying thatS ⊗k M̃ ∈ add M̃ for any composition factor of S of L.

(3.1)

We use induction on the composition length ℓ(L) of L. If ℓ(L) = 1, there is nothing to prove. Hence, we
assume that ℓ(L) ≥ 2 and that the statement for any k G̃-module L′ satisfying ℓ(L′ ) < ℓ(L) is true. Let
T be a simple submodule of L. We get the exact sequence0

T ⊗k M̃L ⊗k M̃

L/T ⊗k M̃

0

(3.2)

obtained by applying the exact functor − ⊗k M̃ to the exact sequence
0

T

L

L/T

0.

By the rigidity of M̃ , the assumption (3.1) and the assumption of this induction, the sequence (3.2) splits,
and we get thatM
M
S ′ ⊗k M̃ ∼S ⊗k M̃ ,L ⊗k M̃ ∼= T ⊗k M̃ ⊕ L/T ⊗k M̃ ∼= T ⊗k M̃ ⊕=S′

S

where S ′ and S run over all composition factors of L/T and L, respectively.
Lemma 3.7. Let M̃ be a rigid k G̃-module. Then the following conditions are equivalent:G̃(1) IndG̃
G ResG M̃ ∈ add M̃ and M̃ is relatively G-projective.

(2) S ⊗k M̃ ∈ add M̃ for each simple k(G̃/G)-module S.
Proof. By Proposition 2.1, we have that
G̃
G̃
G̃
G̃
∼
∼
∼
IndG̃
G ResG M̃ = IndG (kG ⊗k ResG M̃ ) = (IndG kG ) ⊗k M̃ = k(G̃/G) ⊗k M̃ .

7

G̃(1) ⇒ (2). By the assumptions, we have that IndG̃
G ResG M̃ =add M̃ . Hence, by Proposition 2.1 we getthatG̃
S ⊗k M̃ =add S ⊗k IndG̃
G ResG M̃
G̃G̃∼= IndG̃
G (ResG S ⊗k ResG M̃ )⊕ dimk S∼⊗k ResG̃
= IndG̃
G M̃ )G (kGG̃
=add IndG̃
G ResG M̃

=add M̃ ,for any simple k(G̃/G)-module S, which implies that S ⊗k M̃ ∈ add M̃ .
(2) ⇒ (1). By Lemma 3.6, we have that
M
G̃
∼
∼
IndG̃
S ⊗k M̃ ,G ResG M̃ = k(G̃/G) ⊗k M̃ =S

where S runs over all composition factors of the k G̃-module k(G̃/G). Therefore, the assumption impliesG̃
that IndG̃
G ResG M̃ ∈ add M̃ . Moreover, since the trivial k G̃-module kG̃ appears as a composition factorG̃of k(G̃/G), we have that the module M̃ appears as a direct summand of IndG̃
G ResG M̃ , that is M̃ is arelatively G-projective k G̃-module.
We obtain below some conditions that are equivalent to the assumption of Theorem 3.4.
Theorem 3.8. Let M̃ be a support τ -tilting k G̃-module. Then the following conditions are equivalent:(1) M̃ =add IndG̃
G M for some G̃-invariant support τ -tilting kG-module M .
G̃(2) IndG̃
G ResG M̃ ∈ add M̃ and M̃ is relatively G-projective.

(3) S ⊗k M̃ ∈ add M̃ for each simple k(G̃/G)-module S.
Proof. (1) ⇒ (2). Assume that M̃ =add IndG̃
G M for some G̃-invariant support τ -tilting kG-module M .
Then clearly M̃ is a relatively G-projective k G̃-module (see [2, 3.9.1]), and by Proposition 2.2, we have
that
M
M
G̃
G̃
G̃
G̃
G̃
∼
g̃M ) ∼
IndG̃
IndG̃
=
G M ∈ add M̃ .
G ResG M̃ =add IndG ResG IndG M = IndG (g̃∈[G̃/G]g̃∈[G̃/G]G̃
G̃(2) ⇒ (1). Assume that IndG̃
G ResG M̃ ∈ add M̃ and that M̃ is relatively G-projective. Put M := ResG M̃ .
Then by Lemma 2.5 and Theorem 3.4, M is a G̃-invariant support τ -tilting kG-module. We show that
G̃
G̃
G̃
IndG̃
G M =add M̃ , that is add(IndG M ) = add M̃ . By the assumption that IndG ResG M̃ ∈ add M̃ , we haveG̃add(IndG M ) ⊂ add M̃ . On the other hand, since M̃ is relatively G-projective, M̃ is a direct summandG̃G̃G̃of IndG̃
G ResG M̃ = IndG M . Hence, we have add M̃ ⊂ add(IndG M ).
(2) ⇔ (3). Since support τ -tilting k G̃-modules are rigid k G̃-modules, the equivalence follows from
Lemma 3.7.

Corollary 3.9. Let (sτ -tilt kG)G̃ be the subset of sτ -tilt kG consisting of G̃-invariant support τ -tilting
kG-modules and (sτ -tilt k G̃)⋆ the subset of sτ -tilt k G̃ consisting of support τ -tilting k G̃-modules satisfying
the equivalent conditions of Theorem 3.8. Then the induction functor IndG̃G induces a poset isomorphism(sτ -tilt kG)G̃∼(sτ -tilt k G̃)⋆

(3.3)

IndG̃
G M.

