QI WANG Abstract. As the first attempt to classify τ-tilting finite two-point algebras, we have determined the τ-tilting finiteness for minimal wild two-point algebras and some tame two-point algebras.
1. I NTRODUCTION
In this paper, we always assume that Λ is a finite-dimensional basic al- gebra over an algebraically closed field K . In particular, the representation type of Λ is divided into representation-finite, (infinite-)tame and wild. 7-tilting theory is introduced by Adachi, Iyama and Reiten [7], in which they constructed support τ -tilting modules as a generalization of the clas- sical tilting modules. We recall that a right A-module M is called sup- port τ-tilting if Hom Λ M ( M, τ M ) = 0 and | M | = | Λ M | taking over Λ M := Λ/Λ(1 – e)Λ, where e is an idempotent of Λ such that the simple summands of eΛ/(erad Λ) are exactly the simple composition factors of M. Moreover, a support τ -tilting Λ -module M is called τ -tilting if Λ M = Λ . This wider class bijectively corresponds to the class of two-term silting complexes, func- torially finite torsion classes, left finite semibricks and so on. We refer to [7] and [2] for details. We are interested in τ-tilting finite algebras studied in [13], that is, al- gebras with only finitely many pairwise non-isomorphic basic (support) τ- tilting modules. It is obvious that a representation-finite algebra is τ -tilting finite. Also, it is not difficult to find a tame or a wild algebra which is 7-tilting finite. The 7-tilting finiteness for several classes of algebras has been determined, such as algebras with radical square zero [1], preprojec- tive algebras of Dynkin type [19], Brauer graph algebras [3], biserial algebras [20] and classical Schur algebras [25]. In particular, it has been proved in some cases that τ -tilting finiteness coincides with representation-finiteness, including gentle algebras [22], cycle-finite algebras [21], tilted and cluster- tilted algebras [27], simply connected algebras [24], quasi-tilted algebras, locally hereditary algebras, etc. [4]. We notice that local algebras, i.e., algebras with only one simple module, are always τ -tilting finite. This motivates us to study the algebras with exactly two simple modules (up to isomorphism), which are called two-point algebras in this paper. We point out that Aihara-Kase [6] and Kase [18] have
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got some interesting results. For example, Kase [18, Theorem 6.1] showed that one can always find a τ -tilting finite two-point algebra such that the Hasse quiver of the poset of pairwise non-isomorphic basic support τ -tilting modules, is isomorphic to a t-gon ( t ≥ 4). Besides, it is well-known that Kronecker algebra K( ● ￣ ↑ ● ) is τ-tilting infinite (we present a proof in Lemma 3.2 for the convenience of readers). Thus, A is τ-tilting infinite if the quiver of A contains multiple arrows. It is worth mentioning that two-point algebras are fundamental if we con- sider the representation type of general algebras, and the representation type of two-point algebras has been determined for many years. We may review these results here: the maximal representation-finite two-point algebras are classified by Bongartz and Gabriel [10], the tame two-point algebras are clas- sified by several authors in [11, 12, 15, 16], and the minimal wild two-point algebras are classified by Han [16]. As we mentioned above, τ-tilting finiteness is related to representation type in some classes of algebras. Thus, in order to find a complete classifica- tion of τ-tilting finite two-point algebras, it will be useful to determine the 7-tilting finiteness of all minimal wild two-point algebras. We recall that a complete list of minimal wild two-point algebras is given by Han [16], which is displayed by Table W in his paper. (See also Appendix A of this paper.) Then, the first main result in this paper is presented as follows. Theorem 1.1. Let W i be a minimal wild two-point algebra from Table W.
(1) W 1 , W 2 , W 3 and W 5 are τ-tilting infinite. (2) W 4 and W 6 ～ W 34 are τ-tilting finite. Moreover, we have τ-tilting W i -modules and the type of H (sτ-tilt W i ) is defined in Definition 4. According to Theorem 1.1, most of the minimal wild two-point algebras are τ -tilting finite so that one may expect to give a complete classification of 7-tilting finite tame two-point algebras. This should also be useful toward the complete classification of τ -tilting finite two-point algebras. However, it is difficult at this moment to give a complete result on tame two-point algebras, because the tameness of two-point algebras depends on the tech- nique called degeneration , and it is still open to finding the relation between r-tilting finiteness and degeneration. We may give a partial result on tame two-point algebras. We recall from [16] (see also Proposition 3.1 of this paper) that all tame two-point algebras can degenerate to a finite set (Table T in [16]) of two-point algebras. Then, we check the τ -tilting finiteness for algebras in Table T 1 as follows. This is the second main result in this paper.
