Schemes of modules over gentle algebras and laminations of surfaces

We study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by dropping acyclicity, and by incorporating band modules. A special class of gentle algebras are Jacobian algebras arising from triangulations of unpunctured marked surfaces. For these we obtain a bijection between the set of generically tau-reduced decorated irreducible components and the set of laminations of the surface. As an application, we get that the set of bangle functions (defined by Musiker-Schiffler-Williams) in the upper cluster algebra associated with the surface coincides with the set of generic Caldero-Chapoton functions.


Introduction and main results
1.1. Overview. We study some geometric aspects of the representation theory of gentle algebras. This class of finite-dimensional algebras was defined by Assem and Skowroński [AS], who were classifying the iterated tilted algebras of path algebras of extended Dynkin type A. Gentle algebras are special biserial, which implies that their module categories can be described combinatorially, see [WW] and also [BR].
The irreducible components of the affine schemes of modules over gentle algebras are easy to classify (see Proposition 7.1). As a first main result, we describe all smooth points of these schemes, and we show that most components are generically reduced.
A special class of gentle algebras are Jacobian algebras arising from triangulations of unpunctured marked surfaces (S, M). For these we obtain a bijection between the set of generically τ -reduced decorated irreducible components and the set of laminations of the surface. This bijection is compatible with the parametrization of these two sets via g-vectors and shear coordinates. This bijection has some application to cluster algebras, a class of combinatorially defined commutative algebras discovered by Fomin and Zelevinsky [FZ]. Initially meant as a tool to describe parts of Lusztig's dual canonical basis of quantum groups in a combinatorial way, cluster algebras turned out to appear at numerous different places of mathematics and mathematical physics. The generically τ -reduced decorated components parametrize the generic Caldero-Chapoton functions, which belong to the coefficient-free upper cluster algebra U (S,M) associated with (S, M). In many cases, these generic Caldero-Chapoton functions are known to form a basis, called the generic basis, of U (S,M) , see for example [GLS] and [Q]. We use the bijection mentioned above to show that the generic basis coincides with Musiker-Schiffler-Williams' bangle basis (see [MSW2,Corollary 1.3]) of the coefficient-free cluster algebra A (S,M) associated with (S, M). It is known in most cases (for example, if |M| ≥ 2) that A (S,M) = U (S,M) , see [Mu1,Mu2].
In the following subsections we describe our results in more detail.
1.2. Gentle algebras. Let Q = (Q 0 , Q 1 , s, t) be a quiver. Thus by definition, Q 0 and Q 1 are finite sets, where the elements of Q 0 and Q 1 are the vertices and arrows of Q, respectively. Furthermore, s and t are maps s, t : Q 1 → Q 0 , where s(a) and t(a) are the starting vertex and terminal vertex of an arrow a ∈ Q 1 , respectively. A loop in Q is an arrow a ∈ Q 1 with s(a) = t(a).
A basic algebra A = KQ/I is a gentle algebra provided the following hold: (i) For each i ∈ Q 0 we have |{a ∈ Q 1 | s(a) = i}| ≤ 2 and |{a ∈ Q 1 | t(a) = i}| ≤ 2. (ii) The ideal I is generated by a set ρ of paths of length 2. (iii) Let a, b, c ∈ Q 1 such that a = b and t(a) = t(b) = s(c). Then exactly one of the paths ca and cb is in I. (iv) Let a, b, c ∈ Q 1 such that a = b and s(a) = s(b) = t(c). Then exactly one of the paths ac and bc is in I.
A gentle algebra A = KQ/I is a Jacobian algebra in the sense of [DWZ1] if and only if the following hold: (v) Q is connected. (vi) Q does not have any loops. (vii) Let a, b ∈ Q 1 such that s(a) = t(b) and ab ∈ I. Then there exists an arrow c ∈ Q 1 with s(c) = t(a) and t(c) = s(b) such that bc, ca ∈ I.
The gentle Jacobian algebras are exactly the Jacobian algebras associated to triangulations of unpunctured marked surfaces. This follows from [ABCP, Section 2].

