STABLE ENDOMORPHISM ALGEBRAS OF MODULES OVER SPECIAL BISERIAL ALGEBRAS

JAN SCHRÖER AND ALEXANDER ZIMMERMANN

Abstract. We prove that the stable endomorphism algebra of a module without self-extensions over a special biserial algebra is a gentle algebra. In particular, it is again special biserial. As a consequence, any algebra which is derived equivalent to a gentle algebra is gentle.

Dedicated to Idun Reiten on the occasion of her 60th birthday

1. Introduction

We consider finite-dimensional (associative) algebras over an algebraically closed ground field k. One of the most important classes of tame algebras is the class of special biserial algebras. These algebras occur naturally in many different contexts, see the introduction of [17] or [18] for more details. We shall prove that for a special biserial algebra A and any A-module M without self-extensions the endomorphism algebra of M modulo the ideal of endomorphisms factoring through a projective A-module is not only special biserial again, but is even a gentle algebra. Gentle algebras are special biserial algebras satisfying certain minimality conditions.

The result is interesting and surprising in its own right, since for arbitrary algebras this quotient of the endomorphism algebra of modules without selfextensions will be very different from the original algebra. Nevertheless, our main motivation came from different considerations, namely the theory of derived equivalences. Let mod(A) be the category of finite-dimensional right A-modules and let D^{b}(A) be the derived category of bounded complexes of finite-dimensional A-modules. Two algebras A and B are said to be derived equivalent if the categories D^{b}(A) and D^{b}(B) are equivalent as triangulated categories. The stable module category \underline{mod}(A) is the category with the same objects as mod(A), and whose morphisms are equivalence classes of A-module homomorphisms modulo those which factor through a projective module.

Rickard shows that two algebras A and B are derived equivalent if and only if there exists a so-called tilting complex T in D^{b}(A) such that B ≃ End_{D^{b}(A)}(T), see [10],[12]. Now, Happel proved in [8] that there is a full and faithful embedding F : D^{b}(A) ⟶ \underline{mod}(RA) of triangulated categories, where RA is the repetitive algebra of A. Since for a tilting complex T one gets Hom_{D^{b}(A)}(T, T [1]) = 0, the algebra B is isomorphic to the endomorphism algebra in \underline{mod}(RA) of a module M = F T without selfextensions. Thus, studying stable endomorphism algebras of modules without self-extensions is of central importance when one is interested in derived equivalences.

2000 Mathematics Subject Classification: 16E30, 16G20, 18E30.

We gratefully acknowledge support from the Volkswagen Foundation (RIP Program at Oberwolfach).

In our case, since an algebra is gentle if and only if its repetitive algebra is special biserial, (see [11] and also [14] and [15]), we get as a corollary to our main result that the class of gentle algebras is closed under derived equivalences. Examples of classes of algebras closed under derived equivalence are very rare and show that gentle algebras deserve much attention.

To be more precise, let us recall some definitions: Let Q be a (not necessarily finite) quiver. For an arrow α : a → b, let s(α) = a be its starting point and e(α) = b its end point. A path of length n ≥ 1 in Q is a sequence α_{1} ⋯ α_{n} of arrows such that e(α_{i}) = s(α_{i+1}) for all 1 ≤ i ≤ n−1. A relation for Q is a non-zero k-linear combination of paths of length at least 2 having the same starting point and the same end point. Let ρ be a set of relations for Q. Then (Q, ρ) is called special biserial if the following hold:

(1) Any vertex in Q is the starting point of at most two arrows and also the end point of at most two arrows;

(2) Given an arrow β, there is at most one arrow α with e(α) = s(β) and αβ ∉ ρ, and there is at most one arrow γ with e(β) = s(γ) and βγ ∉ ρ;

(3) Each infinite path in Q contains a subpath which is in ρ. A special biserial pair (Q, ρ) is gentle if additionally the following hold:

(4) All elements in ρ are paths of length 2;

(5) Given an arrow β, there is at most one arrow α′ with e(α′) = s(β) and α′β ∈ ρ, and there is at most one arrow γ′ with e(β) = s(γ′) and βγ′ ∈ ρ.

A k-algebra is called special biserial, or gentle, if it is Morita equivalent to an algebra kQ/(ρ) for (Q, ρ) special biserial, or gentle, respectively. In these cases, kQ/(ρ) is finite-dimensional if and only if Q contains only finitely many vertices. Here kQ is the path algebra of Q, and (ρ) is the ideal generated by the elements in ρ. Recall that kQ has as a basis the set of all paths in Q including a path e_{i} of length 0 for each vertex i in Q. By ‘modules’ we always mean finite-dimensional right modules.

Here is our main result.

Theorem 1.1. Let A be a special biserial algebra, and let M be an Amodule. If Ext_{A}^{1}(M, M) = 0, then \underline{End}_{A}(M) is a gentle algebra.

Let T [i] be the usual shift of a complex T in D^{b}(A) by i degrees to the left (we adopt the convention of [10]). Then, this theorem has the following consequence.

Corollary 1.2. Let A be a finite-dimensional gentle algebra, and let T be a complex in D^{b}(A). If Hom_{D^{b}(A)}(T, T [1]) = 0, then End_{D^{b}(A)}(T) is a gentle algebra. In particular, any algebra B which is derived equivalent to A is gentle, and (up to Morita equivalence) there are only finitely many such algebras B.

Classical examples of gentle algebras are hereditary algebras of type 𝔸_{n} and 𝔸̃_{n}. It was known before that all algebras which are derived equivalent to these examples are gentle again. For a complete classification of the derived equivalence classes of these examples see [1] and [3]. Recently, Vossieck classified all algebras A such that D^{b}(A) is discrete. In this case, it turns out that A is either derived hereditary of Dynkin type or a gentle algebra, see [19]. Using Vossieck’s result, the derived equivalence classes of algebras with discrete derived category have been classified in [4].