We observe that Theorem 1.1 and Theorem 1.2 are useful to determine the τ -tilting finiteness for several other classes of algebras, such as tame two- point distributive algebras [15], two-point symmetric special biserial algebras [8] and so on. We have given an easy observation in Proposition 3.9. This paper is organized as follows. In Section 2, we review some basic concepts of τ -tilting theory and silting theory. Besides, we list some reduc- tion theorems that we will use and carry out several explicit computations. In Section 3, we give the proofs of Theorem 1.1 and Theorem 1.2.
2. P RELIMINARIES
We refer to [9] for the background on the representation theory of finite- dimensional algebras and the basic knowledge of quiver representations. Let mod A be the category of finitely generated right A-modules and proj Λ the full subcategory of mod Λ consisting of projective Λ -modules. For any M ∈ mod Λ , we denote by add( M ) (respectively, Fac( M )) the full subcategory of mod A whose objects are direct summands (respectively, factor modules) of finite direct sums of copies of M . We often describe Λ - modules via their composition series. For example, each simple module S i is written as i, and then 1 = S 1 is an indecomposable A-module M with a unique simple submodule S 2 such that M / S 2 ~ S 1 . We denote by Λ°ץthe opposite algebra of Λ and by | M | the number of isomorphism classes of indecomposable direct summands of M . Let τ be the Auslander-Reiten translation on the module category. Note that it is not functorial. Definition 1 ([7, Definition 0.1]). Let M ∈ mod Λ. Then, (1) M is called τ-rigid if Hom Λ ( 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 there exists an idempotent e of Λ such that M is a τ-tilting (Λ/ΛεΛ)-module. Let (M, P) be a pair with M E mod A and P E proj A. We recall that ( M, P ) is said to be a support τ -tilting pair if M is τ -rigid, Hom Λ ( P, M ) = 0 and | M | + | P | = | Λ |. This is actually an equivalent definition for support τ-tilting modules, i.e., ( M, P ) is a support τ-tilting pair if and only if M is a τ-tilting (Λ/ΛeΛ)-module and P = eΛ. We denote by τ-rigid Λ (respectively, sτ-tilt Λ) the set of isomorphism classes of indecomposable τ -rigid (respectively, basic support τ -tilting) Λ - modules.  It is known from [7] that any τ -rigid Λ -module is a direct summand of someｧ-tilting Λ-module. Definition 2. A finite-dimensional algebra Λ is called τ-tilting finite if it has only finitely many pairwise non-isomorphic basic τ -tilting modules. Otherwise, Λ is called τ-tilting infinite. Let C be an additive category and X, Y objects of C. A morphism f : X → Z with Z ∈ add(Y) is called a minimal left add(Y)-approximation of X if it satisfies: ● every h ∈ Homc(Z, Z) that satisfies h ◦ f = f is an automorphism, ● Hom c ( f , Z ′ ) : Hom c ( Z , Z ′ ) — → Hom c ( X , Z ′ ) is surjective for any Z' E add(Y), where add(Y) is the category of all direct summands of finite direct sums of copies of Y . We recall the concept of left mutation which is the core of 7-tilting theory. Definition-Theorem 2.1 ([7, Definition 2.19, Theorem 2.30]). Let T = M ⊕ N be a basic support τ-tilting Λ-module with an indecomposable direct summand M  satisfying M # Fac(N).  We take an exact sequence with a minimal left add( N )-approximation f : M → N' → coker f → 0. We call μ M (T) := (coker f) ⊕ N the left mutation of T with respect to M , which is again a basic support τ-tilting Λ-module. (The right mutation μ H (T) can be defined dually.) In the above, Zhang has shown in [26] that coker f is either 0 or inde- composable. Moreover, one can show that coker f cannot be projective. We may construct a directed graph H(s r -tilt A) by drawing an arrow from T 1 to T 2 if T 2 is a left mutation of T 1 . On the other hand, we can regard sr-tilt A as a poset with respect to a partial order <. For any M,N E sr-tilt A, we say that N < M if Fac(N) C Fac(M). Then, the directed graph H(sr-tilt A) is exactly the Hasse quiver on the poset sr-tilt A, see [7, Corollary 2.34]. The following statement implies that an algebra Λ is τ -tilting finite if we can find a finite connected component of H (s r -tilt A). Proposition 2.2 ([7, Corollary 2.38]). If the Hasse quiver H(sr-tilt A) contains a finite connected component ∆, then H(sτ-tilt Λ) = ∆. 