Mathematische Zeitschrift
, Volume OnlineFirst – May 21, 2021

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- Springer Journals
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- 1432-1823
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- 10.1007/s00209-020-02687-2
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In this paper, motivated by a τ -tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a ﬁnite dimensional algebra. As an application we describe properties of torsion classes containing band modules. Furthermore, we show that a special biserial algebra is τ -tilting ﬁnite if and only if no band module is a brick. We also recover a criterion for the τ -tilting ﬁniteness of Brauer graph algebras in terms of the Brauer graph. Keywords Band modules · τ -tilting theory · Bricks · Special biserial algebras · Brauer graph algebras Mathematics Subject Classiﬁcation 16G20 · 16S90 · 05E15 · 16W20 · 16D90 · 16P10 1 Introduction An important breakthrough in the representation theory of ﬁnite-dimensional algebras was the systematic introduction of quivers, a powerful tool bringing linear algebra into the theory. In particular, every ﬁnite-dimensional algebra over an algebraically closed ﬁeld K is Morita equivalent to a quotient KQ/I of a path algebra KQ of a quiver Q by an admissible ideal I S. Schroll and H. Trefﬁnger are supported by the EPSRC through the Early Career Fellowship, EP/P016294/1. S. Schroll and Y. Valdivieso are supported by the Royal Society through the Newton International Fellowship NIF\R1\ 180959. H. Trefﬁnger is also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy-EXC-2047/1-390685813. Hipolito Trefﬁnger hipolito.trefﬁnger@hcm.uni-bonn.de Sibylle Schroll schroll@math.uni-koeln.de Yadira Valdivieso yvd1@leicester.ac.uk Department of Mathematics, Universität zu Köln, Weyertal 86-90 50931, Köln, Germany School of Mathematics and Actuarial Sciences, University of Leicester, Leicester LE1 7RH, UK Hausdorff Center of Mathematics, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany 123 S. Schroll et al. [15]. Presentations of algebras in terms of quivers with relations encode much homological and geometric information. As a consequence, quiver representations now play an important role in many areas of mathematics such as, for example, algebraic geometry, mathematical physics, and mirror symmetry. Since the introduction of quivers to representation theory, much work has been done to show that many families of algebras deﬁned by homological properties can, in fact, be characterised by properties of their quivers and relations. One important such family of algebras is that of special biserial algebras. This class contains many well-known families of algebras, such as gentle algebras which play a central role in cluster theory [4] and homological mirror symmetry of surfaces [16] and Brauer graph algebras which originate in the modular representation theory of ﬁnite groups [13]. For these algebras much of the representation theory is encoded in the combinatorics of their quivers and relations. For example, the isomorphism classes of ﬁnitely generated indecomposable modules are given by certain words in the alphabet consisting of the arrows and formal inverses of arrows of their quivers [11,31]. This naturally divides the indecomposable modules over these algebras into two classes, the so-called string modules and inﬁnite families of band modules. The morphisms between string and band modules can also be described in terms of word combinatorics in the quiver [12,17]. One of the motivating observations of this paper is that for any ﬁnite-dimensional algebra given by quiver and relations, that is algebras which are not necessarily special biserial, we can still consider the combinatorics of string and band modules. We show that in the general case, these modules still encode signiﬁcant information on the module categories of the algebras. Even though, in this case there are (possibly inﬁnitely many) indecomposable modules that cannot be described in terms of string and band modules. More recently, the theory of cluster algebras has given new impetus to representation theory with the introduction of many new cluster-inspired representation theoretic concepts. An excellent example of this is the introduction of τ -tilting theory, inspired by mutation in cluster algebras [2]. Since its introduction in 2014, τ -tilting theory has been intensively studied and led to the introduction of many new concepts promising to relate representation theory with other areas of mathematics such as Hall algebras, Donladson–Thomas invariants and Riemannian geometry. From a representation theoretic point of view, one of the important results in [2]are explicit bijections between functorially