The paper is organised as follows. In Section 2 we recall definitions and give a survey of known results. The main theorem and its corollary are proved in Section 3. Finally, in Section 4 we give some examples.

Although we often write maps on the left hand side, we compose them as if they were on the right. Thus the composition of a map θ followed by a map ϕ is denoted θϕ.

2. Known facts on special biserial algebras

In this section, we recall some basic facts on special biserial algebras, string modules and homomorphisms between string modules. As a main reference we use [5], see also there for further references.

Let Q = (Q_{0}, Q_{1}) be a quiver with set of vertices Q_{0} and set of arrows Q_{1}. Let ρ be a set of relations for Q such that (Q, ρ) is special biserial, and let A = kQ/(ρ) be the corresponding special biserial algebra. Without loss of generality we can assume that ρ contains only zero relations, i.e. paths, and relations of the form p−q with p and q paths, which are called commutativity relations. If r = p − q is a commutativity relation, then we say that p and q are contained in r. By ρ^{+} we denote the set ρ together with all paths which are contained in commutativity relations in ρ.

Given an arrow α in Q denote by α^{−} a formal inverse where s(α^{−}) = e(α) and e(α^{−}) = s(α). Also let (α^{−})^{−} = α. The set of formal inverses of arrows is denoted by Q_{1}^{−}. We use the word ‘arrow’ only for element in Q_{1}, we refer to elements in Q_{1}^{−} always as inverse arrows. A string (for (Q, ρ)) of length n ≥ 1 is a sequence c_{1} ⋯ c_{n} of arrows or inverse arrows with the following properties:

(1) e(c_{i}) = s(c_{i+1}) for 1 ≤ i ≤ n − 1;

(2) c_{i} ≠ c_{i+1}^{−} for 1 ≤ i ≤ n − 1;

(3) For all 1 ≤ i < j ≤ n neither c_{i}c_{i+1} ⋯ c_{j} nor c_{j}^{−} ⋯ c_{i+1}^{−}c_{i}^{−} are contained in ρ^{+}.

Let C = c_{1} ⋯ c_{n} be a string. Define s(C) = s(c_{1}) and e(C) = e(c_{n}). Let C^{−} = c_{n}^{−} ⋯ c_{1}^{−} be the inverse string of C. Additionally, we define for every vertex i in Q two strings 1_{(i,1)} and 1_{(i,−1)} of length 0 with s(1_{(i,t)}) = e(1_{(i,t)}) = i and 1_{(i,t)}^{−} = 1_{(i,−t)} for t = −1, 1. The length of an arbitrary string C is denoted by |C|. We call C direct if |C| = 0 or C = c_{1} ⋯ c_{n} with c_{i} ∈ Q_{1} for all i, and C is called inverse if C^{−} is direct.

Similarly as in [5], one can define two maps σ, ϵ : Q_{1} → {−1, 1} which satisfy the following properties:

(1) If α_{1} ≠ α_{2} are arrows with e(α_{1}) = e(α_{2}), then ϵ(α_{1}) = −ϵ(α_{2});

(2) If β_{1} ≠ β_{2} are arrows with s(β_{1}) = s(β_{2}), then σ(β_{1}) = −σ(β_{2});

(3) If α and β are arrows with e(α) = s(β) and αβ ∉ ρ^{+}, then ϵ(α) = −σ(β);

(4) Let i be a vertex such that there is only one arrow α with e(α) = i, and only one arrow β with s(β) = i. If αβ ∈ ρ^{+}, then ϵ(α) = σ(β).

Note that σ and ϵ are defined in [5] without condition (4). The set of all strings is denoted by S. We extend σ and ϵ to maps

σ, ϵ : S → {−1, 1}

as follows: For an arrow α define σ(α^{−}) = ϵ(α) and ϵ(α^{−}) = σ(α). If C = c_{1} ⋯ c_{n} is a string of length n ≥ 1, then let σ(C) = σ(c_{1}) and ϵ(C) = ϵ(c_{n}). For a string 1_{(i,t)} of length 0 define σ(1_{(i,t)}) = t and ϵ(1_{(i,t)}) = −t.

Let C = c_{1} ⋯ c_{n} and D = d_{1} ⋯ d_{m} be strings of length at least 1. If CD = c_{1} ⋯ c_{n}d_{1} ⋯ d_{m} is a string, then we say that the concatenation of C and D is defined. For an arbitrary string C let 1_{(s(C),t)}C = C if σ(C) = t, and let C1_{(e(C),t)} = C if ϵ(C) = −t. Otherwise, the concatenation with a string of length 0 is not defined.

One uses σ and ϵ to give a certain ‘orientation’ to strings. One of the main properties of σ and ϵ is the following:

Lemma 2.1. If C and D are strings such that the concatenation CD is defined, i.e. CD is again a string, then e(C) = s(D) and ϵ(C) = −σ(D).