2.1. Silting theory. We denote by C b (proj A) the category of bounded complexes of projective A-modules and by Kb(proj A) the corresponding homotopy category which is triangulated. Besides, we denote by ～h the homotopy equivalence in K b (proj A). For any T ∈ K b (proj A), let thick T be the smallest full subcategory of K b (proj A) containing T , which is closed under cones, [±1], direct summands and isomorphisms. Definition 3 ([5, Definition 2.1]). A complex T ∈ K b (proj Λ) is called presilting if HomKb(proj Λ)(T,T[i]) = 0 for any i > 0. A presilting complex T is called silting if thick T = K b (proj A). Similar to the left mutation of support τ -tilting modules, we recall the irreducible left silting mutation ([5, Definition 2.30]) of silting complexes. Let T = X ⊕ Y be a basic silting complex in K b (proj A) with an indecom- posable summand X . We take a minimal left add( Y )-approximation π and a triangle Then, cone(π) is indecomposable and μ - χ ( T ) := cone(π) ⊕ Y is again a basic silting complex in K b (proj A), see [5, Theorem 2.31]. We call μ χ (T) the irreducible left mutation of T with respect to X . A complex in K b (proj A) is called two-term if it is homotopy equivalent to a complex T concentrated in degree 0 and −1, i.e.,
We denote by 2-silt A the set of isomorphism classes of basic two-term silting complexes in K b (proj Λ). Similarly, there is a partial order ≤ on the set 2-silt A which is introduced by [5, Theorem 2.11]. For any S,T ∈ 2-silt A, we say that S < T if Homkb(proj Λ)(T,S[i]) = 0 for any i > 0. We denote by H (2-silt A) the Hasse quiver of 2-silt A, which is compatible with the irreducible left mutations of silting complexes. Proposition 2.3 ([7, Theorem 3.2]). There exists a poset isomorphism be- tween sτ-tilt Λ and 2-silt Λ. More precisely, the bijection is given by
where (M,P) is the corresponding support τ-tilting pair and P 1 → P 0 → M −→ 0 is a minimal projective presentation of M . Since the poset s -tilt Λ has the unique maximal element Λ and the unique minimal element 0, we can define the type of H (s r-tilt A) (equivalently, H(2-silt Λ)) as follows. Definition 4. Let A be a τ-tilting finite algebra. We say that the Hasse quiver H (s r-tilt Λ) is of type H m,n if it is of the form
このMoreover, we have Hm,n ~ Hn,m· We have the following equivalent condition for Λ to be τ-tilting finite. Proposition 2.4 ([13, Corollary 2.9]). An algebra Λ is τ-tilting finite if and only if one of (equivalently, any of) the sets τ-rigid Λ, sτ-tilt Λ and 2-silt Λ is finite. 2.2. Reduction theorems. There are some reduction theorems. First, we review the brick- τ -rigid correspondence introduced by Demonet, Iyama and Jasso [13]. Recall that M E mod A is called a brick if End Λ ( M ) = K . We denote by brick A the set of isomorphism classes of bricks in mod A. Lemma 2.5 ([13, Theorem 4.2]). Let Λ be a finite-dimensional algebra. Then, A is τ-tilting finite if and only if the set brick A is finite. Let Λ 1 , Λ 2 be two algebras. We call Λ 2 a quotient or quotient algebra of Λ 1 if there exists a surjective K -algebra homomorphism φ : Λ 1 → Λ 2 . Corollary 2.6. Suppose that Λ 2 is a quotient algebra of Λ 1 . If Λ 2 is τ -tilting infinite, then Λ₁ is also τ-tilting infinite. Proof. There exists a K -linear fully-faithful functor F : mod Λ 2 → mod Λ 1 , and F sends a brick in mod Λ 2 to a brick in mod Λ 1 . Then, the statement follows from Lemma 2.5. Lemma 2.7 ([7, Theorem 2.14]). There exists a poset anti-isomorphism between st-tilt Λ and st-tilt Λ°Ρ Lemma 2.8 ([14, Theorem 1]). Let I be a two-sided ideal generated by Then, central elements which are contained in the Jacobson radical of A. there exists a poset isomorphism between sτ-tilt Λ and sτ-tilt ( Λ/Ι ).
Then, φ ◦ ψ = Id τ τ  and
because the difference Id T 1 − ψ ◦ ϕ is null-homotopic as follows.