M

In particular, the induction functor IndG̃G induces the poset monomorphism(sτ -tilt kG)G̃sτ -tilt k G̃

M

IndG̃
G M.
8

(3.4)

Proof. By [16, Theorem 3.2], the map (3.4) is well-deﬁned. Moreover, by the exactness of the functor
G̃
G̃
IndG̃
G , if N ≤ M in sτ -tilt kG then IndG N ≤ IndG M in sτ -tilt k G̃ for any support τ -tilting kG-modules
N and M . Therefore, the map (3.4) is a poset homomorphism. By the exactness of the functor ResG̃
G,
Lemma 2.5 and Theorem 3.4, the map(sτ -tilt k G̃)⋆(sτ -tilt kG)G̃

M̃

ResG̃
G M̃

(3.5)

is well-deﬁned and a poset homomorphism.
We show that the map (3.4) restricts to a bijection (3.3). By the deﬁnition of (sτ -tilt k G̃)⋆ and theabove argument, the map (3.3) is well-deﬁned and a poset homomorphism. For any relatively G-projectiveG̃
support τ -tilting k G̃-module M̃ with IndG̃
G ResG M̃ ∈ add M̃ , we have thatG̃IndG̃G ResG M̃ =add M̃ .

Moreover, for any G̃-invariant support τ -tilting kG-module M , we have that
M
G̃
∼
ResG̃
xM =add M.
G IndG M =x∈[G̃/G]

Therefore, the maps (3.3) and (3.5) are inverse to each other.
The latter assertion immediately follows from the fact that the map (3.4) is the composition of theposet isomorphism (3.3) and the inclusion map (sτ -tilt k G̃)⋆sτ -tilt k G̃.

As an application of Theorem 3.8, we consider the case that G̃/G is a p-group.
Theorem 3.10. Let G̃ be a ﬁnite group and G a normal subgroup of G̃ of p-power index in G̃. ThenG̃
the induction functor IndG̃G induces an isomorphism as posets between (sτ -tilt kG) and sτ -tilt k G̃, whereG̃(sτ -tilt kG) is the subset of sτ -tilt kG consisting of G̃-invariant support τ -tilting kG-module.
Proof. By Corollary 3.9, the map (3.3) is a poset isomorphism. It is enough to show that (sτ -tilt k G̃)⋆ =sτ -tilt k G̃. It is clear that (sτ -tilt k G̃)⋆ ⊂ sτ -tilt k G̃. To prove the reverse inclusion, take an arbitrary
support τ -tilting k G̃-module M̃ . Since G̃/G is a p-group, the only simple k(G̃/G)-module is the trivialk(G̃/G)-module. Hence, the condition (3) of Theorem 3.8 is satisﬁed in our situation because kG̃/G ⊗k M̃
is isomorphic to M̃ . Therefore, we conclude that sτ -tilt k G̃ ⊂ (sτ -tilt k G̃)⋆ .
Here we reproduce from [16] an example with an error corrected; in Figure 2 below, the centereddirect-sum of three non-projectives is corrected.
Example 3.11. Let k be an algebraically closed ﬁeld of characteristic p = 2. We consider that the casethat G is the alternating group A4 of degree 4 and G̃ is the symmetric group S4 of degree 4. The algebras
kA4 and kS4 are Brauer graph algebras associated to the Brauer graphs in Figure 1(a) and Figure 1(b),respectively:
Now we draw the Hasse diagram H(sτ -tilt kA4 ) of the poset sτ -tilt kA4 as follows:

9

3

2
2′

multiplicity: 2

1′ = k S 4

k A4 = 1(a) The Brauer graph of kA4(b) The Brauer graph of kS4

Figure 1: Brauer graphs10H(sτ -tilt kA4 ) :

3

3

2

1

3

2

1

3

3

2

1

P1 ⊕ 1 3 2 1 3 ⊕ P 3

2 ⊕ P 2 ⊕ P3

2
2 ⊕ P2 ⊕ 3

2 ⊕P ⊕ 2
2
1
3

P1 ⊕ P2 ⊕ P3

3

2

1

3

3
2 ⊕ 2 ⊕ P3

P1 ⊕ P2 ⊕ 2 1 3 2 1

P1 ⊕ 1 3 2 1 3 ⊕ 13

3 ⊕
3 1
1
1 2 3 ⊕ P3

P1 ⊕ 12 ⊕ 2 1 3 2 1

2 ⊕P ⊕
1 2
2
1
2 3 1

3
2
2 ⊕ 2 ⊕ 3

3 ⊕ 3 ⊕P
3
1
2

3 ⊕
3 1
1
1
1 2 3 ⊕ 3

P1 ⊕ 12 ⊕ 13

2 ⊕ 1 ⊕
1 2
1
2
2 3 1

3 ⊕ 2
2
3

3 ⊕ 3 ⊕3
1
2

3 ⊕ 1
1
3

1 ⊕ 12 ⊕ 13

2 ⊕ 1
1
2

3

11
2 ⊕2⊕ 21
3

2 ⊕ 23
2

3 ⊕3
2

3 ⊕3
1

1 ⊕ 133

1 ⊕ 12
1

0
Figure 2: The Hasse diagram of sτ -tilt kA4

2 ⊕2
1

The enclosed support τ -tilting modules in Figure 2 are all the invariant support τ -tilting modules under
the action of S4 . Next, we draw the Hasse diagram H(sτ -tilt kS4 ) of poset sτ -tilt kS4 as follows:H(sτ -tilt(kS4 )) :P1 ′ ⊕ P2 ′
2′
2′