For each string C, we define a map

v_{C} : {1, ⋯, |C| + 1} ⟶ Q_{0}

as follows. If C = 1_{(i,t)} is a string of length 0, then let v_{C}(1) = i. If C = c_{1} ⋯ c_{n} is of length at least 1, then let

v_{C}(i) = \{ \begin{matrix}s(c_{i}) : & 1 ≤ i ≤ |C|; \\ e(c_{n}) : & i = |C| + 1. \end{matrix}

For each string C we construct an A-module M(C) as follows. First, assume that C = c_{1} ⋯ c_{n} is a string of length n ≥ 1. Fix a basis {z_{1}, ⋯, z_{n+1}} of M(C). Let e_{j} be a path of length 0 in Q. Then define

z_{i} · e_{j} = \{ \begin{matrix}z_{i} : & i ∈ v_{C}^{−1}(j); \\ 0 : & otherwise. \end{matrix}

Given an arrow α in Q let

z_{i} · α = \{ \begin{matrix}z_{i+1} : & α = c_{i}, 1 ≤ i ≤ n; \\ z_{i−1} : & α^{−} = c_{i−1}, 2 ≤ i ≤ n + 1; \\ 0 : & otherwise. \end{matrix}

Next, assume C = 1_{(i,t)} is a string of length 0. Then let {z_{1}} be a basis of M(C). For each path e_{j} of length 0 in Q let

z_{1} · e_{j} = \{ \begin{matrix}z_{1} : & j = i; \\ 0 : & otherwise. \end{matrix}

For all α in Q_{1} define z_{1} · α = 0.

Modules of the form M(C) are called string modules, and {z_{1}, ⋯, z_{n+1}} is called the canonical basis of M(C). The construction of string modules goes back to Gelfand and Ponomarev, see [7] and also [5]. One can easily check that string modules are always indecomposable. For strings C_{1} and C_{2} define C_{1} ∼ C_{2} if C_{1} = C_{2} or C_{1} = C_{2}^{−}. Two string modules M(C_{1}) and M(C_{2}) are isomorphic if and only if C_{1} ∼ C_{2}.

For each vertex i, let S_{i} = M(1_{(i,1)}). Then {S_{i} | i ∈ Q_{0}} is a complete set of isomorphism classes of simple A-modules. Furthermore, let P_{i} be the corresponding indecomposable projective A-module with top(P_{i}) = S_{i}.

For a string C define P(C) = {(D, E, F) | D, E, F ∈ S and DEF = C}. We call (D, E, F) in P(C) a factor string of C if the following hold:

(1) Either |D| = 0 or D = d_{1} ⋯ d_{n} where d_{n} ∈ Q_{1}^{−};

(2) Either |F | = 0 or F = f_{1} ⋯ f_{m} where f_{1} ∈ Q_{1}. Dually, we call (D, E, F) a substring of C if the following hold:

(1) Either |D| = 0 or D = d_{1} ⋯ d_{n} where d_{n} ∈ Q_{1};

(2) Either |F | = 0 or F = f_{1} ⋯ f_{m} where f_{1} ∈ Q_{1}^{−}. Let fac(C) be the set of factor strings of C, and by sub(C) we denote the set of substrings of C. If C_{1} and C_{2} are strings, then we call a pair ((D_{1}, E_{1}, F_{1}), (D_{2}, E_{2}, F_{2})) in fac(C_{1}) × sub(C_{2}) admissible if E_{1} ∼ E_{2}. Denote the set of admissible pairs in fac(C_{1}) × sub(C_{2}) by A(C_{1}, C_{2}).

For each a = ((D_{1}, E_{1}, F_{1}), (D_{2}, E_{2}, F_{2})) in A(C_{1}, C_{2}) we define a homomorphism

f_{a} : M(C_{1}) ⟶ M(C_{2})

as follows: If E_{1} = E_{2}, then let

f_{a}(z_{|D_{1}|+i}) = z_{|D_{2}|+i}

for 1 ≤ i ≤ |E_{1}| + 1, and all other canonical basis vectors of M(C_{1}) are mapped to 0. If E_{1} = E_{2}^{−}, then let

f_{a}(z_{|D_{1}|+i}) = z_{|D_{2}E_{2}|+2−i}

for 1 ≤ i ≤ |E_{1}| + 1, and all other canonical basis vectors are mapped to 0. Such homomorphisms are called graph maps. It is proved in [6] that the graph maps {f_{a} | a ∈ A(C_{1}, C_{2})} form a basis of the homomorphism space Hom_{A}(M(C_{1}), M(C_{2})).

There is the following multiplicative behaviour of graph maps: The composition f_{a}f_{b} of graph maps is either 0 or a graph map. This fact is very important and has numerous applications.

If a = ((D_{1}, E_{1}, F_{1}), (D_{2}, E_{2}, F_{2})) is admissible, then we call a and f_{a} oriented if E_{1} = E_{2}. We say that a and f_{a} are left-sided, or right-sided if they are oriented and |D_{1}| = |D_{2}| = 0, or |F_{1}| = |F_{2}| = 0, respectively. In case a is left-sided or right-sided, one calls a and f_{a} one-sided. Finally, a and f_{a} are weakly one-sided if a or ((D_{1}, E_{1}, F_{1}), (F_{2}^{−}, E_{2}^{−}, D_{2}^{−})) are one-sided, and they are two-sided if they are not weakly one-sided.

Let C_{1} = D_{1}E_{1}F_{1} and C_{2} = D_{2}E_{2}F_{2}. A graph map f_{a} can always be transformed to an oriented graph map. This is done by composing f_{a} with one of the canonical isomorphisms M(C_{1}^{−}) → M(C_{1}) or M(C_{2}) → M(C_{2}^{−}). Note that these are both graph maps. Since there are two possibilities, we define a(l) = a(r) = a if a is oriented, and a(l) = ((F_{1}^{−}, E_{1}^{−}, D_{1}^{−}), (D_{2}, E_{2}, F_{2})) and a(r) = ((D_{1}, E_{1}, F_{1}), (F_{2}^{−}, E_{2}^{−}, D_{2}^{−})), otherwise. It follows that a(l) is one-sided if and only if a(r) is one-sided if and only if a is weakly one-sided. The next lemma follows directly from the definitions.