ﬁnite torsion classes, support τ -tilting modules and 2-term silting complexes in the bounded derived category of a ﬁnite dimensional algebra. Accordingly an algebra is called τ -tilting ﬁnite if has ﬁnitely many support τ -tilting modules. Furthermore, for a τ -tilting ﬁnite algebra, there are only ﬁnitely many torsion classes and all are functorially ﬁnite [14]. A further characterisation of τ -tilting ﬁnite algebras is via bricks in their module category. An object in the module category of an algebra is called a brick if its endomorphism algebra is a division ring. By [14], an algebra is τ -tilting ﬁnite if and only if there are ﬁnitely many bricks in its module category. This immediately implies that if the module category of an algebra contains a band module which is a brick then the algebra is τ -tilting inﬁnite (see Proposition 3.1). The representation theory of a τ -tilting ﬁnite algebra is considerably easier to understand than that of a τ -tilting inﬁnite algebra. For example, the support of the scattering diagram of a τ -tilting ﬁnite algebra is completely determined by its support τ -tilting modules [9,10] and the stability manifold of a ﬁnite dimensional algebra A is contractible if the algebra is silting-discrete which implies, in particular, that the heart of any bounded t-structure of the derived category of A is a module category over a τ -tilting ﬁnite algebra [23]. 123 On band modules and τ -tilting ﬁniteness Therefore, a classiﬁcation of τ -tilting ﬁnite algebras is almost as important in today’s representation theory as the determination of representation ﬁnite algebras was in the last century. Much of the motivation in the early days of representation theory of ﬁnite dimensional algebras stems from this quest of determining algebras of ﬁnite representation type with an important role being played by the ﬁrst and second Brauer-Thrall Conjectures, which were proved subsequently by Ro˘ıter [27] and Auslander [6] for the ﬁrst and by Nazarova and Ro˘ıter [22] and Bautista [7] for second. We now state a τ -tilting analogue of the ﬁrst and second Brauer–Thrall Conjectures. Conjecture 1 (First τ - Brauer-Thrall Conjecture) Let A be a ﬁnite dimensional algebra over a ﬁeld K . If there exists a positive integer n such that dim M ≤ n for every ﬁnite dimensional A-module which is a brick, then A is τ -tilting ﬁnite. Conjecture 1 immediately implies the ﬁrst Brauer-Thrall Conjecture in the case of a τ - tilting inﬁnite algebra. We note that this conjecture has also been stated in [21, Conjecture 6.6] and it has been shown to hold for any ﬁnite dimensional algebra over any ﬁeld in [29]. Conjecture 2 (Second τ -Brauer-Thrall Conjecture) Let A be a ﬁnite dimensional algebra over a ﬁeld K . Suppose that A is τ -tilting inﬁnite. Then there is a positive integer d such that there are inﬁnitely many bricks M where dim M = d. We note that the second τ -Brauer-Thrall Conjecture is not a direct translation of the second Brauer-Thrall Conjecture. A direct translation of the latter to a τ -tilting version would state that if an algebra is τ -tilting inﬁnite then there exist inﬁnitely many integer vectors d such that there exist inﬁnitely many isomorphism classes of bricks of dimension vector d. Clearly this does already not hold in the case of the Kronecker algebra. However, even though the formulation of Conjecture 2 is not a direct generalisation of the classical second Brauer-Thrall Conjecture, if Conjecture 2 holds for a τ -tilting inﬁnite algebra A then it follows from [30] that the classical second Brauer-Thrall Conjecture holds for A. Note that the choice of the name ‘τ -Brauer-Thrall Conjectures’ is guided by the fact that the conjectures are concerned with the τ -tilting ﬁniteness of the algebras. We show by exploiting the interplay between the combinatorics of bricks and band modules that both the ﬁrst and the second τ -Brauer-Thrall Conjectures hold for special biserial algebras over an algebraically closed ﬁeld. More precisely, we show the following. Theorem 1.1 (Theorem 5.1)Let A = KQ/I be a special biserial algebra over an alge- braically closed ﬁeld K . Then A is τ -tilting ﬁnite if and only if no band module of A is a brick. Corollary 1.2 Conjectures 1 and 2 hold for special biserial algebras. In order to prove Theorem 1.1 we heavily rely on the following property of endomorphisms of band modules. For notation see Section 2. Theorem 1.3 (Theorem 3.2) Let A be a ﬁnite dimensional algebra over an algebraically closed ﬁeld K and let b be a band. Suppose that there exists a non-trivial nilpotent endomor- ∞ ∞ phism f : M (b,λ, 1) → M (b,λ, 1) induced by a subword w of b .Then w is a proper subword of some rotation of b. As a consequence of Theorem 1.3, we show the following results on torsion classes based on the band modules they contain. 