123 (2) We define ϕ : T 2 → T 2 and ψ : T 2 → T 2 as follows,
Then, φ ◦ ψ = Idτ z  and
In fact, the difference Id T 2 − ψ ◦ ϕ is null-homotopic as follows.
Therefore, we have T 1 ~ h T 1 and T 2 ~ h T 2 . □ 3. M AIN R ESULTS In this section, we will prove our main results mentioned in the intro- duction. But before that, let us review the complete classification for the representation type of two-point algebras.  We refer to Appendix A for Table T and Table W. Proposition 3.1 ([16, Theorem 1, Theorem 2]). Let A be a two-point al- gebra. Up to isomorphism and duality, Λ is representation-finite or tame if and only if Λ degenerates to a quotient algebra of an algebra from Table T, and Λ is wild if and only if Λ has a minimal wild algebra from Table W as a quotient algebra. We denote by rad(A) the Jacobson radical of A and by C (A) the center of A. As explained in [14], although A has a complicated structure, its quotient algebra
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Table  Λ
may have a simpler structure.  Moreover, by Lemma 2.8, we know that #sr-tilt Λ = #sr-tilt Λ and the Hasse quivers H(sr-tilt Λ) and H(sr-tilt Λ) are of the same type. Then, by using this strategy and Corollary 2.6, we can restrict the algebras (except for W 4 , T 20 and T 21 ) in Table W and Table T to a small list (i.e., Table Λ) of two-point algebras. (In Table Λ, by an algebra Λ i , we mean the bound quiver algebra KQ/I i , where I is the admissible ideal generated by the relation ( i ).) As a preparation for proving Theorem 1.1 and Theorem 1.2, we need to determine the τ-tilting finiteness of Λ in Table Λ .  We remark that H(s τ -tilt Λ 1 ) is of type H 1,2 and H(s τ -tilt Λ 6 ) is of type H 2,2, and we omit the details to show this. It is well-known that the Kronecker algebra Λ 2 admits infinitely many (classical) tilting modules, so that it is obviously τ -tilting infinite. Since the construction of tilting modules is not mentioned in this paper, we give a different proof here for the convenience of readers. Similar to the notion of minimal representation-infinite algebra, we call an algebra Λ minimal τ -tilting infinite if Λ is τ -tilting infinite, but any proper quotient algebra of Λ is τ-tilting finite. (11) α² = β² = μν = νμ = αμ = βν = 0; (12) α² = β² = μν = νμ = βν = να = αμβ = 0.
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Lemma 3.2. The Kronecker algebra Λ 2 is minimal τ-tilting infinite. Proof. It is easy to check that
with k ∈ K is a brick in mod A2. Since the family ( Mk ) k ∈ K consists of infinitely many pairwise non-isomorphic bricks, Λ 2 is τ-tilting infinite by Lemma 2.5. Besides, the minimality is obvious. Lemma 3.3. The two-point algebras Λ 3 and Λ 4 are τ-tilting finite. Proof. Since Λ 3 is a quotient algebra of Λ 4 , by Corollary 2.6, it suffices to show that A 4 is 7-tilting finite. We show that the poset 2-silt A 4 has a finite connected component and hence, it exhausts all two-term silting complexes in K b (proj Λ 4 ) by Proposition 2.2 and Proposition 2.3. Then, Λ 4 is τ-tilting finite following from Proposition 2.4. Let P 1 and P 2 be the indecomposable projective Λ 4 -modules. We have
We show that H(2-silt Λ 4 ) is of type H 1,5 as follows,
where
126 Since Hom Λ 4 ( P 2 , P 1 ) = e 1 Λ 4 e 2 = Κ μ ⊕ Κ μ β ⊕ Κ μ β 2 and Hom Λ 4 ( P 1 , P 2 ) = 0, it is not difficult to compute the left mutations μ ī j ( Λ 4 ) and μ ī j j ( Λ 4 ). According to the bijection introduced in Proposition 2.3, one can find the corresponding two-term silting complexes. We only show details for the rest of the steps. (1) Let T 2 = X ⊕ Y := (0 → P 1 ) ⊕ (P 2 → P ⊕ 3 ). Then, μ T 2 ) does not belong to 2-silt A4 and therefore, we ignore this mutation. To compute μ χ ( T 2 ), we take a triangle
We may check that π is a minimal left add( Y )-approximation. In fact, ● if we compose π with the endomorphism
( f ), then k 1 = 1 and k 2 = k 3 = 0. Hence, π is indeed a minimal left add( Y )-approximation. By Lemma 2.9,
Then, π is a minimal left add( X )-approximation. (In fact, we have EndKb(proj  Λ 4 ) = K since End Λ 4 (P 1 ) = K . Furthermore, EndKb(proj ∧ 4 ) (X ⊕ 3 ) ～ Mat(3, 3, K).
