τ -tilting finiteness of two-point algebras
Qi Wang
Osaka University

May 24, 2019

Outline
Introduction
Motivations
Preliminaries
Basics of τ -tilting theory and silting theory
Reduction theorems
Main Results
Applications
References

Auslander-Reiten translation
Let Λ be a finite dimensional basic algebra over an algebraicallyclosed field K . For a Λ-module M with a minimal projective
presentation
dd10
P1 −→
P0 −→
M −→ 0,

we have
TrM := coker HomΛ (d1 , Λ).
The Auslander-Reiten translation is defined byτ M := DTrM,
where D = HomK (−, K ).Introduction

τ -tilting theory was introduced by Adachi-Iyama-Reiten as ageneralization of the classical tilting theory via mutations.
They introduced support τ -tilting modules for finite dimensional
algebras and constructed the (left or right) mutation of support
τ -tilting modules, which has the following nice properties:
I Mutation (left or right) is always possible.
I There is a partial order on the set of (isomorphism classes of)basic support τ -tilting modules such that its Hasse quiver
realizes the support τ -tilting mutations.

Motivations
We call an algebra Λ τ -tilting finite if there are only finitely many(isomorphism classes of) basic support τ -tilting Λ-modules.
Recently, many scholars have studied the support τ -tilting modulesof various algebras. For example,
I Gentle algebras (Plamondon, 2018).
I Tilted and cluster-tilted algebras (Zito, 2019).
I Algebras with radical square zero (Adachi, 2016).
I Brauer graph algebras (Adachi-Aihara-Chan, 2018).
I Preprojective algebras of Dynkin type (Aihara-Mizuno, 2016,Mizuno, 2014).

Note that local algebras are τ -tilting finite, and any idempotent
truncation of a τ -tilting finite algebra is also τ -tilting finite. Thus,
considering τ -tilting finiteness of two-point algebras is fundamental
for determining τ -tilting finiteness of general algebras.
Furthermore, we have to consider quivers that contain loops.
Because Gabriel quivers have loops in general and most of theprevious cases are dealing with algebras which do not have loops.

Basics of τ -tilting theory
We denote by |M| the number of (isomorphism classes of)indecomposable direct summands of M.

Definition 1.1 (Adachi-Iyama-Reiten, 2014)
Let M be a right Λ-module and P ∈ proj Λ.
(1) M is called τ -rigid if HomΛ (M, τ M) = 0.
(2) M is called τ -tilting if M is τ -rigid and |M| = |Λ|.
(3) M is called support τ -tilting if M is a τ -tilting (Λ/ hei)-module,
where e is an idempotent of Λ.
(4) (M, P) is called a support τ -tilting pair if M is τ -rigid,
HomΛ (P, M) = 0 and |M| + |P| = |Λ|.
We denote by τ -rigid Λ (respectively, sτ -tilt Λ) the set of(isomorphism classes of) indecomposable τ -rigid (respectively, basicsupport τ -tilting) Λ-modules.

ExampleLet Λ = K ( 1 oa/2 )/{ab = ba = 0} and we denoteb

S1 = 1, S2 = 2, P1 = 12, P2 = 21.
Then,τ (S1 ) = S2 , τ (S2 ) = S1 , τ (P1 ) = 0, τ (P2 ) = 0.
Thus,
I S1 , S2 , P1 and P2 are τ -rigid modules.
I P1 ⊕ P2 , P1 ⊕ S1 and S2 ⊕ P2 are τ -tilting modules.
I S1 and S2 are also support τ -tilting modules.
I (S1 , P2 ) and (S2 , P1 ) are support τ -tilting pairs.

Mutation
We denote by add(M) (respectively, Fac(M)) the full subcategory
whose objects are direct summands (respectively, factor modules)of finite direct sums of copies of M.

Definition 1.2 (Adachi-Iyama-Reiten, 2014)
Let T = M ⊕ N be a basic τ -tilting module, where M ∈/ Fac(N) is
an indecomposable summand. We take an exact sequence with aminimal left add(N)-approximation π:π

M −→ N 0 −→ U −→ 0,we call µ−
M (T ) := U ⊕ N the left mutation of T with respect to M.

Remark
π is called a minimal left add(N)-approximation if N 0 ∈ add(N) andit satisfies the following conditions:(i) every h : N 0 → N 0 that satisfies h ◦ π = π is an automorphism.
M

π/ N0π# h ' id

N0(ii) for any N 00 ∈ add(N) and g : M → N 00 , there exists
f : N 0 → N 00 such that f ◦ π = g .
M

π/ N0∀g# ∃fN 00

ExampleLet Λ = K ( 1 o/ab2 )/{ab = ba = 0}, then P1 ⊕ P2 is a τ -tilting

module. We consider the left mutation with respect to P2 ,πP2 −→ P1 −→ cokerπ −→ 0,a·where π : eb2 −→ ea1 is a minimal left add(P1 )-approximation,
cokerπ = S1 and µ−
P2 (Λ) = P1 ⊕ S1 .
In fact, we have the following mutation quiver of sτ -tilt Λ.
P1 ⊕ P2

S2 ⊕ P2/ P1 ⊕ S1/ S1/ S2/0Proposition 1.3 (Adachi-Iyama-Reiten, 2014)
If the mutation quiver Q(sτ -tilt Λ) contains a finite connectedcomponent, then it exhausts all support τ -tilting modules.