123 S. Schroll et al. Theorem 1.4 (Theorem 4.1) Let K be an algebraically closed ﬁeld and let A be a ﬁnite dimensional K -algebra containing a band module M (b,λ, 1),for some λ ∈ K . (1) If M (b,λ, 1) ∈ T for some torsion class T ,then M (b,λ, n) ∈ T , for all n ∈ N. (2) If M (b,λ, 1) is not a brick and M (b,λ, 1) is in some torsion class T ,then M (b,μ, 1) ∈ T for all μ ∈ K . (3) If M (b,λ, 1) is a brick then there exist inﬁnitely many distinct torsion classes T ,for ∗ ∗ μ ∈ K , such that T contains M (b,η, 1),for η ∈ K , if and only if η = μ. A different classiﬁcation of τ -tilting ﬁnite special biserial algebras has recently been obtained in [20] based on the classiﬁcation of minimal representation inﬁnite special biserial algebras in [25]. Gentle algebras are a subclass of special biserial algebras. For these algebras a similar criterion to the one obtained in this paper was given in [24]. We ﬁnish the paper in Sect. 6 with an application of our criterion to obtain a characterisation of τ -tilting ﬁnite Brauer graph algebras, giving a new proof of [1, Theorem 6.7]. 2 Background In this section we ﬁx some notation and deﬁnitions which will be used throughout the paper. We ﬁx an algebraically closed ﬁeld K,and A a ﬁnite dimensional K -algebra, which is Morita equivalent to KQ/I for some ﬁnite quiver Q and admissible ideal I in KQ (see [15]). We call elements of a generating set of I relations. Note that we use the same notation for elements in KQ and elements in KQ/I with the implicit understanding that the latter are representatives in their equivalence class. Given an A-module M, we call the top of M, denoted topM, the largest semisimple quotient of M. Similarly, we call the socle of M, denoted socM, the largest semisimple submodule of M. For a quiver Q,let Q be the set of vertices of Q and Q the set of arrows. If α is an 0 1 arrow of Q,wedenoteby s(α) ∈ Q and by t (α) ∈ Q the source and target point of α, 0 0 respectively. For every arrow α : i → j,wedeﬁne α ¯ : j → i to be its formal inverse. Let Q be the set of formal inverses of the elements of Q . We refer to the elements of Q as direct arrows 1 1 and to the elements of Q as inverse arrows. In general, by abuse of notation, we assume that α = α.A walk is a sequence α ...α of elements of Q ∪ Q such that t (α ) = s(α ) for 1 n 1 1 i i +1 every i = 1,..., n − 1 and such that α = α . We say that a walk is of length t if it is the i +1 i composition of exactly t elements in Q ∪ Q . 1 1 We recall that a string in A is by deﬁnition a walk w in Q such that no subpath w of w or subpath w of w is a summand in a relation in I.A band b is deﬁned to be a cyclic string such that every power b is a string, but b itself is not a proper power of some string. A string w = α ...α is a direct (respectively, inverse)if α is a direct arrow (respectively, inverse 1 n i arrow) for every i = 1,..., n. Given a string w,the string module M (w) corresponds to the quiver representation induced by replacing each vertex in w by a copy of the ﬁeld K and every arrow in w by the identity map. In asimilarway,given aband b = α ...α , a non-zero element λ in K and n ∈ N,the 1 t band module M (b,λ, n) is obtained from the band b by replacing each vertex by a copy of the K -vector space K and every arrow α for 1 ≤ i < t by the identity matrix of dimension n,and α by a Jordan block of dimension n and eigenvalue λ (we refer [11] for the precise deﬁnition). 123 On band modules and τ -tilting ﬁniteness Remark 2.1 If A is not special biserial and b is a band then the induced band module M (b,λ, n) might not lie in a homogenous tube of rank 1. For example, this is the case for any band in the 3-Kronecker algebra. Let w = α ...α ...α ...α be a string and let u = α ...α . Then we say that u is a 1 i j t i j submodule substring if α is direct and α is inverse, noting that if i is equal to 1 or if j i −1 j +1 is equal to t, we still consider u a submodule substring. We say that u is a quotient substring if α is inverse and α is direct, and noting again that if i is equal to 1 or if j is equal to i −1 j +1 t, we still consider u a quotient substring. Given two strings v, w such that they have a common substring u which is a quotient substring of v and a submodule substring of w,thenby[12] there is a non-zero map from M (v) to M (w) and the maps of this form give a basis of Hom (M (v), M (w)). Maps between bands are slightly different from maps between strings [17], see also, for example, [18]. For completeness we recall the construction of a basis morphism between bands. We extend the deﬁnition of strings to inﬁnite strings, noting that the only inﬁnite ∞ ∞ strings we will be considering are the strings of the form b which are the inﬁnite strings formed