Lemma 2.2. Let f_{a} and f_{b} be weakly one-sided graph maps. If f_{a(l)} and f_{b(r)} are both left-sided, or both right-sided, then f_{a(l)}f_{b(r)} is again left-sided, or right-sided, respectively. In particular, f_{a}f_{b} is again weakly one-sided. Otherwise, f_{a}f_{b} is 0 or two-sided.

In [16] we use the term ‘perfect’ instead of ‘oriented’. Note also that we use left modules in [16] instead of right modules. Thus strings are written in the opposite way. One of the important steps in our work is to prove a stable version of the following proposition.

Proposition 2.3. Let A = kQ/(ρ) be a special biserial algebra, and let C_{i}, 1 ≤ i ≤ t, be strings for (Q, ρ). If each a ∈ A(C_{i}, C_{j}), 1 ≤ i, j ≤ t, is weakly one-sided, then End_{A}(⨁_{i=1}^{t} M(C_{i})) is a gentle algebra.

Proof. For A a string algebra, this statement is proved in [16, Proposition 4.8]. Now assume A is special biserial, and let p − q be a commutativity relation in ρ. If M is a direct sum of string modules, then M · p and M · q are both 0. Thus we can define B = A/(p_{i}, i ∈ I) where the p_{i} are all paths which are contained in commutativity relations. Clearly, B is a string algebra, and End_{A}(M) is isomorphic to End_{B}(M).

3. Proof of the main result

Throughout this section, we assume that A = kQ/(ρ) is special biserial, with ρ containing only zero relations or commutativity relations.

By \underline{Hom}_{A}(M, N) we denote the A-module homomorphisms from M to N modulo the homomorphisms which factor through projective A-modules. Let \underline{mod}(A) be the stable category of finite-dimensional A-modules. Here the objects are the same as in mod(A), but the morphism space from M to N is \underline{Hom}_{A}(M, N). For a homomorphism f let \underline{f} be the corresponding morphism in \underline{mod}(A).

It is known that any finite-dimensional indecomposable A-module is a string module, a band module, or a non-serial projective-injective module, see [5]. If P is an indecomposable non-serial projective-injective A-module, then the radical rad(P) is isomorphic to a string module M(CD) with C a direct and D an inverse string. Similarly, the socle factor P/soc(P) is isomorphic to a string module M(C′D′) with C′ an inverse and D′ a direct string.

If N is a band module, then Ext_{A}^{1}(N, N) ≠ 0 since band modules lie in homogeneous tubes of the Auslander-Reiten quiver of A, i.e. τ_{A}N = N, where τ_{A} is the Auslander-Reiten translation. For basic facts on Auslander-Reiten theory we refer to [2] or [13].

So, if M is an A-module with Ext_{A}^{1}(M, M) = 0, then M is a direct sum of string modules and of non-serial projective-injectives. Since we are interested in stable homomorphisms, it is enough to consider only string modules.

For the rest of this section, let C_{1} and C_{2} be (not necessarily different) strings for (Q, ρ).

Lemma 3.1. Let f : M(C_{1}) → M(C_{2}) be a homomorphism which factors through a direct sum of non-serial projective-injective A-modules. Then f is a linear combination of two-sided graph maps.

Proof. Clearly, it is enough to prove the lemma for the case that f factorizes through an indecomposable non-serial projective-injective A-module. Thus, let P be an indecomposable non-serial projective-injective A-module, and let f_{1} : M(C_{1}) → P and f_{2} : P → M(C_{2}) homomorphisms such that f = f_{1}f_{2}. By ι : rad(P) → P we denote the canonical radical inclusion, and by π : P → P/soc(P) the projection of P onto its socle factor. Clearly, f_{1} must factor through rad(P), and f_{2} must factor through P/soc(P). Thus there are homomorphisms g and h such that f_{1} = gι and f_{2} = πh. The following picture describes the situation:

[figure: M(C_{1}); M(C_{2}); f; g; h; R_{1}; R_{2}; α; β; ι; γ; δ; R_{1}; R_{2}; α; β; π; γ; δ; R_{1}; R_{2}; rad(P); P; P/soc(P)]

Now g = ∑_{i} λ_{i}f_{a_{i}} and h = ∑_{j} μ_{j}f_{b_{j}} where a_{i} is of the form

((D_{1i}, E_{1i}E′_{1i}^{−}, F_{1i}), (D_{2i}, E_{2i}, F_{2i}))

with E_{1i} and E′_{1i} being direct strings, and b_{j} is of the form

((D_{3j}, E_{3j}^{−}E′_{3j}, F_{3j}), (D_{4j}, E_{4j}, F_{4j}))

with E_{3j} and E′_{3j} being direct strings. Thus we get

f = ∑_{i,j} λ_{i}μ_{j}f_{a_{i}}ιπf_{b_{j}}.

It is straightforward to check that for all i, j the composition f_{a_{i}}ιπf_{b_{j}} is either 0 or a sum of either one or two graph maps which are two-sided.

Lemma 3.2. Let f_{a_{i}} : M(C_{1}) → M(C_{2}), 1 ≤ i ≤ s, be pairwise different graph maps, which are weakly one-sided, with \underline{f_{a_{i}}} ≠ 0. Then the \underline{f_{a_{i}}} are linearly independent in \underline{Hom}_{A}(M(C_{1}), M(C_{2})).