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Secondly, λ ০ π = π for λ ∈ Mat(3,3, K ) implies that λ is the identity. Thus, π is indeed a minimal left add( X )-approximation.) By applying Lemma 2.9 twice, we have
Then,
then all elements of Homk b (proj Λ 4 ) (X,Y) are obtained; if
Then,
128 Then, π is a minimal left add( X )-approximation because of
Then, we have
(5) Let T21212 = X ⊕ Y := (P2 − → P1) ⊕ (P2 − → 0). Then, it is clear that μγ (T21212) does not belong to 2-silt Λ 4 and
To sum up the above, we have constructed a finite connected component in H (2-silt A 4 ). We deduce that H (2-silt A 4 ) is of type H 1,5. By Proposition 2.3, this is equivalent to saying that H(sr-tilt A 4 ) is of type H 1,5. Remark. Let P 1 and P 2 be the indecomposable projective Λ 3 -modules. Then,
Lemma 3.4. The two-point algebra Λ 5 is minimal τ-tilting infinite. Proof. Note that Λ 5 is a gentle algebra and it is representation-infinite by Hoshino and Miyachi's result [17, Theorem A]. Besides, Plamondon [22, Theorem 1.1] showed that a gentle algebra is τ -tilting finite if and only if it is representation-finite. Therefore, Λ 5 is τ-tilting infinite. For the minimality, we may consider

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since the socle of Λ 5 is Καμβ ⊕ Κ μβ ⊕ Κ β and any proper quotient Λ 5/Ι of Λ 5 satisfies αμβ ∈ I . We denote by P 1 and P 2 the indecomposable projective Λ5-modules, then
By direct calculation, we find that H(sr-tilt Â5) ~ H(2-silt Â5)) is of type H 1,4 as follows,
□ This implies that Λ 5 is τ -tilting finite and Λ 5 is minimal τ -tilting infinite. Lemma 3.5. The two-point algebras Λ 7 , Λ 8 , Λ 9 and Λ 10 are τ-tilting finite. Proof. Note that Λ 7 , Λ 8 and Λ 9 are quotient algebras of Λ 10 . By Corollary The indecomposable 2.6, it suffices to show that Λ 10 is τ-tilting finite. projective modules of Λ 10 are
Since
and
we know that the computation of the left mutation sequence started at P 1 is similar to that of Λ 4 , and the computation of the left mutation sequence started at P 2 is similar to that of Λ 4 P . By Lemma 2.7 and the calculation in Lemma 3.3, we deduce that the Hasse quiver H (2-silt A10) is as follows, 130 t-TILTING FINITENESS OF TWO-POINT ALGEBRAS
Λ 7, Λ 8, Λ 9 and Λ 10 are τ -tilting finite. Next, we determine the type of the Hasse quiver H(s r -tilt Λ i ) for i = 7,8,9. (1) The indecomposable projective A7-modules are
We give the Hasse quiver H (2-silt Λ 7 ) by direct calculation as follows,
Hence, H (s r-tilt A 7 ) ~ H (2-silt A 7 ) is of type H 2,3. (2) The indecomposable projective A 8 -modules are
and the Hasse quiver H (2-silt Λ 8 ) is given as follows,
Hence, H(s r -tilt A 8 ) ~ H(2-silt A 8 ) is of type H 3,3. (3) Let Q 1 and Q 2 be the indecomposable projective A 9 -modules. Then,
Since Hom Λ 9 (Q 1 , Q 2 ) = e 2 Λ 9 e 1 = K ν and Hom Λ 9 (Q 2 , Q 1 ) = e 1 Λ 9 e 2 = K μ ⊕ K α 2 μ , the computation of the left mutation sequence started at Q 2 is similar to that of Λ α ρ . Then, by Lemma 2.7 and the calculation in Lemma 3.3, we deduce that H (2-silt A 9 ) is presented as follows,
αμ  μ  ρ ), f2 = (αμ μ ) and f3 = (α²μ αμ μ ).  By Proposition where f1 = ■ 2.3, we conclude that H(sr-tilt A9) ~ H(2-silt A9) is of type H2,5. 132
t-tilting finiteness of two-point algebras
Lemma 3.6. The two-point algebras Λ 11 and Λ 12 are τ-tilting finite. Proof. (1) The indecomposable projective Λ 11 -modules are
We calculate the Hasse quiver H (2-silt A 11 ) directly as follows,
Thus, H (s τ -tilt Λ 11 ) ~ H (2-silt Λ 11 ) is of type H 3,3. (2) The indecomposable projective A12-modules are
and the Hasse quiver H (2-silt Λ 12 ) is shown as follows,