by inﬁnitely many compositions of a band b with itself. Let b and c be two bands, λ, μ ∈ K and n, m two positive integers. If b is different from c or λ is different from μ, a basis of Hom (M (b,λ, n), M (c,μ, m)) is given by maps f induced by pairs (w, φ),where w is a string of ﬁnite length which is a quotient (w,φ) ∞ ∞ ∞ ∞ substring of b and a submodule substring of c and φ is an element of a given basis n m of Hom (K , K ). If b = c and λ = μ, a basis of Hom (M (b,λ, n), M (b,λ, m)) is given by the maps induced by the pairs (w, φ) as above and maps f induced by K -linear maps ψ ∈ n m Hom (K , K ) such that for every vertex i in Q, ( f ) : (M (b,λ, n)) → (M (b,λ, m)) K ψ i i i is equal to a diagonal by blocks nk × mk-matrix, where the diagonal blocks correspond to the maps ψ and where k is the number of times the vertex i is the start of a letter in b.We ∞ ∞ note, in particular, that the maps f do not correspond to substrings of b which are at the same time submodule strings and quotients strings. In the case m = n = 1, the only non-zero basis element of the form f is the identity map. Example 2.2 Let A be the algebra given by the quiver ••• and the ideal β δ I = αδ, βγ . Observe that b = γ δ and c = αγ δγ δ β are bands in A and that γ δγ δ is a ∞ ∞ ∞ ∞ quotient string of b and a submodule substring of c .Let M (b,λ, n) and M (c,μ, m) be two band modules associated to b and c respectively, with λ, μ ∈ K and n, m ∈ N. Then, n m for any morphism φ ∈ Hom (K , K ), the pair (γ δγ δ, φ) gives rise to a basis element f ∈ Hom (M (b,λ, n), M (c,μ, m)) induced by the diagram in Figure 1 with the (γ δγ δ,φ) n m following notation: U = K and V = K and is the (n × n)-Jordan block of eigenvalue λ and is the (m × m)-Jordan block of eigenvalue μ. Then the morphism f : M (b,λ, n) → M (c,μ, m) is as follows. (γ δγ δ,φ) 0 UU 0 1 A B 100 C 01 0 K 010 3 2 VV V 0 0 123 S. Schroll et al. Fig. 1 Diagram of (γ δγ δ, φ) ⎡ ⎤ −1 −1 ⎣ ⎦ where A = 0 , B = φ ,and C = . −2 −2 Remark 2.3 Let b beabandin A. Then every non-zero non-trivial endomorphism of the band module M (b,λ, 1) is a linear combination of maps that are determined by pairs (w, Id ), where w is a string of ﬁnite length which is at the same time a quotient substring and a ∞ ∞ submodule substring of b . For ease of notation, we denote the map f by f . (w,Id ) w 3 Band modules and their endomorphisms In this section, we begin by recalling that if an algebra A contains a band module which is a brick then A is τ -tilting inﬁnite. This is a direct consequence of [14, Theorem 1.4]. Proposition 3.1 Let A = KQ/I be a ﬁnite dimensional algebra. If there exists a band module M which is a brick, then A is τ -tilting inﬁnite. Proof Since K is algebraically closed and hence inﬁnite, for any band b there is an inﬁnite family {M (b,λ, n) : λ ∈ K , n ∈ N} of non-isomorphic indecomposable modules. By hypothesis, there exists a band b, λ ∈ K ,and n ∈ N such that M (b,λ, n) is a ∼ ∼ brick. Since End (M (b,λ, n)) End (M (b,λ , n)) K,for all λ, λ ∈ K ,wehave = = A A that M (b,λ , n) is a brick for all λ ∈ K . In particular, this implies that there is an inﬁnite number of bricks in modA and by [14] A is τ -tilting inﬁnite. Motivated by Proposition 3.1, we show some general results on endomorphisms of band modules. Theorem 3.2 Let A a ﬁnite dimensional algebra and let b be a band. Suppose that there exists a non-trivial nilpotent endomorphism f : M (b,λ, 1) → M (b,λ, 1) induced by the ∞ ∞ subword w of b .Then w is a proper subword of some rotation of b. Proof Since f is a non-trivial nilpotent endomorphism of M (b,λ, 1),wehavethat w is both ∞ ∞ a quotient substring and a submodule substring of b . This is equivalent to the existence ∞ ∞ of letters α, β, γ, δ ∈ Q ∪ Q such that αwβ and γwδ are subwords of b . 123 On band modules and τ -tilting ﬁniteness Suppose the existence of a positive integer n such that w can be written as w = (b ) w for some rotation b of b, where the length of w is strictly smaller than the length of b.Since w is a subword of b, there exists a rotation b of b such that w = w (b ) . Hence w = w w w ∞ ∞ n n for some subword w of b and we have w w = (b ) and w w = (b ) . n ∞ ∞ From the fact that αwβ = αw w w β = αb w β is a subword of b we can conclude that the last letter of w is α. But, at the same time, from γwδ = γw w w δ = γ b w δ we can conclude that the last letter of w is γ , a contradiction. Thus the length of w cannot be greater than the length of b. Proposition 3.3 Let b be a band and suppose that there exists a non-trivial nilpotent endo- ∞ ∞ morphism f : M (b,λ, 1) → M (b,λ, 1) induced by the subword w of b .If α is a letter of w then α appears at least twice in b. Proof Suppose that b = α α ...