On band modules and $\tau$-tilting finiteness

In this paper 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 $\tau$-tilting finite if and only if no band module is a brick.

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 D b (A) of a finite dimensional algebra A. 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 [10]. 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 [10], an algebra A 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 modules which is a brick then the algebra is τ -tilting infinite (see Proposition 5.1).
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 [5,6] 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 [16].
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.
In this paper we will exploit the interplay between the combinatorics of band modules and bricks. As an application to the case of special biserial algebras we give necessary and sufficient conditions for the τ -tilting finiteness of these algebras.
More precisely, we show the following. As a consequence of Theorem 1.1, we show the following results on torsion classes based on the band modules they contain. We note that in the first part of Theorem 1.2, the result holds for any finite dimensional algebra, whereas in the second part we only consider the class of special biserial algebras. (1) Suppose that the band module M (b, λ, 1) is not a brick for some λ ∈ k * . If M (b, λ, 1) is in some torsion class T , then M (b, λ ′ , n) ∈ T , for all λ ′ ∈ k * and all n ∈ N. (2) Suppose that the band module M (b, λ, 1) is a brick and that A is special biserial. Then there exists an infinite family of distinct torsion classes T µ , µ ∈ k * , such that M (b, λ, 1) ∈ T µ if and only if λ = µ.
As a further application of Theorem 3.5, we obtain the following characterisation of τ -tilting finite special biserial algebras. Theorem 1.3 (Theorem 5.2). 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.
A different classification of τ -tilting finite special biserial algebras has recently been obtained in [15] based on the classification of minimal representation infinite algebras in [18]. We also note that gentle algebras are a subclass of special biserial algebras. For these algebras a similar criterion to the above was given in [17].
We finish the paper in Section 6 with an application of our criterion to Brauer graph algebras, obtaining a new proof of the fact that a Brauer graph algebra with Brauer graph G is τ -tilting finite if and only if G contains at most one cycle and if that cycle is of odd length. This has originally been shown in [1,Theorem 6.7].
Acknowledgements: The authors would like to thank Rosanna Laking and Jan Schröer for helpful conversations regarding band modules and their morphisms. The authors also would like to thank Alexandra Zvonareva for comments on an earlier version of the paper.

Background
In this section we fix some of the notation and definitions which will be used throughout this paper.
We fix an algebraically closed field k, and A a basic finite dimensional k-algebra, which is Morita equivalent to kQ/I for some finite quiver Q and admissible ideal I in kQ (see [11]). Furthermore, unless otherwise stated, an algebra given by quiver and relations kQ/I is assumed to be finite dimensional and the ideal I is assumed to be admissible. Note that we use the same notation for elements in kQ and elements in kQ/I with the implicit understand 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 . Given f ∈ End A (M ), we say that f is non-trivial if f is non-zero and is not a scalar multiple of the identity.
For a quiver Q, let Q 0 be the set of vertices of Q and Q 1 the set of arrows. If α is an arrow of Q, we denote by s(α) ∈ Q 0 and by t(α) ∈ Q 0 the source and target point of α, respectively.
For every arrow α : i → j, we defineᾱ : j → i to be its formal inverse. Let Q 1 be the set of formal inverses of the elements of Q 1 . We refer to the elements of Q 1 as direct arrows and to the elements of Q 1 as inverse arrows. A walk is a sequence α 1 . . . α n of elements of Q 1 ∪ Q 1 such that t(α i ) = s(α i+1 ) for every i = 1, . . . , n − 1 and such that α i+1 = α −1 i . We recall that a string in A is by definition a walk w in Q avoiding the zero relations and such that neither w nor w −1 is a summand in a relation. A band b is defined to be a cyclic string such that every power b n is a string, but b itself is not a proper power of some string c. A string w = α 1 . . . α n is a direct (inverse) if α i is a direct arrow (resp. inverse arrow) for every i = 1, . . . , n.
Given a string w, the string module M (w) is obtained by replacing each vertex in w by a copy of the field k and every arrow in w by the identity map. In a similar way, given a band b = α 1 . . . α t , a non-zero element λ in k * and n ∈ N, the 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 n and every arrow α i for 1 ≤ i < t by the identity matrix of dimension n, and α t by a Jordan block of dimension n and eigenvalue λ (we refer [7] for the precise definition).
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.
. . α t be a (finite or infinite) string and let u = α i . . . α j . Then we say that u is a submodule substring if α i−1 is direct and α j+1 is inverse. We say that u is a factor substring if α i−1 is inverse and α j+1 is direct.
Given two strings v, w such that they have a common substring u which is a submodule string of w and a factor string of v, by [8] there is a map from M (v) to M (w) and the maps of this form give a basis of Hom A (M (v), M (w)).
Maps between bands are slightly different from maps between strings [13], see also, for example, [14]. For completeness we recall the construction of a basis morphism between bands. We first define ∞ b ∞ to be the string formed by infinitely many composition 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 A (M (b, λ, n), M (c, µ, m)) is given by maps φ (w,g) induced by pairs (w, g) (detailed in an example below), where w is a string of finite length which is a factor substring of ∞ b ∞ and a submodule substring of ∞ c ∞ and g ∈ Hom k (k n , k m ).
If b = c and λ = µ, a basis of Hom A (M (b, λ, n), M (b, λ, m)) is given maps induced by the pairs (w, g) as above and maps f h induced by k-linear maps h ∈ Hom k (k n , k m ) such that for every vertex Observe that b = γδ and c = αγδγδ β are bands in A and that γδγδ is a submodule string of ∞ b ∞ and a factor 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, for any morphism g ∈ Hom k (k n , k m ), the pair (γδγδ, g) gives rise to a basis element φ (γδγδ,g) ∈ Hom A (M (b, λ, n), M (c, µ, m)) induced by the diagram in Figure 1 with the following notation: U = k n and V = k m and Ψ is the (n × n)-Jordan block of eigenvalue λ and Φ is the (m × m)-Jordan block of eigenvalue µ. Figure 1. Diagram of (γδγδ, g) Remark 2.3. Let b be a band in A. Then we have that every non-zero nilpotent endomorphism of the band module M (b, λ, 1) is a linear combination of maps that are determined by pairs (w, 1), where w is a string of finite length which is at the same time a factor substring and a submodule substring of ∞ b ∞ .