Thus, H (s τ -tilt Λ 12 ) ~ H (2-silt Λ 12 ) is of type H 2,4. Lastly, we summarize the number #s -tilt Λ i and the type of H (s -tilt Λ i ) for Λ i with i ≠ 2,5 as follows,
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3.1. The proof of Theorem 1.1. First, one can easily find that W 1 , W 2 , W 3 and W 5 have A 2 as a quotient algebra and therefore, they are 7-tilting infinite. It is also not difficult to find that W 4 is 7-tilting finite and H(st-tilt W 4 ) is of type H 1,2. We may distinguish the case W 14 . Note that ναμ ∈ C ( W 14 ) and the indecomposable projective modules of W14 := W14/ < ναμ > are
Then, we find that W 14 is a quotient algebra of Λ 10 by α 2 μ . Thus, by similar calculation with Λ 10 in the proof of Lemma 3.5, one can check that H(s τ -tilt W 14 ) is of type H 3.5 . By Lemma 2.8, we deduce that H(s τ -tilt W 14 ) is of type H3.5. Next, we show that W 6 ~ W 34 (except for W 14 ) are 7-tilting finite by determining the type of H (s r-tilt W i ) for i = 6, 7, . . . , 34 ( i ≠ 14). In order to do this, we can apply Lemma 2.8 to construct a two-sided ideal I generated by elements in C(W i ) ∩ rad(W i ) such that s τ -tilt A ~ s τ -tilt (W i /I). Then, we can find the type of H (sr-tilt W i ) following Table A. Here, we compute the center of an algebra by GAP as follows, see [28].
In particular, we point out that although Λ 7̸= W 21 ⁄ I , but s τ -tilt Λ 7 ∼ s τ -tilt W 21 .  To see this, one may check that νμ +μν ∈ C ( W 21 ) and therefore, 3.2. The proof of Theorem 1.2. Similar to the proof of Theorem 1.1, one can check that T 1 has Λ 2 as a quotient algebra, T 3 and T 17 have Λ 5 as a quotient algebra. Hence, T 1 , T 3 and T 17 are τ -tilting infinite. We may also apply Lemma 2.8 to construct a two-sided ideal I generated by elements in C(T i ) ∩ rad(T i ) such that s τ -tilt B ~ s τ -tilt (T i /I). We have
Case (T 9 ). Since νμ, αμν + μνα + ναμ ∈ C (T 9 ) and αμν ∈ C (T 9 ) with Tg := Tg/ < νμ, ναμ, αμν + μνα > Then, we have μν ∈ C(T 9 / < αμν >) and therefore, sr-tilt T 9 ~ sr-tilt  (T 9 / < μν,νμ,ναμ >) ~ sr-tilt Λ 8 . Case (T20). For any k ∈ K/{0}, we have μν + νμ ∈ C(T20) such that
Thus, H(s r-tilt T 20 ) ~ H(s r-tilt T 20 ) ~ H(2-silt T 20 ) is of type H 2,3. Case (T 21 ). For any k 1 , k 2 ∈ K /{0}, we have μν + νμ ∈ C (T 21 ). Similarly, we have s τ -tilt T 21 ~ s τ -tilt (T 21 / < μν, νμ >) and the corresponding Hasse quiver is of type H2.2- 3.3. Other applications. At the end of this paper, we give two easy ob- servations. First, we have Proposition 3.7. Let A be a connected two-point algebra without loops. Then, Λ is τ-tilting finite if and only if it is representation-finite. Proof. By our assumption, the quiver Q of A does not contain loops. If Q contains multiple arrows, then Λ has the Kronecker algebra Λ 2 as a quo- tient algebra and hence, A is 7-tilting infinite. Then, we deduce that if A is τ-tilting finite, then Q is either ● →→● or ● →→●. On the other hand, any finite-dimensional algebra with quiver ● ≤ ≥ ● or ● representation-finite from Bongartz and Gabriel [10]. Second, we determine the τ -tilting finiteness of two-point symmetric spe- cial biserial algebras.  We refer to [23] for the basic concepts and properties of symmetric special biserial algebras, or equivalently, Brauer graph algebras. In [8], the authors classified two-point symmetric special biserial algebras up to Morita equivalence, so that we can determine their τ-tilting finiteness. Proposition 3.8 ([8, Theorem 7.1]). Let Λ be a two-point symmetric special biserial algebra. Then, Λ is Morita equivalent to one of Bi = KQ/Ii below.