α . By Theorem 3.2 and without loss of generality we can 1 2 t assume that b = wu, where the ﬁrst letter of u is direct and the last letter of u is inverse, for some subword u of b of length at least one and that this occurrence of w in b is a quotient ∞ ∞ substring of b .Since b starts with w, there exists a r < t such that w = α α ...α . 1 2 r On the other hand, w is also a submodule string of b and by Theorem 3.2 there exist strings v and v such that b = vwv , where the last letter of v is direct and the ﬁrst letter of v is inverse. We choose v to be the shortest possible, which implies that the length s of v is strictly less than t. Furthermore, s ≥ 1 since the last letter of v is direct, while the last letter of u is inverse. Now w = α α ...α . Hence α = α for all 1 ≤ i ≤ r.If s + r < t,the result s+1 s+2 s+r i s+i follows at once. Otherwise, α = α if s + i ≤ t and α = α if s + i > t. In particular, i s+i i s+i −t since 0 < s < t, α and α are distinct letters of b = α α ...α . i s+i −t 1 2 t Proposition 3.4 Let w be a string in A containing a substring α ...α α such that 1 k 1 b = α ...α is a band. Then top(M (b,λ, 1)) is a direct summand of top(M (w)), and 1 k soc(M (b,λ, 1)) is a direct summand of soc(M (w)), for every λ ∈ K . Proof Up to cyclic permutation b can be written as b = w w ...w w where w is 1 2 2t −1 2t 2i −1 a direct string and w is an inverse string for all 1 ≤ i ≤ t, for some positive integer t.Then 2i soc(M (b,λ, 1)) is the direct sum of the simple modules S(t (w )) = S(s(w )) for all 1 ≤ 2i −1 2i i ≤ t and top(M (b,λ, 1)) is the direct sum of the simple modules S(t (w )) = S(s(w )) 2i 2i +1 for all 0 ≤ i ≤ t − 1. If α is not the ﬁrst letter of one of the w , then the result follows. Now suppose that 1 i α is the ﬁrst letter of w for some 0 ≤ i ≤ 2t. Without loss of generality assume that α 1 i 1 is the ﬁrst letter of the direct string w .Then α is the last letter of the inverse string w . 1 k 2t It immediately follows that soc(M (b,λ, 1)) is a direct summand of soc(M (w)). Finally, to prove that top(M (b,λ, 1)) is a direct summand of top(M (w)), it is enough to observe that S(s(w )) is a direct summand of top(M (w)), corresponding to the substring α α . 1 k 1 Proposition 3.5 Let b be a band in A with no repeated letters. Then for all λ ∈ K , every non-trivial nilpotent endomorphism f ∈ End (M (b,λ, 1)) induces a map f : top(M (b,λ, 1)) → soc(M (b,λ, 1)). In particular, the image of every non-trivial nilpotent endomorphism f is semisimple. Proof Let f ∈ End (M (b,λ, 1)) be a non-zero endomorphism. By Remark 2.3 f is given by a linear combination of morphisms of the form f given by a submodule substring w 123 S. Schroll et al. ∞ ∞ ∞ ∞ of b which is at the same time a quotient substring of b .Since b has no repeated letters, by Proposition 3.3 every w corresponds to a vertex. Therefore, every summand of f is induced by a simple module which is a direct summand of both the top and socle of M (b,λ, 1). 4 Bands and torsion classes In this section we study torsion classes containing band modules. We show that if a torsion class contains a band module which is not a brick then the torsion class contains all band modules in the same inﬁnite family. Furthermore, we show that if a band module M (b,λ, 1) is a brick then, for any μ ∈ K with μ = λ, the minimal torsion class containing M (b,λ, 1) is distinct from the minimal torsion class containing M (b,μ, 1).Moreprecisely,weshow the following. Theorem 4.1 Let A be a ﬁnite dimensional algebra containing a band module M (b,λ, 1), for some λ ∈ K . (1) If M (b,λ, 1) ∈ T for some torsion class T then M (b,λ, n) ∈ T , for all n ∈ N. (2) If M (b,λ, 1) is not a brick and M (b,λ, 1) is in some torsion class T then M (b,μ, 1) ∈ T for all μ ∈ K . (3) If M (b,λ, 1) is a brick then there exist inﬁnitely many distinct torsion classes T ,for ∗ ∗ μ ∈ K , such that T contains M (b,η, 1),for η ∈ K , if and only if η = μ. Proof Let M (b,λ, 1) be a band module in a torsion class T . (1) This follows from the fact that T is closed under extensions and M (b,λ, n) is an extension of M (b,λ, 1) by M (b,λ, n − 1). (2) Suppose M (b,λ, 1) is not a brick. By Theorem 3.2, there exists a string module M (w) which is at the same time a quotient and a submodule of M (b,λ, 1) giving rise to a short exact sequence 0 −→ M (w) −→ M (b,λ, 1) −→ N −→ 0 where N is the string module M (b,λ, 1)/M (w).Both M (w) and N are quotients of M (b,λ, 1) and thus since T is closed under quotients, we have M (w) ∈ T and N ∈ T . Moreover, we have that the string modules M (w) and N are quotients of M (b,μ, 1),for any μ ∈ K , since the construction of string modules M (w) and N is independent of the parameter λ. Moreover, for every μ ∈ K we have a short exact sequence 0 −→ M (w) −→ M (b,μ, 1) −→ N −→ 0. Then the result follows from the fact that T is closed under extensions. (3) Recall that given a band b such that M (b,λ, 1) is a brick for some λ ∈ K , then M (b,μ, 1) is a brick for all μ ∈ K . Moreover, we have that, for η ∈ K ,Hom (M (b,η, 1), M (b,μ, 1)) = 0 if and only if μ = η.Deﬁne T to be A μ Filt(Fac(M(b, ¯, 1))), the class of all A-modules ﬁltered by quotients of an element in add(M (b,μ, 1)). We claim that, for η ∈ K , M (b,η, 1) is not in T if μ = η. Suppose to the contrary that M (b,η, 1) ∈ T . Then, M (b,η, 1) is ﬁltered by objects in Fac(M (b,μ, 1)). Hence there is a submodule 0 = L of M (b,η, 1) which is in Fac(M (b,μ, 1)) and there is a non-zero map from M (b,μ, 1) to M (b,η, 1) with μ = η, a contradiction. 