Band modules and their endomorphisms
In this section, we study some general results about bands and band modules in the module category of an algebra A = kQ/I. In particular, we focus on the endomorphism algebras of band modules.
Proposition 3.1. Let A be a finite dimensional algebra. Suppose that w is a string that has a repeated direct (resp. inverse) letter α such that there is an inverse (resp. direct) letter β in between the two copies of α. Then there is a subword w ′ of w which is a band.
Proof. By hypothesis and without loss of generality we can write w as w = w 0 αw 1 βw 2 αw 3 , such that α does not appear in either w 1 nor w 2 . Set w ′ = αw 1 βw 2 . Clearly s(α) = t(w 2 ). Since w ′ is not a directed string and w 2 α is a substring of w, any power (w ′ ) n is a string. Moreover, w ′ is not a proper power of any string c, because α appears exactly once in w ′ . This finishes the proof.
Proposition 3.2. Let A be an algebra and M (b, λ, n) a band module with λ ∈ k * and n ∈ Z >0 . If M (b, λ, n) is a brick, then n = 1.
Proof. Let M (b, λ, n) be a band module such that n ≥ 2 and λ ∈ k * . Then there is a morphism is not a brick, as claimed.
Proof. Up to cyclic permutation b can be written as b = w 1 w 2 . . . w 2t−1 w 2t where w 2i−1 is a direct string and w 2i is an inverse string for all 1 ≤ i ≤ t, for some positive integer t. Then soc (M (b, λ, 1)) is the direct sum of the simple modules S(t(w 2i−1 )) = S(s(w 2i )) for all 1 ≤ i ≤ t and top (M (b, λ, 1)) is the direct sum of the simple modules S(t(w 2i )) = S(s(w 2i+1 )) for all 0 ≤ i ≤ t − 1.
If α 1 is not the first letter of one of the w i , then the result follows. Now Suppose that α 1 is the first letter of w i for some 0 ≤ i ≤ 2t. Without loss of generality assume that α 1 is the first letter of the direct string w 1 . Then α k is the last letter of the inverse string w 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 1 )) is a direct summand of top(M (w)), corresponding to the substring α k α 1 .
Proposition 3.4. Let b be a band such that any repeated letter in b appears in the same direct or inverse substring of b. Then there exists a band c with no repeated letters such that soc(M (b, λ, 1)) is isomorphic to soc(M (c, λ, 1)) and top(M (b, λ, 1)) is isomorphic to top(M (c, λ, 1)), for every λ ∈ k * .
Proof. Suppose b = w 1 w 2 . . . w t−1 w t where without loss of generality we can assume that w i is a direct string when i is odd and w i is inverse when i is even.
Assume that α appears twice in some direct string w i . Then the path l going from α to itself is a cycle in Q. Now we can construct a band b 1 from b by deleting the cycle l. Moreover, we have soc (M (b, λ, 1)) ∼ = soc (M (b 1 , λ, 1)) and top(M (b, λ, 1)) ∼ = top (M (b 1 , λ, 1)).
We repeat this process until there are no more repeated letters, using similar arguments if there are repeated letters in an inverse string w j . In this way we construct a band c as in the statement.
Theorem 3.5. Let A = kQ/I be an algebra and suppose that there exists a band in A. Then there is a band b in A such that, for all λ ∈ k * , every non-trivial nilpotent endomorphism f ∈ End A (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 of M (b, λ, 1) is semisimple.
The proof of Theorem 3.5 directly follows from the next Lemma.
In particular, the image of every non-trivial nilpotent endomorphism f is semisimple.
Proof. Let f ∈ End A (M (b, λ, 1)) be a non zero endomorphism. By Remark 2.3 f is given by a linear combination of morphisms, which are given by submodule strings of ∞ b ∞ which are at the same time factor strings of ∞ b ∞ . Since b has no repeated letters, 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).
Proof of Theorem 3.5. By Propositions 3.1 and 3.3, we can assume without loss of generality that there exists a band in A with no repeated letters. The result then follows from Lemma 3.6.  M (b, λ, 1) is not a brick. Then there exists a simple module S which is at the same time a direct summand of top (M (b, λ, 1)) and soc (M (b, λ, 1)).