1′

2′
2′

1′

1′

2′
2

1′

1′

⊕ P2 ′′

2′

P1′ ⊕ 1′′2

1′
1′ ⊕ ′
1
′
1
2′

′

2

⊕ 22′
′

2′
2′

1′
1′

0
Figure 3: The Hasse diagram of sτ -tilt kS4
The functor IndSA44 takes each enclosed S4 -invariant support τ -tilting kA4 -module in Figure 2 to that in
Figure 3 with the same square.
G̃−1(sτ -tilt k G̃) := {M ∈ sτ -tilt kG | IndG̃
Remark 3.12. Let (IndG̃G M ∈ sτ -tilt k G̃}. Then (sτ -tilt kG)G)−1(sτ -tilt k G̃) by [16, Theorem 3.2]. On the other hand, they do not coincide in
is contained in (IndG̃
G)
general. Moreover, though the poset homomorphism(sτ -tilt kG)G̃sτ -tilt k G̃

M

IndG̃
GM

is a monomorphism by Corollary 3.9, the one−1(IndG̃(sτ -tilt k G̃)G)sτ -tilt k G̃

M

IndG̃
GM

is not a monomorphism in general.
For example, for p = 2, the alternating group A4 of degree 4 and the symmetric group S4 of degree
4, a kA4 -module M := 1 ⊕ 12 is a support τ -tilting kA4 -module, where 1 denotes the trivial kA4 -moduleand 2 a non-trivial kA4 -module. Also, it holds that σM ∼= 1 ⊕ 13 for σ ∈ S4 \ A4 , where 3 denotes
the non-trivial simple kA4 -module not isomorphic to 2. Therefore, we have that M ̸∈ (sτ -tilt kA4 )S4 .
1′′
However, IndS4 M ∼= 1′ ⊕ 1′ is a support τ -tilting kS4 -module, where 1′ denotes the trivial kS4 -module
A4

1

2′

and 2 the simple kS4 -module of dimension 2. This implies that M ∈ (IndSA44 )−1 (sτ -tilt kS4 ). Moreover,
1′
1′
1′
′
′
for N := 1 ⊕ 1 ⊕ 1 ∈ (sτ -tilt kA4 )S4 , it holds that IndS4 N ∼= 1′ ⊕ 1′ ⊕ 1′ =add 1′ ⊕ 1′ (=add IndS4 M ).
′

2

3

A4

Therefore, the map

1

2′(IndSA44 )−1 (sτ -tilt kS4 )sτ -tilt kS4

M

IndSA44 M

is not a monomorphism.
12

2′

1

2′

A4

At the end of this section, we discuss a feature of vertices of indecomposable τ -rigid k G̃-modules.
Lemma 3.13. Let G̃ be a ﬁnite group. Then the trivial k G̃-module kG̃ is a τ -rigid if and only if G̃ has
no normal subgroups of index p.
Proof. By [21, Chap. I, Corollary 10.13], there exists a normal subgroup of G̃ of index p if and only if
Ext1kG̃ (kG̃ , kG̃ ) ̸= 0. Also, by the simplicity of the trivial k G̃-module and Auslander-Reiten duality, we
have that
HomkG̃ (kG̃ , τ kG̃ ) ∼= HomkG̃ (kG̃ , τ kG̃ ) ∼= D Ext1kG̃ (kG̃ , kG̃ ).
Therefore, we get the result.
Theorem 3.14. Let G̃ be a ﬁnite group. Then any indecomposable τ -rigid k G̃-module has a vertex
contained in a Sylow p-subgroup of G̃ properly if and only if G̃ has a proper normal subgroup of p-powerindex.
Proof. Assume that G̃ has no proper normal subgroup of p-power index. Then by Lemma 3.13, the trivial
k G̃-module, whose vertex is a Sylow p-subgroup of G̃, is a τ -rigid module.
Conversely, assume that G̃ has normal subgroup of p-power index. In this case, there exists a normal
subgroup G of G̃ of index p. Let X̃ be an arbitrary τ -rigid k G̃-module. Then, X̃ is a direct summandof a support τ -tilting k G̃-module M̃ by [1, Theorem 2.10], that is, X̃ is relatively G-projective. Also,
there exists a G̃-invariant support τ -tilting kG-module M such that M̃ =add IndG̃
G M by Theorem 3.10.
G̃
Hence, X̃ is a direct summand of IndG M . Therefore, X̃ has a vertex contained in a Sylow p-subgroup ofG̃ properly.

4Preliminaries for block versions of the main results

We recall the deﬁnition of blocks of group algebras. Let G be a ﬁnite group. The group algebra kG hasa unique decomposition
kG = B0 × · · · × Bl
into the direct product of indecomposable algebras. We call each indecomposable direct product component Bi a block of kG and the decomposition above the block decomposition. We remark that any block
Bi is a two-sided ideal of kG.
For any indecomposable kG-module U , there exists a unique block Bi of kG such that U = Bi U and
Bj U = 0 for all j ̸= i. Then we say that U lies in the block Bi or simply U is a Bi -module. We denote
by B0 (kG) the principal block of kG, in which the trivial kG-module kG lies.
Let G be a normal subgroup of a ﬁnite group G̃, B a block of kG and B̃ a block of k G̃. We say thatB̃ covers B (or that B is covered by B̃) if 1B 1B̃ ̸= 0, where 1B and 1B̃ denote the respective identity
element of B and B̃.
Proposition 4.1 ([2, Theorem 15.1, Lemma 15.3]). With the notation above, the following are equivalent:(1) The block B̃ covers B.
(2) There exists a non-zero B̃-module Ũ such that ResG̃G Ũ has a non-zero direct summand lying in B.
(3) For any non-zero B̃-module Ũ , there exists a non-zero direct summand of ResG̃G Ũ lying in B.
(4) For any non-zero B̃-module Ũ and indecomposable direct summand V of ResG̃G Ũ , there exists g̃ ∈ G̃such that V lies in the block g̃Bg̃ −1 .
(5) The block B is a direct summand of B̃ as a (kG, kG)-bimodule.
(6) The block B̃ is a direct summand of k G̃B G̃ as a (k G̃, k G̃)-bimodule.
We denote by IG̃ (B) the inertial group of B in G̃, that isonIG̃ (B) := g̃ ∈ G̃ g̃Bg̃ −1 = B .