123 On band modules and τ -tilting ﬁniteness Remark 4.2 The inﬁnite family of torsion classes in Theorem 4.1(3) is such that two torsion classes in that family are not comparable in the poset of torsion classes of A. Remark 4.3 The results in Theorem 4.1(1) and (2) serve as an indication of why τ -tilting ﬁnite algebras of inﬁnite and of even wild representation type exist. Examples of τ -tilting ﬁnite wild algebras are preprojective algebras of Dynkin type with at least six vertices [19] and wild contraction algebras [5]. 5 -tilting ﬁniteness for special biserial algebras In this section we apply the results of Section 3 to show that the converse of Proposition 3.1 holds for special biserial algebras. More precisely, we show the following. Theorem 5.1 Let A = KQ/I be a special biserial algebra. Then A is τ -tilting ﬁnite if and only if no band module of A is a brick. Recall that an algebra KQ/I is special biserial if every vertex in Q is the start of at most two arrows, and the end of at most two arrows and if for every α ∈ Q there is at most one β ∈ Q such that αβ ∈ / I and there is at most one γ ∈ Q such that γα ∈ / I . 1 1 In order to prove Theorem 5.1, we ﬁrst show the following lemma. Lemma 5.2 Let A = KQ/I be a special biserial algebra. Then all but ﬁnitely many strings w contain a subword of the form αβvα,for α, β ∈ Q and some string v, such that βvα is a band and α is not a letter of v. Proof Let w be a string, x beavertexin Q and α, β ∈ Q such that t (α) = t (β) = x.Since A is special biserial, the simple module S associated to the vertex x is a direct summand of the socle soc(M (w)) of the string module M (w) if and only if either w starts with α or β,if w ﬁnishes with α or β,orif αβ or βα is a subword of w. Suppose that the dimension of soc(M (w)) is greater or equal to 2n + 3, where n is the number of non-isomorphic simple A-modules. This implies that there are 2n +1 simple direct summands of soc(M (w)) given by a subword of the form γ δ of w,for some γ, δ ∈ Q . Then, by the pigeon hole principle, without loss of generality we can suppose that αβ appears at least twice as a subword of w. Then there exist strings v, w and w such that w = w αβvαβw . 1 2 1 2 Furthermore, we can assume that there is no occurrence of α in v. We claim that βvα is a band in A.First,notethat s(β) = t (α). Moreover, βvα starts with an inverse letter and ﬁnishes with a direct letter, so there are no subwords of any rotation βvα which are in I . By construction the letter α appears only once in βvα. As a consequence, βvα is not the concatenation of several copies of a smaller band. Hence, βvα is a band. Since the quiver has ﬁnitely many arrows and A is ﬁnite dimensional, there are only ﬁnitely many strings w such that dim (soc(M (w))) ≤ 2n + 3. Proof of Theorem 5.1 Suppose that no band module in the module category of A is a brick. Then any brick must be a string. We claim that there are only ﬁnitely many string modules which are bricks. By Lemma 5.2 we have that all but ﬁnitely many strings w are of the form w = w αβvαw 1 2 where b = βvα is a band in A. We claim that the string module M (w) of any string w of this form is not a brick. By hypothesis, M (b,λ, 1) is not a brick. So, there exists a non-trivial nilpotent endomorphism f : M (b,λ, 1) → M (b,λ, 1) which is realised by a substring ∞ ∞ w of b by Remark 2.3.Now,byTheorem 3.2 the submodule substring and the quotient substring w of b are included in some rotation of b = βvα. But Proposition 3.3 implies that α is not a letter of w since α only appears once in b. Hence, both copies of w are subwords 123 S. Schroll et al. of βv. As a consequence, w is both a quotient and a submodule substring of w. Thus, there is a non-trivial nilpotent endomorphism f : M (w) → M (w) that is realised by the substring w of w.Inother words, M (w) not a brick. The converse follows from Proposition 3.1. As a consequence of Theorem 5.1 we obtain a proof of both τ -Brauer-Thrall conjectures for special biserial algebras. Corollary 5.3 Conjectures 1 and 2 hold for special biserial algebras. Proof We show Conjecture 1 by contraposition. Suppose that a special biserial algebra A is τ -tilting inﬁnite. By Theorem 5.1 there exists a band module M (b,λ, 1) which is a brick. ∞ ∞ Hence there is no subword of b which is both a submodule substring and a quotient substring. Let b = α α ...