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. In the case of special biserial algebras, 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). Theorem 4.1. Let A be an algebra and T be a torsion class in modA. Suppose that b is a band in A such that M (b, λ, 1) ∈ T , for some λ ∈ k * . If M (b, λ, 1) is not a brick then M (b, λ ′ , n) ∈ T , for all λ ′ ∈ k * and all n ∈ N.
Proof. Let M (b, λ, 1) ∈ T be a band module which is not a brick. Corollary 3.7 implies the existence of a simple module S which is at the same time a direct summand of top(M (b, λ, 1)) and soc(M (b, λ, 1)). Then there is a short exact sequence where N is the string module M (b, λ, 1)/S. Both S and N are quotients of M (b, λ, 1) and S ∈ T and N ∈ T because torsion classes are closed under quotients.
Since M (b, λ, 1), we have that M (b, λ ′ , n) is not a brick for all λ ′ ∈ k * and all n ∈ N. Moreover, we have that S and the string module N are quotients of M (b, λ ′ , 1) because the top and the socle of a band module are independent of the parameter λ ′ ∈ k * . Hence, for every λ ′ ∈ k * we have a short exact sequence   M (b, λ, 1) is a brick for some λ ∈ k * . Then there exists an infinite family of distinct torsion classes T µ , µ ∈ k * such that M (b, λ, n) ∈ T µ for any n ∈ N if and only if λ = µ.

τ -tilting finiteness for special biserial algebras
In this section we apply the results of the Section 3 to construct a necessary and sufficient criterion to determine the τ -tilting finiteness of special biserial algebras. We note a criterion for τ -tilting finiteness of biserial algebras has been determined in [15] together with a description of the minimal τ -tilting infinite special biserial algebras.
For gentle algebras, a subclass of special biserial algebras, τ -tilting finiteness has been determined in [17] and has been shown to coincide with the gentle algebras of finite representation type.
We start by stating a direct consequence of [10, Theorem 1.4].
Proposition 5.1. 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, 1) : 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, 1) is a brick. Since End A (M (b, λ, 1)) ∼ = End A (M (b, λ ′ , 1)), for all λ, λ ′ ∈ k * , we have that M (b, λ, 1) is a brick for all λ ∈ k * . In particular, this implies that there is an infinite number of bricks in modA and by [10] A is τ -tilting infinite.
We will see now that for special biserial algebras the converse of the above also holds. Proof. Suppose that no band module in modA is a brick. Then we have that any brick in modA is a string. We claim that there are only finitely many string modules which are bricks.
Given that A is finite dimensional, we have in particular that Q has finitely many arrows. Since all objects in modA are finite dimensional as k-vector spaces, by the pigeon hole principle all but finitely many strings are as in the statement of Proposition 3.1. Thus all but finitely many strings w have a substring b which is a band. By Proposition 3.1 we can assume without loss of generality that every repeated letter in b appears in the same direct or inverse string.