//1 ,//2 I2：μιν2 = ν2μ1 = μ2ν1 = ν1μ2 = 0 Q : • (μ 1ν 1 )m = (μ 2ν 2 )m , (ν 1μ 1 )m = (ν 2μ 2 )m , m , n ≥ 1 I3 : μ1ν2 = ν1μ1 = μ2ν1 = ν2μ2 = 0, (μινιμενշ) n = (μշνշμινι) n , (νιμενշμι) n = (νշμινιμε) n , n ≥ 1. I d : αμ = να = 0,α m = (μν) n ,m ≥ 2,n ≥ 1. I5 : α² = νμ = 0, (αμν) n = (μνα) n ≥ 1. ) β Q : α{ •ベI 6 : αμ = μβ = βν = να = 0, αm = (μν) n , βr = (νμ) n , m, r ≥ 2, n ≥ 1. I7 : α² = νμ = μβ = βν = 0, (αμν)² = (μνα)²,βm = (ναμ)²,π ≥ 2,π ≥ 1. I 8 : α 2 = β 2 = μν = νμ = 0, (ναμβ) n = (βναμ) n , (αμβν) n = (μβνα) n , n ≥ 1. Proposition 3.9. Let Bi be a two-point symmetric special biserial algebra. Then, Bi is τ-tilting finite if i = 1,4,5,6,7; τ-tilting infinite if i = 2,3,8. Moreover, we have
Proof. One can easily check that B 2 and B 3 have Λ 2 as a quotient algebra, and Bs has A5 as a quotient algebra. Therefore, B2, B3 and Bs are 7-tilting infinite. We show the remaining case by case. Case ( B 1 ). If n = 1, then μν, νμ ∈ C ( B 1 ). If n ≥ 2, then μν + νμ ∈ C ( B 1 ). Both of them satisfy s r-tilt B 1 ~ sr-tilt (B 1 / < μν,νμ >) ~ sr-tilt Λ 6 . Case ( B 4 ). If n = 1, then α, νμ ∈ C ( B 4 ). If n ≥ 2, then α, μν + νμ ∈ C(B 4 ). Both of them satisfy sr-tilt B4 ~ sr-tilt (B4/ < α,μν,νμ >) ~ sr-tilt Λ6. Case (B 5 ). If n = 1, then μν,ναμ ∈ C (B 5 ). If n ≥ 2, then αμν + μνα + ναμ ∈ C ( B 5 ) and αμν ∈ C ( B 5 ) such that μν ∈ C ( B 5 / < αμν >), where B5 := B5/ < ναμ,αμν + μνα >. Hence, s τ -tilt B 5 ~ s τ -tilt (B 5 / < μν,ναμ >) ~ s τ -tilt Λ 8 . Case ( B 6 ). If n = 1, then α,β ∈ C ( B 6 ). If n ≥ 2, then α,β,μν + νμ ∈ C(B 6 ). Both of them satisfy sr-tilt B 6 ~ sr-tilt (B 6 / < α,β,μν,νμ >) ~ sr-tilt Λ 6 . Case (B7). If n = 1, then β, μν ∈ C (B7). If n ≥ 2, then β, αμν + μνα + ναμ ∈ C(B 7 ) and αμν ∈ C(B 7 ) such that μν ∈ C(B 7 ) < αμν >), where B7 := B7 / < β,ναμ,αμν + μνα >. Thus, s τ-tilt B τ ~ s τ-tilt (B τ / < β,μν,ναμ >) ~ s τ-tilt Λ 8 . ■ Appendix A. Table T and Table W introduced in [16] Table T (1) νιμι = ν2μ2 = (ℓ1μ1+ℓ2μ2)(k1ν1+k2ν2) = (ℓ3μ1+ℓ4μ2)(k3ν1+k4ν2) = 0, where k 1 , k 2 , k 3 , k 4 , ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 ∈ K  and k 1 k 4̸= k 2 k 3 , ℓ 1 ℓ 4̸= ℓ 2 ℓ 3 .