1.3.
Smooth locus and generic reducedness of module schemes. Let Q be a quiver with Q 0 = {1, . . . , n}, and let A = KQ/I be a basic algebra. For d ∈ N n let Irr (A, d) be the set of irreducible components of the affine scheme mod (A, d)  Irr (A, d).
The group acts on the K-rational points of mod (A, d) by conjugation, where d = (d 1 , . . . , d n ). The orbit of M ∈ mod (A, d) is denoted by O M . The orbits in mod (A, d) correspond bijectively to the isomorphism classes of A-modules with dimension vector d.
For Z ∈ Irr(A, d) let Z • be the interior of Z. These are all M ∈ Z such that M is not contained in any other irreducible component of mod (A, d). Obviously Z • is a non-empty, open, irreducible subset of mod (A, d).
where T M is the tangent space of M at the affine scheme mod (A, d). Otherwise, M is singular. Let smooth (A, d) denote the set of smooth points of mod (A, d).
For each gentle algebra A we obtain a complete description of smooth points of mod (A, d) for all d, see Theorem 7.6. As a consequence we get the following neat characterization for the case of gentle Jacobian algebras. Note that the inclusion ⊆ in Theorem 1.1 is true for arbitrary basic algebras A. The other inclusion ⊇ is wrong in general. For example, it fails for most gentle algebras which are not Jacobian algebras.
M is the tangent space of M at the reduced affine scheme mod(A, d) red associated with mod (A, d). We call mod (A, d) reduced if mod (A, d) = mod (A, d) red . This is the case if and only if M is reduced for all M ∈ mod (A, d).
An irreducible component Z ∈ Irr (A) is generically reduced provided Z contains a dense open subset U such that each M ∈ U is reduced.
Theorem 1.2 (Generic reducedness). Let A be a gentle algebra without loops. Then each Z ∈ Irr(A) is generically reduced.
We prove a slightly more general version of Theorem 1.2 where we characterize all generically reduced components for arbitrary gentle algebras, see Theorem 7.4.
For acyclic gentle algebras, Theorem 1.2 is a consequence of [DS]. Here τ A denotes the Auslander-Reiten translation of A.
For each Z ∈ Irr (A) there is a dense open subset U ⊆ Z such that the maps c A , e A and h A are constant on U . These generic values are denoted by c A (Z), e A (Z) and h A (Z).

It follows that
idempotents of A by e 1 , . . . , e n . Let a ∈ Q 1 . Then we are in one of the following two cases: (i) There is no arrow b ∈ Q 1 with s(a) = t(b) such that ab ∈ I. In this case, the 3-dimensional subalgebra of A spanned by e s(a) , e t(a) and a is called a 2-block of A. (ii) There are arrows b, c ∈ Q 1 with s(a) = t(b), s(c) = t(a) and s(b) = t(c) such that ab, ca, bc ∈ I. In this case, the 6-dimensional subalgebra of A spanned by e s(a) , e s(b) , e s(c) , a, b and c is called a 3-block of A.
In the special case where the quiver Q consists just of a single vertex, we call A itself a 1-block. A ρ-block of A is a subalgebra which is either a 1-block, 2-block or 3-block. (Note that the ρ-blocks are not necessarily unital subalgebras, i.e. the unit of a ρ-block of A does in general not coincide with the unit of A.) We say that a vertex j ∈ Q 0 or an arrow a ∈ Q 1 belongs to a ρ-block A i of A if e j ∈ A i or a ∈ A i , respectively. Note that each arrow of Q belongs to exactly one ρ-block of A, and each vertex of Q belongs to at most two ρ-blocks.
In Section 4 we extend this observation to arbitrary basic algebras A = KQ/I. This reduces the study of schemes of modules over gentle algebras to schemes of complexes.
Our next result characterizes the generically τ -reduced components of a gentle Jacobian algebra in terms of the generically τ -reduced components of its ρ-blocks.
The fact that the generic reducedness or the smooth locus of a component Z relate to the generic reducedness or the smooth locus of the components π i (Z) does not come as a surprise. The following result however is somewhat unexpected, since the Auslander-Reiten translation for A is quite different from the Auslander-Reiten translations for the ρ-blocks of A.
Theorem 1.4. Let A = KQ/I be a gentle Jacobian algebra, and let A 1 , . . . , A t be its ρ-blocks. For an irreducible component Z ∈ Irr(A) the following are equivalent: One might ask if Theorem 1.4 holds for arbitrary finite-dimensional K-algebras using of course a generalized definition for ρ-blocks.
1.5. Band components. The indecomposable modules over a gentle algebra A (or more generally, over a string algebra) are either string modules or band modules, see [BR, WW] for details. The band modules occur naturally in K * -parameter families. An irreducible component Z ∈ Irr (A) is a string component if it contains a string module whose orbit is dense in Z, and Z is a band component if it contains a K * -parameter family of band modules whose union of orbits is dense in Z.