α where α is a direct letter and α is an inverse letter. Then, 1 2 t 1 t for every positive integer n, every proper quotient substring of b is a quotient substring of ∞ ∞ n b . Likewise, for every positive integer n, every proper submodule substring of b is a ∞ ∞ n submodule substring of b . Then, it follows from [12]that M (b ) is a brick for all positive integer n. Thus Conjecture 1 holds for special biserial algebras. Suppose that A is special biserial τ -tilting inﬁnite. Again, by Theorem 5.1 there exists a band module M (b,λ, 1) which is a brick. Then, for any μ ∈ K , M (b,μ, 1) is a brick. Since K is an algebraically closed ﬁeld, Conjecture 2 follows. 6 Characterisation of -tilting ﬁnite Brauer Graph Algebras An algebra A is said to be symmetric if A = Hom (A, k) as A-A-bimodule. It directly follows from the deﬁnition of a Brauer graph algebra that it is special biserial and it was shown in [26,28], see also [3], that every symmetric special biserial algebra is a Brauer graph algebra. A Brauer graph is a ﬁnite undirected connected graph, possibly with multiple edges and loops, in which every vertex is equipped with a cyclic ordering of the edges incident with it and a strictly positive integer, its multiplicity. We brieﬂy recall here that given a Brauer graph the corresponding Brauer graph algebra is given by a quotient of a path algebra of a quiver Q by an ideal I generated by relations. The vertices in Q are in bijection with the edges of the Brauer graph and the arrows of Q are induced by the cyclic orderings of the edges at each vertex of the Brauer graph. Each vertex in the Brauer graph gives rise to a cycle in the quiver, sometimes referred to as a special cycle. The relations of the Brauer graph algebra can be read directly from the special cycles and the Brauer graph. For a precise deﬁnition and the construction of a symmetric special biserial algebra from a Brauer graph we refer the reader to standard textbooks such as [8]. Before we start, we need to ﬁx some notation. A cycle C in a Brauer graph G is a set of vertices {v ,...,v } and a set of edges {e ,..., e } such that e is incident with v and v , 1 n 1 n n 1 n and e is incident with v and v for all 1 ≤ i ≤ n − 1. A cycle C is minimal if all its i i −1 i vertices are distinct. We say that a cycle C is odd (respectively, even)ifitisaminimalcycle with an odd (respectively, even) number of vertices. As an application of Theorem 5.1,wegiveanew proof of thecharacterisationin[1]ofthe τ -tilting ﬁniteness of Brauer graph algebras in terms of their Brauer graph. More precisely, we show the following. Theorem 6.1 [[1] ,Theorem 6.7] Let A = KQ/I be a Brauer graph algebra with Brauer graph G. Then A is τ -tilting ﬁnite if and only if G has no even cycles and at most one odd cycle. 123 On band modules and τ -tilting ﬁniteness In order to show Theorem 6.1, we ﬁrst show the following lemmas. Lemma 6.2 Let A = KQ/I be a Brauer graph algebra with Brauer graph G. If G has an even cycle, then there is a band b such that M (b,λ, 1) is a brick. Proof By hypothesis G has an even cycle C with pairwise distinct vertices v ,...,v and 1 2t edges e ,..., e in G such that e is incident with v and v for all i and e is incident 1 2t i i i +1 2t with v and v .Now,deﬁne w to be the shortest direct path in Q from e to e if i is even 2t 1 i i i +1 and the shortest inverse path from e to e if i is odd. Then it is easy to see that the word i i +1 w = w w ...w is a band. 1 2 2t We claim that M (w, λ, 1) is a brick. Indeed, by construction w has no repeated letters. Then, Proposition 3.5 implies that every non-trivial nilpotent endomorphism f of M (w, λ, 1) factors through a map f : top(M (w, λ, 1)) → soc(M (w, λ, 1)). By construction, we have that t t ∼ ∼ top(M (w, λ, 1)) = S(e ) and soc(M (w, λ, 1)) = S(e ) 2 j 2 j −1 j =1 j =1 where all the S(e ) are distinct. Then M (w, λ, 1) has no non-trivial nilpotent endomorphisms and thus it is a brick. Remark 6.3 If a Brauer graph G is not simply laced then it has a cycle of length 2 and it follows from the previous lemma that it contains a band module which is a brick. Lemma 6.4 Let A = KQ/I be a Brauer graph algebra with Brauer graph G. If G has two odd cycles, then there is a band b such that M (b,λ, 1) is a brick. Proof Suppose that G has two odd cycles. Then there exist two sets of vertices, v ,...,v 1 2t +1 and v ,...,v , and two set of edges e ,..., e and e ,..., e in G such that e is 1 2t +1 i 1 2r +1 1 2r +1 incident with v and v for all i and e is incident with v and v , e