μ
Table W
Q. WANG 138 (2) α 6 = α²μ = 0; (9) α² = μνμ = νμν = (ναμ) n = 0, n ≥ 1, n ∈ N; (10) α³ = μν = νμ = ναμ = 0; (11) α³ = μν,νμ = να² = α²μ = 0; (12) α 4 = μν,να = α 2 μ = 0; (13) α m = να = αμ = (μν) n = 0, m ≥ 2, n ≥ 1, m, n ∈ N; (14) α² = μν,ναμ = 0; (15) α³ = μν,να = α²μ = 0; (16) α³ = μν,να = νμ = 0; (3) α² = β² = 0; (4) α² = βn = μβ = 0, n ≥ 2, n ∈ N; (5) α m = β n = αμ = μβ = 0, m,n ≥ 2,m,n ∈ N; (6) α² = β³ = 0, αμ = μβ²; (7) α³ = β 6 = 0, αμ = μβ ; (8) α 4 = β 4 = 0,αμ = μβ ; (17) α² = β² = νμ = μν = 0; (18) α² = βm = νμ = μβ = βν = (ναμ)n = 0, m ≥ 2,n ≥ 1,m,n ∈ N; (19) αm = βn = (νμ)r = αμ = να = μβ = βν = 0, m,n ≥ 2,r ≥ 1,m,n,r ∈ N; (20) α² = μν, β² = νμ, βν = 0, αμ = kμβ, k ∈ K/{0}; (21) α m = β n = 0,β 2 = νμ,να = βν, k 1α 2 = μν,αμ = k 2μβ, k1, k2 ∈ K/{0}, m, n ≥ 2, m, n ∈ N. μ 1 ,μ 2 ,μ 3 //1 ,//2 V1 , V2 (3) νշμ1 = ν1μ2,μ1ν1 = μ2ν1 = μ1ν2 = μ2ν2 = ν1μ1 = 0; α2α1 = α1μ = α2μ = 0; α (2) μιν = μշν = 0; (4) α² = α² = α1α2 = 0, (5) α² = αμ1 = αμ2 = 0; & V
(14) α³ = μν = νμ = α²μ = 0; (15) α 3 = μν = αμ = 0; (16) α³ = μν = ναμ = να² = α²μ = 0; (17) α 4 = μν = νμ = αμ = να 3 = 0; (18) α 4 = μν = νμ = 0, ναμ = να² = α²μ = 0; (8) α² = β³ = αμ = 0; (19) α 5 = μν = νμ = να = α 2 μ = 0; (9) α³ = β³ = αμ = μβ² = 0; (20) α² = να = νμν = αμν = 0; (10) α² = β4 = αμ = μβ2 = 0; (21) α² = να = μνμ = 0; (11) α² = β³ = αμβ = μβ² = 0; (12) α 4 = β 5 = μβ 2 = 0, αμ = μβ ; (22) α 3 = νμ = να = αμν = α 2 μ = 0; (13) α³ = β 7 = μβ 2 = 0, αμ = μβ ; (23) α² = μν,α³ = α²μ = 0, (24) α 4 = νμ = ναμ = να 2 = 0, α³ = μν; (25) α³ = β² = νμ = μν = να = μβ = βν = α²μ = 0; (26) α² = β² = νμ = αμ = να = βν = 0; (27) α² = μν, β² = νμ = αμ = μβ = βνα = 0; (28) α² = μν, β² = νμ = αμ = βν = 0; (29) α² = μν, β² = νμ = να = μβ = 0; (30) α² = μν, β² = νμ = να = βν = αμβ = 0; (31) αμ = μβ,α² = β³ = μν = να = βν = μβ² = 0; (32) αμ = μβ,α² = β² = να = βν = μνμ = νμν = 0; (33) αμ = μβ,α³ = β³ = νμ = μν = να = βν = μβ² = α²μ = 0; (34) αμ = μβ,α³ = β² = νμ = να = βν = α²μ = 0;
A CKNOWLEDGEMENT
I am grateful to my supervisor Prof. Susumu Ariki for suggesting this project and for his kind guidance during the writing of this paper. A lot of thanks to Dr. Kengo Miyamoto for teaching me numerous knowledge of 7-tilting theory, to Prof. Takuma Aihara and Prof. Ryoichi Kase for many useful discussions. I appreciate the anonymous referee very much for being patient with me and giving me many useful suggestions. Finally, I would like to mention my sincere gratitude to Prof. Toshiaki Shoji. Without his recommendation, I could not imagine studying abroad.
Q. WANG
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