Proof. Assume that ∑_{i=1}^{s} λ_{i}\underline{f_{a_{i}}} = 0 with λ_{i} ≠ 0 for some i, and define f = ∑_{i=1}^{s} λ_{i}f_{a_{i}}. Thus there is a projective A-module P = ⨁_{i=1}^{l} P (i), P (i) indecomposable for all i, and homomorphisms f_{1} = [f_{11}, ⋯, f_{1l}] : M(C_{1}) → P and f_{2} = [f_{21}, ⋯, f_{2l}]^{t} : P → M(C_{2}) such that f = f_{1}f_{2}. Thus f_{1}f_{2} = ∑_{i=1}^{l} f_{1i}f_{2i}. If P (i) is not a non-serial projective-injective for some i, then f_{1i} and f_{2i} are both linear combinations of graph maps. By the multiplicativity of graph maps, we get that f_{1i}f_{2i} is a linear combination of graph maps as well, and each of these factors through P (i). Otherwise, if P (j) is a non-serial projective-injective for some j, Lemma 3.1 yields that f_{1j}f_{2j} is a linear combination of two-sided graph maps. But different graph maps are linearly independent, and the f_{a_{i}} are by assumption weakly one-sided. So all f_{a_{i}} with λ_{i} ≠ 0 must factor through a projective, a contradiction.

Lemma 3.3. Let f_{a} and f_{b} be weakly one-sided graph maps. If f_{a(l)} and f_{b(r)} are both left-sided, or both right-sided, and if \underline{f_{a}} ≠ 0 ≠ \underline{f_{b}}, then \underline{f_{a}f_{b}} ≠ 0.

Proof. Without loss of generality assume that f_{a} and f_{b} are both rightsided. Define f_{c} = f_{a}f_{b}. For the sake of brevity we just write ‘1’ instead of 1_{(e(E),−ϵ(E))}. Thus c is of the form

((D_{1}, E, 1), (D_{2}, E, 1)).

Assume that \underline{f_{a}} ≠ 0 ≠ \underline{f_{b}}. Note that this implies that M(D_{1}E) and M(D_{2}E) are both not projective. We have to show that \underline{f_{a}f_{b}} ≠ 0.

Assume that f_{c} = f_{1}f_{2} is a factorization of f through a projective module P. Since f_{c} is weakly one-sided, and since different graph maps are linearly independent, it follows from Lemma 3.1 that we can assume that P does not contain non-serial projective-injective direct summands. Using the multiplicativity of graph maps, we can assume without loss of generality that P = M(L^{−}RE) with L and R direct strings, |R| ≥ 1, and f_{1} and f_{2} right-sided graph maps. In particular, E must be a direct string. Note that D_{2} = D′_{2}R for some string D′_{2}. We know that a is of the form

((D_{11}, D_{12}E, 1), (D_{3}, D_{12}E, 1)),

and b is of the form

((D_{4}, D_{22}E, 1), (D_{21}, D_{22}E, 1)).

Now let D_{122} be a direct string of maximal length such that D_{12} = D_{121}D_{122} for some string D_{121}, and let D_{222} be a maximal direct string such that D_{22} = D_{221}D_{222} for some string D_{221}. Since f_{c} = f_{a}f_{b}, we get that |D_{12}| = 0 or |D_{22}| = 0.

If |D_{22}| = 0, then f_{b} clearly factors through P = M(L^{−}RE), a contradiction. Thus, assume |D_{22}| ≥ 1. This implies |D_{12}| = 0. If now |R| ≤ |D_{222}|, then f_{a} factors through P. Otherwise, if |R| > |D_{222}|, then D_{22} = D_{222}, since D_{2} = D′_{2}R and since b is admissible. This implies that f_{b} factors through P. This finishes the proof.

Proposition 3.4. Let A = kQ/(ρ) be a special biserial algebra, and let C_{i}, 1 ≤ i ≤ t, be strings for (Q, ρ). If each f_{a}, a ∈ A(C_{i}, C_{j}), 1 ≤ i, j ≤ t, is weakly one-sided or factors through a projective A-module, then \underline{End}_{A}(⨁_{i=1}^{t} M(C_{i})) is a gentle algebra.

Proof. Since we are interested in algebras only up to Morita equivalence, we can assume that the M(C_{i}) are pairwise not isomorphic. We can also assume that the M(C_{i}) are not projective. For the sake of brevity let M = ⨁_{i=1}^{t} M(C_{i}). By definition \underline{End}_{A}(M) = End_{A}(M)/P where P is the ideal of all endomorphisms which factor through projectives.

The graph maps in Hom_{A}(M(C_{i}), M(C_{j})), 1 ≤ i, j ≤ t, form a basis of End_{A}(M). By our assumption, we know that all two-sided graph maps are contained in P. This implies that the weakly one-sided graph maps generate \underline{End}_{A}(M) as a vectorspace. Let B be the set of weakly one-sided graph maps f_{a} in End_{A}(M) which satisfy \underline{f_{a}} ≠ 0. It follows form Lemma 3.2 that \underline{B} = \{\underline{f_{a}} | f_{a} ∈ B\} is a basis of \underline{End}_{A}(M). This basis behaves again multiplicative. Namely, let f_{a} and f_{b} be in B. Then it follows from Lemma 2.2 that \underline{f_{a}f_{b}} = 0, or \underline{f_{a}f_{b}} ∈ \underline{B}.

Now we can compute the quiver with relations of the algebra \underline{End}_{A}(M). The vertices of this quiver are the identity maps M(C_{i}) → M(C_{i}). The arrows are the maps \underline{f_{a}} in \underline{B} which are not of the form \underline{f_{b}}\underline{f_{c}} with \underline{f_{b}} and \underline{f_{c}} non-invertible elements in \underline{B}. Now we proceed as in the proof of [16, Proposition 4.8]. As in [16, Lemmas 4.1, 4.2 and Corollary 4.3] we get that there are at most two arrows ending and at most two arrows starting at each vertex of the quiver of \underline{End}_{A}(M). Thus we get property (1) in the definition of a gentle algebra. Then property (2) and (5) are proved as in [16, Lemma 4.4, 4.5]. However, to get (5) we have to use also Lemma 3.3. Since M is a finite-dimensional module, also \underline{End}_{A}(M) must be finite-dimensional, thus we get (3). Then we copy the proof of [16, Lemma 4.6] and use Lemma 3.3 to show that each path \underline{f_{a_{1}}} ⋯ \underline{f_{a_{l}}}, which is 0 in \underline{End}_{A}(M), has to be of length 2. As in [16, Lemma 4.7] we get that there are no commutativity relations. Thus property (4) holds. This finishes the proof.