An irreducible component Z ∈ Irr(A) is a brick component if it contains a brick,
i.e. an A-module M with dim End A (M ) = 1. In this case, by upper semicontinuity the bricks in Z form a dense open subset of Z.
Theorem 1.5. Let A be a gentle algebra. Then each band component is a brick component.
Using the terminology of [CBS], each irreducible component Z ∈ Irr(A) is a direct sum of uniquely determined indecomposable irreducible components. The string and band components are the only indecomposable components for string algebras.
The generically τ -reduced string components are exactly the components containing an indecomposable τ -rigid module, which is then automatically a string module.
Theorem 1.6. Let A be a gentle algebra. For Z ∈ Irr(A, d) the following are equivalent: (i) Z is a direct sum of band components.
In this case, Z is generically τ -reduced.
Theorem 1.6 is closely related to the seemingly different [CCKW,Proposition 4.3]. The proofs follow the same line of arguments. We thank Ryan Kinser for pointing this out to us.
For acyclic gentle algebras, Theorems 1.5 and 1.6 can be extracted from Carroll and Chindris [CC,Corollary 10] and [CC,Proposition 11], see also [C,Theorem 2]. As a consequence of Theorem 1.5, one gets the known result that a gentle algebra A is representation-finite if and only if mod(A) contains just finitely many bricks, compare [P2, Theorem 1.1].
1.6. Laminations of marked surfaces and generically τ -reduced components. A lamination of an unpunctured marked surface (S, M) is a set of homotopy classes of curves and loops in (S, M), which do not intersect each other, together with a positive integer attached to each class. Let Lam(S, M) be the set of such laminations. (For more precise definitions, we refer to Section 10.) Let T be a triangulation of (S, M), and let A T be the associated gentle Jacobian algebra. A decorated irreducible component is roughly speaking an irreducible component of mod(A T , d) equipped with a certain integer datum. Similarly as before, one defines generically τ -reduced decorated irreducible components. Let decIrr τ (A T ) be the set of all generically τ -reduced decorated components of decmod (A T , (d, v)), where (d, v) runs over all dimension vectors. A precise definition can be found in Section 9.
Theorem 1.7. Let (S, M) be an unpunctured marked surface, and let T be a triangulation of (S, M). Let A = A T be the associated Jacobian algebra. Then there is a natural bijection η T : Lam(S, M) → decIrr τ (A).
In their ground breaking work, Fomin, Shapiro and Thurston [FST] proved that the laminations of (S, M) consisting of curves are in bijection with the cluster monomials of a cluster algebra A (S,M) associated with (S, M). Note that Fomin, Shapiro and Thurston work with cluster algebras with arbitrary coefficient systems, whereas we always assume that A (S,M) is a coefficient-free cluster algebra.
Musiker, Schiffler and Williams [MSW2] defined a set of bangle functions, whose elements are parametrized by Lam(S, M), and which (by results in [MSW1]) contains all cluster monomials. They show that B T forms a basis of A (S,M) provided |M| ≥ 2, see [MSW2,Corollary 1.3].
A result by W. Thurston (see [FT,Theorem 12.3]) says that there is a bijection sending a lamination to its shear coordinate. Combining a theorem by Brüstle and Zhang [BZ,Theorem 1.6] with a result by Adachi, Iyama and Reiten [AIR,Theorem 4.1], one gets a bijection between the laminations in Lam(S, M) which consist only of curves, and the set of generically τ -reduced decorated components in decIrr τ (A T ), which contain a dense orbit. On the other hand, Plamondon [P1] proved that there is a bijection g T : decIrr τ (A T ) → Z n sending a component to its g-vector. Theorem 1.7 extends Brüstle-Zhang's bijection mentioned above to a bijection be the set of generic Caldero-Chapoton functions as defined in [GLS]. As a consequence of more general results in [DWZ2], the set G T is contained in the upper cluster algebra U (S,M) and contains all cluster monomials. Furthermore, by [P1,Theorem 1.3], the set G T is (in a certain sense) independent of the choice of the triangulation T of (S, M).
The proof of the next theorem is based on the bijection from Theorem 1.7.
The diagram in Figure 1 summarizes the situation.