Proposition 1.4 (Adachi-Iyama-Reiten, 2014)
Any τ -rigid module is a direct summand of some τ -tilting modules.

Poset structure
Definition 1.5 (Adachi-Iyama-Reiten, 2014)
For M, N ∈ sτ -tilt Λ, we say M > N if Fac(N) ⊆ Fac(M).

ExampleLet Λ be the algebra given before. The Hasse quiver of sτ -tilt Λ isP1 ⊕ P2>/ P 1 ⊕ S1>/ S1>

S2 ⊕ P2>
>

/ S2>

/0

Proposition 1.6 (Adachi-Iyama-Reiten, 2014)
The mutation quiver Q(sτ -tilt Λ) and the Hasse quiver H(sτ -tilt Λ)coincide.

Basics of silting theory
Let Kb (proj Λ) be the homotopy category of bounded complexes ofprojective Λ-modules. For T ∈ Kb (proj Λ), let thick T be thesmallest full triangulated subcategory containing T , which is closed
under isomorphic classes, cones, [±1] and direct summands.

Definition 1.7 (Aihara-Iyama, 2012)
If thick T = Kb (proj Λ) andHomKb (proj Λ) (T , T [i]) = 0
for any i > 0, then we call T a silting complex.
A complex T in Kb (proj Λ) is called two-term if it is concentrated
in degree 0 and −1. We denote by 2-silt Λ the set of (isomorphismclasses of) basic two-term silting complexes in K b (proj Λ).

Proposition 1.8 (Adachi-Iyama-Reiten, 2014)
There exists a poset isomorphism between sτ -tilt Λ and 2-silt Λ.
More precisely, the bijection is given bysτ -tilt Λ / 2-silt ΛM[f ,0]/ (P1 ⊕ P −→P0 )where (M, P) is a support τ -tilting pair and f : P1 → P0 is aminimal projective presentation of M.

By the previous proposition, we have the poset structure on 2-silt Λin the following example.

Example/aLet Λ = K ( 1 o2 )/{ab = ba = 0}. The mutation quiverbQ(2-silt Λ) and the Hasse quiver H(2-silt Λ) also coincide.
0−→P1
⊕
0−→P2





0−→P1
⊕
aP2 −→P1



b



P1 −→P2⊕
P1 −→0b

P1 −→P2⊕0−→P2

P2 −→0⊕aP2 −→P1

P1 −→0⊕P2 −→0Reduction theorems
Recall that M is called a brick if EndΛ (M) = K and let brick Λ bethe set of (isomorphism classes of) indecomposable bricks inmod Λ. We call Λ2 a factor algebra of Λ1 if there is a surjective
K -algebra homomorphism φ : Λ1 −→ Λ2 .

Lemma 1.9 (Demonet-Iyama-Jasso, 2015)
There is a bijection between τ -rigid Λ and brick Λ, which is given by
M → M/radB M,
where M is a τ -rigid module and B = EndΛ M.

Corollary 1.10 (W, 2019)
Let Λ2 is a factor algebra of Λ1 . If Λ2 is τ -tilting infinite, so is Λ1 .Lemma 1.11 (Adachi-Iyama-Reiten, 2014)
There exists a poset isomorphism between sτ -tilt Λ and sτ -tilt Λop .

Lemma 1.12 (Eisele-Janssens-Raedschelders, 2018)
Let I be a two-sided ideal generated by elements which arecontained in the center and the radical, then there exists a poset
isomorphism between sτ -tilt Λ and sτ -tilt (Λ/I ).

Lemma 1.13 (Plamondon, 2018)
Any gentle algebra satisfies
τ -tilting finite ⇔ representation-finite.

Main Results

Definition 2.1Let Λ be a τ -tilting finite algebra. The Hasse quiver H(sτ -tilt Λ) isof type Hm,n if it is of the form

Λ

6 41/ 42/ ...

/ 4m(/ 2/ ...

/ n1Note that Hm,n ' Hn,m .

)

5 0.

Theorem 2.2 (W, 2019)
Let Γi be an algebra from the Table Γ.
(1) For i = 1, 3, 4, 5, 6, 7, Γi are minimal τ -tilting infinite.
(2) For the remaining i = 2, 8, 9, . . . , 20, Γi are τ -tilting finite.
Moreover, the number of support τ -tilting modules of Γi andthe type of H(sτ -tilt Γi ) are as follows.
Γi

Γ2

Γ8

Γ9

Γ10

Γ11

Γ12

Γ13

#sτ -tilt Γi87981298

type

H1,5

H2,3

H2,5

H3,3

H5,5

H2,5H3,3

Γi

Γ14

Γ15

Γ16

Γ17

Γ18

Γ19

Γ20

#sτ -tilt Γi

812106

type

H3,3

H5,5

H4,4

H2,2

8
H3,3

H2,4

Sketch of proof(1) Γ1 : We combine the τ -rigid-brick corresponding and the fact:
Mλ = •

λ1//• is a brick.