13

Remark 4.2. For a block B̃ of k G̃ and a block B of kG, the block B̃ covers only B if and only if∼IG̃ (B) = G̃ by [2, Theorem 15.1 (1)]. Since ResG̃
G kG̃ = kG , the principal block B0 (kG) of kG is the only
block of kG covered by the principal block B0 (k G̃) of k G̃ by the equivalence of Proposition 4.1 (1), (3).
Therefore, we have that IG̃ (B0 (kG)) = G̃.
Proposition 4.3. Let G be a normal subgroup of a ﬁnite group G̃, B a block of kG and U an indecomposable B-module. Then the following hold:G̃(1) For a block B̃ of k G̃ covering the block B, the module ResG̃
G B̃IndG U has a direct summand isoG̃morphic to U . In particular, the B̃-module B̃IndG U is non-zero.

(2) Any indecomposable direct summand Ṽ of IndG̃G U lies in a block of k G̃ covering B.
Proof. Let U be an indecomposable B-module. By the equivalence of Proposition 4.1 (1), (5), the blockB̃ has a direct summand B as a (kG, kG)-bimodule. Hence, there exists a (kG, kG)-bimodule B ′ suchthat B̃ ∼= B ⊕ B ′ as a (kG, kG)-bimodule. Therefore, we have that
G̃
G̃
G̃
′
′
∼
∼
∼
∼
ResG̃
G B̃IndG U = ResG B̃(k G̃ ⊗kG U ) = ResG B̃ ⊗kG U = (B ⊕ B ) ⊗kG U = U ⊕ (B ⊗kG U ),which prove (1).
Let Ṽ be an indecomposable direct summand of IndG̃G U lying in a block Ã of k G̃. Since the restrictedG̃
module ResG̃
Ṽ
isa
directsummandofthekG-module
ResG̃
G
G IndG U , we have that the block Ã covers B
by Proposition 2.2 and the equivalences of Proposition 4.1 (1), (2), (4). Hence, we get that (2).
The following is a generalization of [24, Corollary 5.5.6] (or [25, Corollary 9.9.6]).
Proposition 4.4. Let G be a normal subgroup of a ﬁnite group G̃ and B a block of kG. If there exists
an indecomposable B-module X such that IndG̃
G X is an indecomposable k G̃-module, then there exists
only one block of k G̃ covering B.
G̃
Proof. Let Ã and B̃ be a block of k G̃ covering B. The modules ÃIndG̃
G X and B̃IndG X are non-zerodirect summands of the indecomposable k G̃-module IndG̃
G X by Proposition 4.3 (1). Hence, we get that
G̃
G̃
G̃
∼
∼
B̃IndG X = ÃIndG X = IndG X by the indecomposability of IndG̃
G X. Since the non-zero k G̃-module
X
lies
in
the
blocks
Ã
and
B̃,
we
get
that
Ã
=
B̃.
IndG̃
G

Corollary 4.5 ([24, Corollary 5.5.6] or [25, Corollary 9.9.6]). If G̃/G is a p-group, then there exists only
one block of k G̃ covering B.
Proof. It immediately follows from Proposition 4.4 and Green’s indecomposability theorem (for example,see [2, 11, 24]).
Proposition 4.6 ([22, Theorem 6.8.3] or [24, Theorem 5.5.10, Theorem 5.5.12]). Let G be a normal
subgroup of a ﬁnite group G̃ and B a block of kG. Then the following hold:(1) For any block β of kIG̃ (B) covering B, there exists a block B̃ of k G̃ such thatXx1β x−1 = 1B̃ ,x∈[G̃/IG̃ (B)]and then B̃ covers B. Moreover, the correspondence sending β to B̃ induces a bijection between
the set of blocks of kIG̃ (B) covering B and those of k G̃ covering B.
(2) If B̃ corresponds to β under the bijection of (1), then the induction functor
IndG̃
IG̃ (B) : kIG̃ (B)-modk G̃-modrestricts to a Morita equivalence
IndG̃
IG̃ (B) : β-mod14B̃-modand its inverse functor is given by
βResG̃
IG̃ (B) : B̃-mod

β-mod .

Proposition 4.7. Let G be a normal subgroup of a ﬁnite group G̃, B a block of kG, U a B-module, βa block of kIG̃ (B) covering B and B̃ a block of k G̃ covering B such thatXx1β x−1 = 1B̃ .
x∈[G̃/IG̃ (B)]G̃G̃∼Then B̃IndG̃
G U = IndIG̃ (B) βIndG

I (B)

U.

Proof. Let B̃1 = B̃, . . . , B̃e be the all blocks of k G̃ covering B. By Proposition 4.6, we can take β1 =β, . . . , βe the blocks of kIG̃ (B) satisfying the induction functor IndG̃
IG̃ (B) restricts to a Morita equivalence
IndG̃
IG̃ (B) : βi-mod

B̃i-mod

for any i = 1, . . . , e. By Proposition 4.3 (2), we get the following isomorphism:
I (B)

IndGG̃

U∼= β1 IndGG̃

I (B)

I (B)

U ⊕ · · · ⊕ βe IndGG̃

U.