Characterisation of τ -tilting finite Brauer Graph Algebras
An algebra A is said to be symmetric if A ∼ = Hom A (A, k) as A-A-bimodule. It was shown in [19] 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 with a strictly positive integer, its multiplicity. The construction of a symmetric special biserial algebra from a Brauer graph can be found, for example, in [4].
Before we start, we need to fix some notation. A cycle C in a Brauer graph G is a set of vertices {v 1 , . . . , v n } and a set of edges {e 1 , . . . , e n } such that e n is incident with v 1 and v n , and e i is incident with v i−1 and v i for all 1 ≤ i ≤ n − 1. A cycle C is minimal if all its vertices are distinct. We say that a cycle C is odd (resp. even) if it is a minimal cycle with an odd (resp. even) number of vertices.
As an application of Theorem 5.2, we give a new proof of the characterisation in [1] of the 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.
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 1 , . . . , v 2t and edges e 1 , . . . , e 2t in G such that e i is incident with v i and v i+1 for all i and e 2t is incident with v 2t and v 1 . Now, define w i to be the shortest direct path from e i to e i+1 if i is even and the shortest inverse path from e i to e i+1 if i is odd. Then it is easy to see that the word w = w 1 w 2 . . . w 2t is a band in Q.
We claim that M (w, λ, 1) is a brick. Indeed, by construction w has no repeated letters. Then, Lemma 3.6 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 where all the S(e i ) 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 1 , . . . , v 2t+1 and v ′ 1 , . . . , v ′ 2r+1 , and two set of edges e 1 , . . . , e 2t+1 and e ′ 1 , . . . , e ′ 2r+1 in G such that e i is incident with v i and v i+1 for all i and e ′ j is incident with v ′ j and v ′ j+1 , e 2t+1 is incident with v 2t+1 and v 1 and e ′ 2r+1 is incident with v ′ 2r+1 and v ′ 1 . Since G is connected, there exists a set of vertices v ′′ 1 , . . . , v ′′ s and edges e ′′ 1 , . . . , e ′′ s+1 such that e ′′ 1 is incident with v 1 and v ′′ 1 , e ′′ s+1 is incident with v ′′ s and v ′ 1 and e ′′ k is incident with v ′′ k−1 and v ′′ k for all 2 ≤ k ≤ s. Note that if v 1 = v ′ 1 , 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 α i to be the shortest direct path from e i to e i+1 if i is even and the shortest inverse path from e i to e i+1 if i is odd for all 1 ≤ i ≤ 2t. Now let α 2t+1 be the shortest direct path from e 2t+1 to e ′′ 1 . Denote by β i the shortest direct path from e ′′ i to e ′′ i+1 if i is odd and the shortest inverse path from e ′′ i to e ′′ i+1 if i is even for all i between 0 and s − 1. Define γ 0 as the shortest direct path from e ′′ s+1 to e ′ 1 . For all 1 ≤ j ≤ 2r define γ j to be the shortest direct path from e ′ j to e ′ j+1 if j is even and the shortest inverse path from e ′ j to e ′ j+1 if j is odd. Set γ 2r+1 as the shortest inverse path from e ′ 2r+1 to e ′′ s+1 . Finally let α 0 be the shortest direct path from e ′′ 1 to e 1 . By construction w = α 1 . . . α 2t+1 β 0 . . . β s−1 γ 0 . . . γ 2r+1 β s−1 . . . β 0 α 0 is a band and w has no repeated letters. Then, Lemma 3.6 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)). 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.
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 be a band in 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 5.2.
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 1 , . . . , v 2t+1 and edges e 1 , . . . , e 2t+1 in G such that e i is incident with v i and v i+1 for all i and e 2t+1 is incident with v 2t+1 and v 1 . Since G has no even cycle, it is simply-laced and e i is the unique edge incident with v i and v 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 = α 1 . . . α 2t+1 β 1 . . . β 2t+1 , where α i is a direct path from e i to e i+1 if i is even and the inverse path from e i to e i+1 if i is odd for all 1 ≤ i ≤ 2t + 1 and β i is a the inverse path from e i to e i+1 if i is even and the direct path from e i to e i+1 if i is odd for all 1 ≤ i ≤ 2t + 1. Then the simple module S(e i ) 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.