The following lemma is based on an idea of Ringel.

Lemma 3.5. Let f_{a} : M(C_{1}) → M(C_{2}) be a two-sided graph map with a = ((D_{1}, E_{1}, F_{1}), (D_{2}, E_{2}, F_{2})). If D_{1}E_{1}F_{2} and D_{2}E_{1}F_{1} are strings, or if D_{1}E_{1}D_{2}^{−} and F_{2}^{−}E_{1}F_{1} are strings, then Ext_{A}^{1}(M(C_{2}), M(C_{1})) ≠ 0.

Proof. Without loss of generality assume E_{1} = E_{2}, and set E = E_{1}. Thus D_{1}ED_{2}^{−} and F_{2}^{−}EF_{1} cannot be strings. If D_{1}EF_{2} and D_{2}EF_{1} are strings, we get a short exact sequence

0 ⟶ M(D_{1}EF_{1}) ⟶ M(D_{1}EF_{2}) ⊕ M(D_{2}EF_{1}) ⟶ M(D_{2}EF_{2}) ⟶ 0

of A-modules, see [16] for a precise construction. Since a is two-sided, it follows that the middle term of this sequence is not isomorphic to the direct sum of its end terms. Thus the sequence does not split, which implies Ext_{A}^{1}(M(C_{2}), M(C_{1})) ≠ 0.

Corollary 3.6. Let f_{a} : M(C_{1}) → M(C_{2}) be a two-sided graph map with a = ((D_{1}, E_{1}, F_{1}), (D_{2}, E_{2}, F_{2})). If E_{1} is not a direct or an inverse string, then Ext_{A}^{1}(M(C_{2}), M(C_{1})) ≠ 0.

Proposition 3.7. Let f_{a} : M(C_{1}) → M(C_{2}) be a two-sided graph map, and assume Ext_{A}^{1}(M(C_{2}), M(C_{1})) = 0. Then \underline{f_{a}} = 0.

Proof. Let a = ((D_{1}, E_{1}, F_{1}), (D_{2}, E_{2}, F_{2})). Without loss of generality we can assume that E_{1} = E_{2}. Set E = E_{1}. If D_{1}EF_{2} and D_{2}EF_{1} are both strings, then by Lemma 3.5 we get Ext_{A}^{1}(M(C_{2}), M(C_{1})) ≠ 0, a contradiction.

Thus, assume that D_{2}EF_{1} is not a string. This implies that E must be a direct string.

Let F_{11} be a direct string of maximal length such that F_{1} = F_{11}F_{12} for some string F_{12}. Similarly, let D_{22} be a maximal direct string such that D_{2} = D_{21}D_{22} for some D_{21}. Observe that D_{22}EF_{11} is a string if and only if D_{2}EF_{1} is a string. Thus D_{22}EF_{11} is not a string. Since D_{22}E and EF_{11} are both strings, we get that |D_{22}| ≥ 1 and |F_{11}| ≥ 1. Thus f_{a} : M(C_{1}) → M(C_{2}) can be visualised as follows:

[figure: D_{1}; E; F_{11}; F_{12}; f_{a}; D_{21}; D_{22}; E; F_{2}; M(C_{1}); M(C_{2})]

Now let F_{111} be a maximal direct string such that D_{22}EF_{111} is still a string. Thus there exists a (direct) string F_{112} with |F_{112}| ≥ 1 such that F_{11} = F_{111}F_{112}. Next, let D_{212} be a maximal direct string such that D_{21} = D_{211}D_{212}^{−} for some string D_{211}. Let H be a maximal direct string such that D_{212}H is still a string. Define P = H^{−}D_{212}^{−}D_{22}EF_{111}. Since |D_{22}| ≥ 1 and |F_{112}| ≥ 1 we get a graph map f_{b} : M(C_{1}) → M(P) with

b = ((D_{1}, EF_{111}, F_{112}F_{12}), (H^{−}D_{212}^{−}D_{22}, EF_{111}, 1_{(e(F_{111}),−ϵ(F_{111}))})).

The corresponding picture looks as follows:

[figure: D_{1}; E; F_{111}; F_{112}; F_{12}; f_{b}; D_{22}; E; F_{111}; D_{212}; H; M(C_{1}); M(P)]

Observe that D_{211} is either of length 0 or ends with an arrow. Thus we get another graph map f_{c} : M(P) → M(C_{2}) with

c = ((H^{−}, D_{212}^{−}D_{22}E, F_{111}), (D_{211}, D_{212}^{−}D_{22}E, F_{2})),

and the corresponding picture is the following:

[figure: D_{22}; E; F_{111}; D_{212}; H; f_{c}; D_{212}; D_{211}; D_{22}; E; F_{2}; M(P); M(C_{2})]

Next, one checks easily that f_{a} = f_{b}f_{c}. Thus f_{a} factors through M(P). In case M(P) is projective, we get \underline{f_{a}} = 0.

Otherwise, the indecomposable projective module P_{j} with top S_{j}, j = s(D_{22}), must be a non-serial projective-injective module. Assume we are in this case. This implies that there exists a commutativity relation D_{212}Hα− D_{22}EF_{111}β in ρ where α and β are arrows. We know that D_{212}H = γK for some arrow γ and some (direct) string K, and D_{22} = δD′_{22} for some arrow δ and some (direct) string D′_{22}. Thus the radical rad(P_{j}) of P_{j} is isomorphic to M(D′_{22}EF_{111}βα^{−}K^{−}), and the socle factor P_{j}/soc(P_{j}) is isomorphic to M(P). The next picture describes P_{j}, its radical and socle factor together with the canonical inclusion and projection, respectively.