Moreover, by Proposition 2.1 (2), we have that
G̃
G̃
∼
IndG̃
G U = IndIG̃ (B) IndG

I (B)

G̃
U∼
= IndG̃
IG̃ (B) β1 IndG

I (B)

G̃
Since the k G̃-module IndG̃
IG̃ (B) βi IndG

I (B)

I (B)

G̃
U ⊕ · · · ⊕ IndG̃
IG̃ (B) βe IndGU.

U lies in the block B̃i for any i = 1, . . . , e, we get thatG̃G̃∼B̃i IndG̃G U = IndIG̃ (B) βi IndG

I (B)

U.

Therefore, we complete the proof.

5

Block version of main results

In this section we give block versions of our theorems stated in Section 3. Let Λ be a ﬁnite-dimensional
algebra. For Λ-modules M and N , we write M ≤add N if add M ⊂ add N . This relation is clearlyreﬂexive and transitive. Moreover, if M ≤add N and N ≤add M then M =add N for any Λ-modules M
and N . The following is the special case of the block version of Theorem 3.4.
Theorem 5.1. Let G be a normal subgroup of a ﬁnite group G̃, B a block of kG satisfying IG̃ (B) = G̃,G̃
B̃ a block of k G̃ covering B and M̃ a support τ -tilting B̃-module. If it holds that B̃IndG̃
G ResG M̃ ∈ add M̃
and M̃ is relatively G-projective, then we have that ResG̃G M̃ is a support τ -tilting B-module. Moreover, ifG̃(M̃ , P̃ ) is a support τ -tilting pair for B̃ corresponding to M̃ , then (ResG̃G M̃ , ResG P̃ ) is a support τ -tilting
pair for B corresponding to ResG̃G M̃ .
Proof. Let (M̃ , P̃ ) be a support τ -tilting pair for B̃ corresponding to the support τ -tilting B̃-module M̃ .
Our assumption IG̃ (B) = G̃ implies that the block B is the only block of kG covered by B̃ by Remark 4.2.
G̃
Hence, we have that the restricted modules ResG̃G M̃ and ResG P̃ are B-modules by Proposition 4.1 (4).
G̃G̃First, we show that (ResG M̃ , ResG P̃ ) is a τ -rigid pair for B. Since the B̃-module M̃ is a support
τ -tilting B̃-module, it is a τ -rigid B̃-module. Hence, we have that ResG̃G M̃ is a τ -rigid B-module byLemma 3.1. On the other hand, by Proposition 2.1 (4) we have that
G̃
G̃
G̃
∼
HomB (ResG̃G P̃ , ResG M̃ ) = HomkG (ResG P̃ , ResG M̃ )G̃∼= HomkG̃ (P̃ , IndG̃
G ResG M̃ )G̃∼= HomB̃ (P̃ , B̃IndG̃
G ResG M̃ ).

15

G̃
G̃
G̃
Now, by the assumption that B̃IndG̃
G ResG M̃ ∈ add M̃ , we have that HomB̃ (P̃ , B̃IndG ResG M̃ ) = 0
G̃because (M̃ , P̃ ) is a support τ -tilting pair for B̃. Therefore, we conclude that (ResG̃G M̃ , ResG P̃ ) is aτ -rigid pair for B.
G̃
Next, we show that the τ -rigid pair (ResG̃G M̃ , ResG P̃ ) is a support τ -tilting pair for B. We show thatG̃X ∈ add ResG M̃ under the assumption thatG̃G̃HomB (X, τ (ResG̃G M̃ )) = HomB (ResG M̃ , τ X) = HomB (ResG P̃ , X) = 0,G̃which implies that the pair (ResG̃G M̃ , ResG P̃ ) is a support τ -tilting pair for B by Proposition 3.3. Under
these assumptions, we have the following:G̃∼
HomB̃ (B̃IndG̃
G X, τ M̃ ) = HomkG̃ (IndG X, τ M̃ )(τ M̃ is a B̃-module)∼= HomkG (X, ResG̃
G (τ M̃ ))(Proposition 2.1 (5))∼= HomkG (X, τ (ResG̃G M̃ ))(Lemma 3.2)∼
= HomB (X, τ (ResG̃G M̃ ))= 0.

(X and τ (ResG̃G M̃ ) are B-modules)G̃∼
HomB̃ (M̃ , τ (B̃IndG̃
G X)) = HomB̃ (M̃ , B̃IndG (τ X))(Lemma 2.4)∼
= HomkG̃ (M̃ , IndG̃
G (τ X))(M̃ is the B̃-module)∼= HomkG (ResG̃G M̃ , τ X)

(Proposition 2.1 (4))∼= HomB (ResG̃G M̃ , τ X)

(ResG̃
G M̃ and τ X are B-modules)

= 0.
G̃
∼
HomB̃ (P̃ , B̃IndG̃
G X) = HomkG̃ (P̃ , IndG X)(P̃ is a B̃-module)∼= HomkG (ResG̃G P̃ , X)(Proposition 2.1 (4))∼= HomB (ResG̃G P̃ , X)(ResG̃
G P̃ and X are B-modules)= 0.
By these three isomorphisms and the fact that (M̃ , P̃ ) is a support τ -tilting pair for B̃, applying
Proposition 3.3, we have that B̃IndG̃G X ∈ add M̃ . Also, since the block B̃ covers B, the B-module X is aG̃G̃
direct summand of ResG B̃IndG X by Proposition 4.3 (1). Therefore, we have that X ∈ add ResG̃G M̃ .
We wish to restate the assumption of Theorem 5.1 with some equivalent conditions. First, we givethe lemmas which can be applied in case of rigid B̃-modules not only support τ -tilting B̃-modules. The
following lemma is a block version of Lemma 3.6, which is helpful to prove Theorem 5.4.
Lemma 5.2. Let G̃ be a ﬁnite group, B̃ a block of k G̃, M̃ a rigid B̃-module and L a k G̃-module. If it
holds that B̃(S ⊗k M̃ ) ∈ add M̃ for any composition factor S of L, then the following isomorphism as
B̃-modules holds:
M
B̃(L ⊗k M̃ ) ∼B̃(S ⊗k M̃ ),=S