is incident with i i +1 2t +1 j j j +1 v and v and e is incident with v and v .Since G is connected, there exists a set 2t +1 1 2r +1 2r +1 1 of vertices v ,...,v and edges e ,..., e such that e is incident with v and v , e 1 s 1 s+1 1 1 s+1 is incident with v and v and e is incident with v and v for all 2 ≤ k ≤ s. Note that if 1 k k−1 k v = v ,then k = 0. This proof consist of two cases: when k is odd and when k is even. We prove the case of k odd, the case of k even being very similar. Similarly to the proof of Lemma 6.2, we construct a suitable band in Q and we will do so in several steps. First, deﬁne w to be the shortest direct path from e to e if i is even and the shortest i i i +1 inverse path from e to e if i is odd for all 1 ≤ i ≤ 2t.Now let w be the shortest direct i i +1 2t +1 path from e to e . Denote by u the shortest direct path from e to e if i is odd and 2t +1 i 1 i i +1 the shortest inverse path from e to e if i is even for all i between 0 and s − 1. Deﬁne w i i +1 0 as the shortest direct path from e to e .For all1 ≤ j ≤ 2r deﬁne w to be the shortest s+1 1 j direct path from e to e if j is even and the shortest inverse path from e to e if j is j j +1 j j +1 odd. Set w as the shortest inverse path from e to e . Finally let w be the shortest 2r +1 2r +1 s+1 direct path from e to e . By construction w = w ...w u ... u w ...w u ... u w is a band and w 1 2t +1 0 s−1 s−1 0 0 2r +1 0 has no repeated letters. Then, Proposition 3.5 implies that every non-trivial nilpotent endo- morphism f of M (w, λ, 1) factors through a map f : top(M (w, λ, 1)) → soc(M (w, λ, 1)). Furthermore, by construction, we have that top(M (w, λ, 1)) and soc(M (w, λ, 1)) have no-common direct summand, thus implying that M (w, λ, 1) has no non-trivial nilpotent endomorphisms. In other words, M (w, λ, 1) is a brick in modA. 123 S. Schroll et al. We now prove Theorem 6.1. Proof of Theorem 6.1 Let A be a Brauer Graph algebra with Brauer graph G. Recall that every indecomposable non-projective A-module M comes from a string or a band in the Brauer graph. Furthermore, if G contains an even cycle or if G contains two odd cycles then by Lemmas 6.2 and 6.4 the algebra A is τ -tilting inﬁnite. Thus suppose that G contains at most one odd cycle. If G is a tree and all but at most one multiplicity is equal to one then A is of ﬁnite representation type and, in particular, A is τ -tilting ﬁnite. Now suppose that G is a tree and that there are at least two vertices of G with multiplicity strictly greater than one and let b beabandin A. Given that G is a tree, there exists a vertex v in G with multiplicity strictly greater than one such that b = wb ,where w is a direct or inverse path maximal in b starting and ending at the same edge x of G which is incident with v. By maximality of w, we have that the simple module S(x ) associated to x is a direct summand of both soc(M (b,λ, 1)) and top(M (b,λ, 1)),for any λ ∈ K . Hence no band module in modA is a brick. So, A is τ -tilting ﬁnite by Theorem 5.1. The last case to consider is when G has exactly one cycle of odd length 2t + 1. Then there exists a set of vertices v ,...,v and edges e ,..., e in G such that e is incident 1 2t +1 1 2t +1 i with v and v for all i and e is incident with v and v .Since G has no even cycle, i i +1 2t +1 2t +1 1 it is simply-laced and e is the unique edge incident with v and v . i i i +1 Now consider a band b in A.If b is such that there exists a vertex v such that b = wb as in the case of the tree with at least two vertices of of higher multiplicities, then M (b,λ, 1) is not a brick. Otherwise b is of the form b = w ...w u ... u ,where w is a direct path from 1 2t +1 1 2t +1 i e to e if i is even and the inverse path from e to e if i is odd for all 1 ≤ i ≤ 2t + 1 i i +1 i i +1 and u is a the inverse path from e to e if i is even and the direct path from e to e i i i +1 i i +1 if i is odd for all 1 ≤ i ≤ 2t + 1. Then the simple module S(e ) is a direct summand of both top(M (w, λ, 1)) and soc(M (w, λ, 1)).So, M (w, λ, 1) is not a brick and by the same argument as above, A is τ -tilting ﬁnite. Acknowledgements For helpful conversations regarding band modules and their morhpisms, the authors would like to thank Rosanna Laking and Jan Schröer. Particular thanks goes to the latter for suggesting to formulate our results in terms of a τ -tilting version of the Brauer-Thrall Conjectures. The authors also thank Alexandra Zvonareva for her comments, and the referee for pointing out a mistake in an earlier version of the paper. Funding Open Access funding enabled and organized by Projekt DEAL. Y. Valdivieso is also supported by the European Commission under the Marie Sklodowska-Curie Individual Fellowship, MSCA-IF 838316. 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