[figure: K; D′_{22}; E; F_{111}; α; β; ι; D_{212}; D_{22}; H; E; F_{111}; α; β; π; D_{212}; D_{22}; H; E; F_{111}; rad(P_{j}); P_{j}; P_{j}/soc(P_{j}) = M(P)]

If |F_{112}| > 1, then it is easy to check that f_{a} factors through P_{j}. Namely, we have F_{112} = βF′_{112} for some (direct) string F′_{112} of length at least 1. Then define f_{d} : M(C_{1}) → rad(P_{j}) where

d = ((D_{1}, EF_{111}β, F′_{112}F_{12}), (D′_{22}, EF_{111}β, α^{−}K^{−})).

Thus f_{d} looks as follows:

[figure: D_{1}; E; F_{111}; β; F′_{112}; F_{12}; f_{d}; K; D′_{22}; E; F_{111}; α; β; M(C_{1}); rad(P_{j})]

Now one checks easily that

f_{a} = f_{d}ιπf_{c}.

Thus f_{a} factors through a projective module.

Next, assume |F_{112}| = 1. This implies F_{112} = β. Let F_{121} be a maximal direct string such that F_{12} = F_{121}^{−}F_{122} for some string F_{122}. Then there is a graph map f_{e} : M(C_{1}) → rad(P_{j}) with

e = ((D_{1}, EF_{111}βF_{121}^{−}, F_{122}), (D′_{22}, EF_{111}βF_{121}^{−}, R^{−}))

where R is a (direct) string such that RF_{121} = Kα. Thus f_{e} looks as in the following picture:

[figure: D_{1}; E; F_{111}; β; F_{121}; F_{122}; f_{e}; R; D′_{22}; E; F_{111}; F_{121}; β; =; K; D′_{22}; E; F_{111}; α; β; M(C_{1}); rad(P_{j})]

If |F_{121}| < |Hα|, then one easily checks that

f_{a} = f_{e}ιπf_{c}.

Thus, again f_{a} factors through a projective.

It remains to consider the case |F_{121}| ≥ |Hα|. Note that this implies |D_{212}| ≥ 1. It follows that there is some (direct) string F_{1211} such that F_{121} = F_{1211}F_{1212} and D_{212} = D_{2121}F_{1211} for some (direct) strings F_{1212} and D_{2121}. Note that |F_{1212}| ≥ 1. Thus we get a graph map f_{l} : M(C_{1}) → M(C_{2}) with

l = ((D_{1}EF_{111}βF_{1212}^{−}, F_{1211}^{−}, F_{122}), (D_{211}, F_{1211}^{−}, D_{2121}^{−}D_{22}EF_{2})).

We get

f_{a} + f_{l} = f_{e}ιπf_{c}.

Define g = f_{a} + f_{l}. Clearly, we have \underline{g} = 0. The following picture illustrates the situation:

[figure: D_{1}; E; F_{111}; β; F_{1212}; F_{1211}; F_{122}; M(C_{1}); g; D_{211}; F_{1211}; D_{2121}; D_{22}; E; F_{2}; M(C_{2})]

Now we need some Auslander-Reiten theory: If |D_{1}| ≥ 1, then f_{a} factors through τ_{A}^{−1}M(C_{1}). This can be checked by using the construction of Auslander-Reiten sequences for string algebras as explained in [5]. Here we use also that |F_{111}| ≥ 1. Then we use the Auslander-Reiten formula

Ext_{A}^{1}(M(C_{2}), M(C_{1})) ≃ D\underline{Hom}_{A}(τ_{A}^{−1}M(C_{1}), M(C_{2})).

Thus, since Ext_{A}^{1}(M(C_{2}), M(C_{1})) = 0, we get \underline{f_{a}} = 0.

Similarly, if |F_{122}| ≥ 1, then f_{l} factors through τ_{A}^{−1}M(C_{1}). Here we use that |F_{1212}| ≥ 1. Again, since Ext_{A}^{1}(M(C_{2}), M(C_{1})) = 0, we get \underline{f_{l}} = 0. But if \underline{g} = \underline{f_{l}} = 0, then we get also \underline{f_{a}} = 0.

Thus, it remains to consider the case |D_{1}| = |F_{122}| = 0. Then we can construct a short exact sequence

0 ⟶ M(C_{1}) ⟶ P_{j} ⊕ M(D_{211}F_{1211}^{−}) ⊕ M(EF_{2}) ⟶^{h} M(C_{2}) ⟶ 0.

Here

h = [πf_{c}, f_{m}, f_{n}]^{t}

where f_{m} and f_{n} are graph maps with

m = ((1, D_{211}F_{1211}^{−}, 1), (1, D_{211}F_{1211}^{−}, D_{2121}^{−}D_{22}EF_{2}))

and

n = ((1, EF_{2}, 1), (D_{211}F_{1211}^{−}D_{2121}^{−}D_{22}, EF_{2}, 1)).

For the sake of brevity, in the definition of m and n we just wrote ‘1’ for all strings of length 0. Next, one checks easily that h is an epimorphism with kernel isomorphic to M(C_{1}). This sequence does obviously not split, thus Ext_{A}^{1}(M(C_{2}), M(C_{1})) ≠ 0, a contradiction. Altogether, we proved the following: If D_{2}EF_{1} is not a string, then the Proposition holds.