where S runs over all composition factors of L.
Proof. A similar proof of Lemma 3.6 works in this setting. Let L be an arbitrarily k G̃-module and M̃ arigid B̃-module satisfying that
B̃(S ⊗k M̃ ) ∈ add M̃ for any composition factors S of L.

16

(5.1)

We use induction on the composition length ℓ(L) of L. If ℓ(L) = 1, there is nothing to prove. Hence, we
assume that ℓ(L) ≥ 2 and that the statement for any k G̃-module L′ satisfying ℓ(L′ ) < ℓ(L) is true. Let
T be a simple submodule of L. We get the exact sequence
0B̃(T ⊗k M̃ )B̃(L ⊗k M̃ )B̃((L/T ) ⊗k M̃ )

0

(5.2)

obtained by applying the exact functor B̃(− ⊗k M̃ ) to the exact sequence
0

T

L

L/T

0.

By the rigidity of M̃ , the assumption (5.1) and the assumption of the induction, the sequence (5.2) splits,
and we get that
M
M
B̃(L ⊗k M̃ ) ∼B̃(S ′ ⊗k M̃ ) ∼B̃(S ⊗k M̃ ),= B̃(T ⊗k M̃ ) ⊕ B̃((L/T ) ⊗k M̃ ) ∼= B̃(T ⊗k M̃ ) ⊕=S′

S

where S ′ and S run over all composition factors of L/T and L, respectively.
Lemma 5.3. Let G be a normal subgroup of a ﬁnite group G̃, B̃ a block of k G̃ and M̃ a rigid B̃-module.
Then the following conditions are equivalent:G̃(1) B̃IndG̃
G ResG M̃ ∈ add M̃ and M̃ is relatively G-projective.

(2) B̃(S ⊗k M̃ ) ∈ add M̃ for each simple k(G̃/G)-module S.
Proof. By Proposition 2.1, we have that
G̃
G̃
G̃
G̃
∼
∼
∼
B̃IndG̃
G ResG M̃ = B̃IndG (kG ⊗k ResG M̃ ) = B̃((IndG kG ) ⊗k M̃ ) = B̃(k(G̃/G) ⊗k M̃ ).
G̃(1) ⇒ (2). By the assumptions, we have that B̃IndG̃
G ResG M̃ =add M̃ . Hence, by Proposition 2.1 we getthatG̃
B̃(S ⊗k M̃ ) =add B̃(S ⊗k B̃IndG̃
G ResG M̃ )G̃≤add B̃(S ⊗k IndG̃
G ResG M̃ )G̃G̃∼
= B̃IndG̃
G (ResG S ⊗k ResG M̃ )⊕ dimk S∼⊗k ResG̃= B̃IndG̃
G (kGG M̃ )
G̃
=add B̃IndG̃
G ResG M̃

=add M̃ ,for any simple k(G̃/G)-module S, which implies that B̃(S ⊗k M̃ ) ∈ add M̃ .
(2) ⇒ (1). By Lemma 5.2, we have that
M
G̃∼∼B̃IndG̃
B̃(S ⊗k M̃ ),G ResG M̃ = B̃(k(G̃/G) ⊗k M̃ ) =S

where S runs over all composition factors of the k G̃-module k(G̃/G). Therefore, the assumption impliesG̃
that B̃IndG̃
G ResG M̃ ∈ add M̃ . Moreover, since the trivial k G̃-module kG̃ appears as a composition factorG̃of k(G̃/G), we have that the B̃-module M̃ appears as a direct summand of B̃IndG̃
G ResG M̃ , that is M̃ isa relatively G-projective k G̃-module.
We obtain below some conditions that are equivalent to the assumption of Theorem 5.1.
Theorem 5.4. Let G be a normal subgroup of a ﬁnite group, B a block of kG satisfying IG̃ (B) = G̃and B̃ a block of k G̃ covering B. Let M̃ be a support τ -tilting B̃-module. Then the following conditions
are equivalent:17(1) M̃ =add B̃IndG̃
G M for some G̃-invariant support τ -tilting B-module M .
G̃(2) B̃IndG̃
G ResG M̃ ∈ add M̃ and M̃ is relatively G-projective.

(3) B̃(S ⊗k M̃ ) ∈ add M̃ for each simple k(G̃/G)-module S.
Proof. (1) ⇒ (2). Assume that M̃ =add B̃IndG̃
G M for some G̃-invariant support τ -tilting B-module M .
Then clearly M̃ is a relatively G-projective B̃-module, and we get that
G̃
G̃
G̃
G̃
B̃IndG̃
G ResG M̃ =add B̃IndG ResG B̃IndG M
G̃G̃≤add B̃IndG̃
G ResG IndG M
M
∼
g̃M )
= B̃IndG̃G(g̃∈[G̃/G]