(2) Γ4 : This is a (infinite-)tame gentle algebra.
(3) Others: We compute the Hasse quiver H(sτ -tilt Γi ) orH(2-silt Γi ) directly.

Theorem 2.3 (W, 2019)
Let Λ be a two-point algebras with quiver QΛ .
(1) If/•,QΛ = •

then Λ is τ -tilting infinite if and only if it has Γ3 as a factoralgebra.
(2) IfQΛ = •

/•,then Λ is τ -tilting infinite if and only if it has one ofΓ3 , Γ4 , Γ5 , Γ6 and their opposite algebras as a factor algebra.

Sketch of proof

We give an example to explain the main method. Let
Λ = K( 1a/2eb )/I ,where I is an admissible ideal such that Λ is a finite dimensionalalgebra. Then I is either < bn > or < bn , abm > with n > m. Notethat the latter can be reduced to the former by Lemma 1.12.

Application 1

Representation type of two-point algebras has already completelydetermined for many years.

Proposition 3.1 (Han, 2002)
Let A be a two-point algebra. Up to isomorphism and duality, A isrepresentation-finite or (infinite-)tame if and only if A degenerates
to a factor algebra of an algebra from Table T, and A is wild if andonly if A has a wild algebra from Table W as a factor algebra.

Theorem 3.2 (W, 2019)
Let Ti and Wi be algebras from Table T and Table W.
(1) T1 , T3 , T17 , W1 , W2 , W3 and W5 are τ -tilting infinite.
(2) Others are τ -tilting finite. Moreover, #sτ -tilt Ti , #sτ -tilt Wi ,the type of H(sτ -tilt Ti ) and the type of H(sτ -tilt Wi ) are asfollows.
T2

T4

T5

T6

6

5

6

H1,3

H1,2

H1,3

T7

T8

T9

T10

T1158128

H1,2H3,3H5,5H3,3T12

T13

T14

T15

T16

T18

T19

T20

T21

768798676H2,3H2,2H3,3H2,3H2,5H3,3H2,2H2,3H2,2W4W6W756867510

H1,2H1,3

H1,5H1,3H1,4H1,2H3,5W15

W16

W17W18989

8H2,5H3,3H2,5H3,3W25

W26

W27

W28

7H2,3W8W9

W19

W29

W10

W20

H3,3W137810H2,3H3,3H3,5W33

W34

W31

W22

W14W24

W30

W21W12W23

8H2,4W11W32
6H2,4H2,2Application 2Let Λ be a block algebra of Hecke algebras of classical type over analgebraically closed field k of odd characteristic.

Proposition 3.3 (Ariki, 2018)
If Λ is (infinite-)tame, then it is Morita equivalent to one of thebounded quiver algebras Hi = kQi /Ii below.
%

H1 : Q1 = α%•oH2 : Q2 = α%•oH3 : Q3 = α

β , I1 =< α2 , β 2 , αβ, βα >.

•eµ/• , I2 =< αµ, να, α2 − (µν)2 >.

νµ/ν•eβI3 =< αµ, µβ, βν, να, α2 − µν, β 2 − νµ >.
H4 : Q4 = α%

•oµν

/•eβ

I4 =< αµ, µβ, βν, να, α2 − (µν)2 , β 2 − (νµ)2 >.

Theorem 3.4 (W, 2019)
Let Λ be a (infinite-)tame block algebra, then Λ is τ -tilting finite.
Moreover, we haveHi

H1#sτ -tilt Hi

2H2

H36H4

References[1] T. Adachi, Characterizing τ -tilting finite algebras with radicalsquare zero. Proc. Amer. Math. Soc. 144 (2016), no. 11,4673–4685.
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tilting complexes over Brauer graph algebras. Math. Z. 290(2018), no. 1–2, 1–36.
[3] T. Adachi, O. Iyama and I. Reiten, τ -tilting theory. Compos.
Math. 150 (2014), no. 3, 415–452.
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categories. J. Lond. Math. Soc. (2) 85 (2012), no. 3, 633–668.
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11 (2017), no. 6, 1287–1315.
[6] S. Ariki, Tame block algebras of Hecke algebras of classicaltype. Preprint (2018), arXiv, 1805.08341.

[7] L. Demonet, O. Iyama and G. Jasso, τ -tilting finite algebras,bricks and g -vectors. Preprint (2015), arXiv: 1503.00285.
[8] F. Eisele, G. Janssens and T. Raedschelders, A reduction
theorem for τ -rigid modules. Math. Z. 290 (2018), no. 3–4,1377–1413.
[9] Y. Han, Wild two-point algebras. J. Algebra 247 (2002), no.
1, 55–77.
[10] P. G. Plamondon, τ -tilting finite gentle algebras arerepresentation-finite. Preprint (2018), arXiv: 1809.06313.
[11] Q. Wang, τ -tilting finiteness of two-point algebras I. Preprint(2019), arXiv:1902.03737.
[12] S. Zito, τ -tilting finite tilted and cluster-tilted algebras.
Preprint (2019), arXiv: 1902.05866.

Thank you very much for your attention !