Thus, assume now that D_{2}EF_{1} is a string. If D_{1}EF_{2} is also a string, then we apply Lemma 3.5 and get that Ext_{A}^{1}(M(C_{2}), M(C_{1})) ≠ 0, again a contradiction. Otherwise, if D_{1}EF_{2} is not a string, then we proceed as before and get that the Proposition holds. This finishes the proof.

Proof of Theorem 1.1. Let A be a special biserial algebra, and let M be an A-module with Ext_{A}^{1}(M, M) = 0. Thus M does not contain direct summands which are isomorphic to band modules. Furthermore, since we are interested in stable endomorphism algebras, we can assume that M does not contain projective direct summands. Also, since we consider algebras only up to Morita equivalence, we assume that M is a direct sum of pairwise different indecomposable modules. Thus M is isomorphic to a direct sum ⨁_{i=1}^{t} M(C_{i}) of pairwise different non-projective string modules M(C_{i}).

The endomorphism algebra End_{A}(M) has a basis consisting of the graph maps in Hom_{A}(M(C_{i}), M(C_{j})), 1 ≤ i, j ≤ t. By Proposition 3.7 we know that any two-sided graph map M(C_{i}) → M(C_{j}) factorizes through a projective module. Hence we can apply Proposition 3.4 and get that \underline{End}_{A}(M) is a gentle algebra.

Proof of Corollary 1.2. If B is a finite-dimensional k-algebra, then \underline{mod}(B) is a cosuspended category with loop functor Ω, where Ω is the syzygy functor. For B-modules M and N and n ≥ 1 we have

\underline{Hom}_{B}(Ω^{n}(M), N) = Ext_{B}^{n}(M, N)

where the Ext_{B}^{n}(M, N) are the usual groups of module extensions. Now let A be a finite-dimensional k-algebra. Then there is a full and faithful embedding

F : D^{b}(A) ⟶ \underline{mod}(RA)

of triangulated categories, where RA is the repetitive algebra of A. For details we refer to [8, Chapter 2] and [9, Section 3]. For any two bounded complexes M and N of A-modules, we get

Hom_{D^{b}(A)}(M[−1], N) = \underline{Hom}_{RA}(F (M[−1]), F (N))
= \underline{Hom}_{RA}(Ω(F (M)), F (N))
= Ext_{RA}^{1}(F (M), F (N)),

since F is full, faithful and respects the triangulated structure.

Now assume that A is a finite-dimensional gentle algebra, and let T be a complex in D^{b}(A) such that Hom_{D^{b}(A)}(T, T [1]) = 0. Using the above considerations, we get Ext_{RA}^{1}(F (T), F (T)) = 0. Since A is gentle, we know that RA is special biserial. Furthermore, End_{D^{b}(A)}(T) and \underline{End}_{RA}(F (T)) are isomorphic. Thus we can apply Theorem 1.1 and get that End_{D^{b}(A)}(T) is a gentle algebra.

If T is a tilting complex, then by definition Hom_{D^{b}(A)}(T, T [i]) = 0 for every i ≠ 0. But two algebras A and B are derived equivalent if and only if there is a tilting complex T in D^{b}(A) such that B is isomorphic to End_{D^{b}(A)}(T), see [10] or [12]. Thus any algebra which is derived equivalent to a finite-dimensional gentle algebra is gentle again.

It is known that two derived equivalent finite-dimensional algebras A and B have the same number of isomorphism classes simple modules [10, Lemma 6.3.3]. Since there are (up to Morita equivalence) only finitely many gentle algebras with a given number of isomorphism classes of simple modules, also the last statement in Corollary 1.2 holds.

4. Examples and remarks

Example 1. Let A be the algebra k[x, y]/(x^{2}, y^{2}, xy) where k[x, y] is the polynomial algebra in two commuting variables. Observe that A is special biserial. Now let M be the string module M(x^{−}yx^{−}y). One easily checks that Ext_{A}^{1}(M, M) = 0 and End_{A}(M) = k[x_{1}, ⋯, x_{6}]/(x_{i}x_{j}, 1 ≤ i, j ≤ 6), which is obviously not gentle and also far from being special biserial. But we get \underline{End}_{A}(M) = k. Thus Theorem 1.1 cannot be extended to a result on endomorphism algebras, rather than stable endomorphism algebras.

Example 2. Now let A = k[x, y]/(x^{3}, y^{3}, xy) which is again special biserial. Let M be the band module M(x^{−}x^{−}yx^{−}yy, 1, 1), see [5] for the construction of M. Since M is a band module we know that Ext_{A}^{1}(M, M) ≠ 0. The endomorphism algebra End_{A}(M) is a factor of the free algebra in 5 noncommuting variables, and we get \underline{End}_{A}(M) = k. On the other hand, if we interpret M as a module over the algebra B = k[x, y]/(x^{5}, y^{5}, xy), then one easily checks that End_{A}(M) ≃ End_{B}(M) ≃ \underline{End}_{B}(M).

Example 3. The algebra A = k[x]/(x^{2}) is gentle, the simple module S satisfies Ext_{A}^{1}(S, S) ≠ 0, and End_{A}(S) = \underline{End}_{A}(S) = k is gentle.

Remark. There are still several important open problems: One needs criteria to decide when two given gentle algebras are derived equivalent.

Also it is still not clear whether tilting-cotilting equivalence is the same as derived equivalence. This holds in the case of gentle one-cycle algebras, see [3] and [4].

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Jan Schröer Department of Pure Mathematics University of Leeds Leeds LS2 9JT UK

   E-mail address: jschroer@maths.leeds.ac.uk

Alexander Zimmermann Faculté de Mathématiques et CNRS (LAMFA UMR 6140) Université de Picardie Jules Verne 33, rue St Leu 80039 Amiens FRANCE

   E-mail address: alexander.zimmermann@u-picardie.fr