M∼=

B̃IndG̃
GM

g̃∈[G̃/G]=add M̃ .
G̃
Hence, we get B̃IndG̃
G ResG M̃ ∈ add M̃ .
G̃G̃(2) ⇒ (1). Assume that B̃IndG̃
G ResG M̃ ∈ add M̃ and that M̃ is relatively G-projective. Put M := ResG M̃ .
Then by Lemma 2.5, Proposition 4.1 (4), Remark 4.2 and Theorem 5.1, M is a G̃-invariant support τ G̃tilting B-module. We show that B̃IndG̃
G M =add M̃ , that is add(B̃IndG M ) = add M̃ . By the assumptionG̃
G̃
G̃that B̃IndG̃
G M = B̃IndG ResG M̃ ∈ add M̃ , we have that add(B̃IndG M ) ⊂ add M̃ . On the other hand,G̃G̃
since M̃ is relatively G-projective, M̃ is a direct summand of IndG̃
G ResG M̃ = IndG M . Moreover, sinceG̃M̃ lies in B̃, M̃ is a direct summand of B̃IndG M . Hence, we have add M̃ ⊂ add(B̃IndG̃
G M ).
(2) ⇔ (3) Since support τ -tilting B̃-modules are rigid B̃-modules, the equivalence follows from Lemma 5.3.

Corollary 5.5. Let G be a normal subgroup of a ﬁnite group G̃, B a block of kG satisfying IG̃ (B) = G̃and B̃ a block of k G̃ covering B. We denote by (sτ -tilt B)G̃ the subset of sτ -tilt B consisting of G̃invariant support τ -tilting B-modules and by (sτ -tilt B̃)⋆⋆ the subset of sτ -tilt B̃ consisting of support
τ -tilting B̃-modules satisfying the equivalent conditions of Theorem 5.4. Then the functor B̃IndG̃G inducesa poset isomorphism(sτ -tilt B)G̃∼(sτ -tilt B̃)⋆⋆(5.3)

B̃IndG̃G M.

M

In particular, the functor B̃IndG̃G induces the poset monomorphism(sτ -tilt B)G̃sτ -tilt B̃

MB̃IndG̃
G M.

(5.4)

Proof. By [16, Theorem 3.3], the map (5.4) is well-deﬁned. Moreover, by the exactness of the functor
G̃
G̃
B̃IndG̃
G , if N ≤ M in sτ -tilt B then B̃IndG N ≤ B̃IndG M in sτ -tilt B̃. Therefore, the map (5.4) is a poset
homomorphism. By the exactness of the functor ResG̃
G , Lemma 2.5 and Theorem 5.1, the map(sτ -tilt B̃)⋆⋆(sτ -tilt B)G̃

M̃

ResG̃
G M̃

is well-deﬁned and a poset homomorphism.

18

(5.5)

We show that the map (5.4) restricts to a bijection (5.3). By the deﬁnition of (sτ -tilt k G̃)⋆⋆ and theabove argument, the map (5.3) is well-deﬁned and a poset homomorphism. For any relatively G-projectiveG̃
support τ -tilting B̃-module M̃ with B̃IndG̃
G ResG M̃ ∈ add M̃ , we have thatG̃M̃ =add B̃IndG̃
G ResG M̃ .

Moreover, for any G̃-invariant support τ -tilting B-module M , we have that
M
G̃
G̃
G̃
∼
M ≤add ResG̃
g̃M =add M
G B̃IndG M ≤add ResG IndG M =g̃∈[G̃/G]by Proposition 4.3 (1), Proposition 2.2 and the G̃-invariance of M . Therefore, we have that M =add
G̃ResG̃G B̃IndG M . Therefore, the maps (5.3) and (5.5) are inverse to each other.
The latter assertion immediately follows from the fact that the map (5.4) is the composition of theposet isomorphism (5.3) and the inclusion map (sτ -tilt B̃)⋆⋆sτ -tilt B̃.

The following is a block version of Theorem 3.4.
Theorem 5.6. Let G be a normal subgroup of a ﬁnite group G̃, B a block of kG, B̃ a block of k G̃covering B, β the block of kIG̃ (B) satisfyingX

x1β x−1 = 1B̃

x∈[G̃/IG̃ (B)]I (B)and M̃ a support τ -tilting B̃-module. If it holds that βIndGG̃

I (B)ResGG̃

G̃
βResG̃
IG̃ (B) M̃ ∈ add βResIG̃ (B) M̃

I (B)

G̃
and βResG̃
βResG̃
IG̃ (B) M̃ is relatively G-projective, then we have that ResG
IG̃ (B) M̃ is a support τ -tilting
B-module. Moreover, if (M̃ , P̃ ) is a support τ -tilting pair for B̃ corresponding to M̃ , then the pair

I (B)(ResGG̃I (B)G̃βResG̃
IG̃ (B) M̃ , ResG

I (B)is a support τ -tilting pair for B corresponding to ResGG̃

βResG̃
IG̃ (B) P̃ )βResG̃
IG̃ (B) M̃ .

Proof. Since the functor
βResG̃
IG̃ (B) : B̃-mod

β-mod

is a Morita equivalence by Proposition 4.6, the module βResG̃
IG̃ (B) M̃ is a support τ -tilting β-module andG̃
(βResG̃
IG̃ (B) M̃ , βResIG̃ (B) P̃ )is a corresponding support τ -tilting pair for β. Hence, by Theorem 5.1 it immediately follows the consequence.
The following is a block version of Theorem 3.8.
Theorem 5.7. Let G be a normal subgroup of a ﬁnite group G̃, B a block of kG, B̃ a block of k G̃covering B, β the block of kIG̃ (B) satisfyingX

x1β x−1 = 1B̃

x∈[G̃/IG̃ (B)]and M̃ a support τ -tilting B̃-module. Then the following conditions are equivalent:(1) M̃ =add B̃IndG̃
G M for some IG̃ (B)-invariant support τ -tilting B-module M .
I (B)(2) βIndGG̃

I (B)ResGG̃G̃
G̃
βResG̃
IG̃ (B) M̃ ∈ add βResIG̃ (B) M̃ and βResIG̃ (B) M̃ is relatively G-projective.

19G̃(3) β(S ⊗k βResG̃
IG̃ (B) M̃ ) ∈ add βResIG̃ (B) M̃ for each simple k(IG̃ (B)/G)-module S.

Proof. We remark that the module βResG̃
IG̃ (B) M̃ is a support τ -tilting β-module since the functor
βResG̃
IG̃ (B) : B̃-mod

β-mod

(5.6)

is a Morita equivalence by Proposition 4.6.
(1) ⇒ (2). Assume that M̃ =add B̃IndG̃
G M for some IG̃ (B)-invariant support τ -tilting B-module M . By
IG̃ (B)
M is a support τ -tilting β-module. Since the functor[16, Theorem 3.3], the module βIndG
IndG̃
IG̃ (B) : β-mod(5.7)

B̃-modG̃
∼is a Morita equivalence with the inverse functor (5.6). we have IndG̃
IG̃ (B) βResIG̃ (B) M̃ = M̃ . Also, byI(B)the assumption and Proposition 4.7, we get that M̃ =add B̃IndG̃ M ∼βInd G̃ M . Therefore,= IndG̃

G
IG̃ (B)GIG̃ (B)G̃G̃
G̃
we have that IndIG̃ (B) βResIG̃ (B) M̃ =add IndIG̃ (B) βIndG
M . Hence, by the fact that the functor (5.7)
I (B)G̃
is a Morita equivalence again, we have that βResIG̃ (B) M̃ =add βIndGG̃ M . Therefore, we get theconsequence (2) by the equivalence of Theorem 5.4. (1) and (2).
(2) ⇒ (1). Since βResG̃
IG̃ (B) M̃ is a support τ -tilting β-module, there exists an IG̃ (B)-invariant support
I (B)

G̃
τ -tilting B-module M such that βResG̃
IG̃ (B) M̃ =add βIndG
Therefore, by Proposition 4.7, we get that

M by the assumptions and Theorem 5.4.

G̃
G̃
G̃
M̃ ∼
= IndG̃
IG̃ (B) βResIG̃ (B) M̃ =add IndIG̃ (B) βIndG

I (B)

M∼
= B̃IndG̃
G M.

(2) ⇔ (3). Since the support τ -tilting β-module βResG̃
IG̃ (B) M̃ is the rigid β-module, the equivalence
follows from Lemma 5.3.
Corollary 5.8. Let (sτ -tilt B)IG̃ (B) be the subset of sτ -tilt B consisting of IG̃ (B)-invariant support
τ -tilting B-modules and (sτ -tilt B̃)⋆⋆⋆ the subset of sτ -tilt B̃ consisting of support τ -tilting B̃-modules
satisfying the equivalent conditions of Theorem 5.7. Then the functor B̃IndG̃G induces a poset isomorphism(sτ -tilt B)IG̃ (B)(sτ -tilt B̃)⋆⋆⋆M

B̃IndG̃
G M.

(5.8)

In particular, the functor B̃IndG̃G induces the poset monomorphism(sτ -tilt B)IG̃ (B)sτ -tilt B̃

MB̃IndG̃G M.

Proof. Let (sτ -tilt β)⋆⋆ be the subset of sτ -tilt β consisting of support τ -tilting β-modules satisfying theequivalent conditions of Theorem 5.4. Since the functor
IndG̃
IG̃ (B) : β-modB̃-modis a Morita equivalence, we have poset isomorphismssτ -tilt βsτ -tilt B̃

M

IndG̃
IG̃ (B) M

20and(sτ -tilt β)⋆⋆(sτ -tilt B̃)⋆⋆⋆

M

IndG̃
IG̃ (B) M.

(5.9)

By Theorem 5.4, we get the poset isomorphism(sτ -tilt B)IG̃ (B)(sτ -tilt β)⋆⋆
I (B)

M

βIndGG̃

(5.10)M.

By Proposition 4.7, the map (5.8) is the composition of the poset isomorphisms (5.10) and (5.9). Hence,
we complete the proof.
As an application of Corollary 5.8, we consider the case that IG̃ (B)/G is a p-group. The following
theorem is a signiﬁcant generalization of [17, Theorem 1.2] and [10, Theorem 15].
Theorem 5.9. Let G be a normal subgroup of a ﬁnite group G̃, B a block of kG and B̃ a block of k G̃covering B. If the quotient group IG̃ (B)/G is a p-group, then the functor IndG̃G induces an isomorphism
as posets between (sτ -tilt B)IG̃ (B) and sτ -tilt B̃, where (sτ -tilt B)IG̃ (B) is the subset of sτ -tilt B consistingof IG̃ (B)-invariant support τ -tilting B-modules.
Proof. This immediately follows from Corollary 4.5, Theorem 5.7 (3), Corollary 5.8 and the fact that thetrivial k(IG̃ (B)/G)-module is a unique simple k(IG̃ (B)/G)-module.

Acknowledgements
The authors would like to thank the referee for his or her helpful comments.

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Ryotaro KOSHIO
Department of Mathematics, Tokyo University of Science
1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan
E-mail: 1120702@alumni.tus.ac.jp
Yuta KOZAKAI
Department of Mathematics, Tokyo University of Science
1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan
E-mail: kozakai@rs.tus.ac.jp22