1.7.
Overall structure of the article. The article is organized as follows. After the introduction (Section 1), we recall in Section 2 some fundamentals on schemes of modules over basic algebras. Section 3 contains a characterization of generically τ -reduced components for tame algebras. In Section 4 we introduce ρ-block decompositions of schemes of modules and derive some consequences on tangent spaces. Section 5 contains a few facts on the representation theory of gentle algebras. We also recall the definition of rank functions of modules over gentle algebras. Section 6 consists of a detailed study of schemes of complexes. We determine their smooth points, and we describe all rigid and τ -rigid modules over the associated basic algebras. In Section 7 we apply the results obtained in Section 6 and prove Theorems 1.1, 1.2, 1.6 and 1.5. The proofs of Theorems 1.3 and 1.4 can be found in Section 8. In Section 9 we recall some basics on decorated modules and schemes of decorated modules over finite-dimensional algebras. Section 10 contains the proof of Theorem 1.7, and also the proof that under the bijection in Theorem 1.7, shear coordinates and g-vectors are compatible. Theorem 1.8 is proved in Section 11. In Section 12 we illustrate the combinatorics used in Section 11 by an example.

Scheme of modules
In this section, we recall some definitions and elementary facts on the representation theory of basic algebras and on schemes of modules over such algebras. Throughout, let K be an algebraically closed field.
A path in Q is a tuple p = (a 1 , . . . , a m ) of arrows a i ∈ Q 1 such that s(a i ) = t(a i+1 ) for all 1 ≤ i ≤ m−1. Then length(p) := m is the length of p, and we set s(p) := s(a m ) and t(p) := t(a 1 ). Additionally, for each vertex i ∈ Q 0 there is a path e i of length 0, and let s(e i ) = t(e i ) = i. We often just write a 1 · · · a m instead of (a 1 , . . . , a m ). A path p = (a 1 , . . . , a m ) of length m ≥ 1 is a cycle in Q, or more precisely an m-cycle in Q, if s(p) = t(p).
Let KQ be the path algebra of Q, and let m be the ideal generated by the arrows of Q. An ideal I of KQ is admissible if there exists some m ≥ 2 such that m m ⊆ I ⊆ m 2 . In this case, we call A := KQ/I a basic algebra. Clearly, basic algebras are finite-dimensional. By a Theorem of Gabriel, each finite-dimensional K-algebra is Morita equivalent to a basic algebra.
where the p i are pairwise different paths of length at least 2 in Q with s(p i ) = s(p j ) and t(p i ) = t(p j ) for all 1 ≤ i, j ≤ s and λ i ∈ K * for all i.
Each admissible ideal is generated by a finite set of relations.
Let A = KQ/I be a basic algebra. Up to isomorphism, there are n simple A-modules S 1 , . . . , S n corresponding to the vertices of Q. Let P 1 , . . . , P n (resp. I 1 , . . . , I n ) be the projective covers (resp. injective envelopes) of the simple modules S 1 , . . . , S n .
For a path p = (a 1 , . . . , a m ) in Q and a representation M as above, let A representation of a basic algebra A = KQ/I is a representation M of Q, which is annihilated by the ideal I, i.e. for each relation s j=1 λ j p j in I we demand that s j=1 λ j M (p j ) = 0.
In the usual way, we identify the category rep(A) of representations of A with the category mod(A) of finite-dimensional left A-modules.
For d = (d 1 , . . . , d n ) ∈ N n let mod (A, d) be the affine scheme of representations of A with dimension vector d. By definition the K-rational points of mod (A, d) are When there is no danger of confusion, we will just write mod (A, d) for the set of K-rational points of mod (A, d). One can regard mod(A, d) as a Zariski closed subset of the affine space mod (Q, d) The group GL d (K) acts on the K-rational points of mod(A, d) by conjugation. More precisely, for g = (g 1 , . . . , g n ) ∈ GL d (K) and M ∈ mod(A, d) let The following lemma is obvious.
Lemma 2.1. For M ∈ mod(A, d) the following are equivalent: For the following proposition we refer to Gabriel [Ga,Proposition 1.1] and Voigt [V].
The following three results are well known and can be extracted e.g. from [H,Sh1,Sh2].
Proposition 2.8. Let Z ∈ Irr (A, d). Then the smooth points in Z form a (possibly empty) open subset of Z.
Proposition 2.9. Let M ∈ mod (A, d) be contained in at least two different irreducible components. Then M is singular.
The following statement is proved in [G,Proposition 3.7]. It relies on results from [GP].

Generically τ -reduced components for tame algebras
In this section, we characterize the indecomposable τ -reduced components for tame algebras. The proof consists basically of combining some known results in a straightforward manner.
Let A be a finite-dimensional K-algebra. Then A is a tame algebra if for each dimension d there exists a finite number M 1 , . . . , M t of A-K[X]-bimodules M i , which are free of rank d as K[X]-modules, such that all but finitely many d-dimensional A-modules are isomorphic to for some 1 ≤ i ≤ t and some λ ∈ K.
The following lemma is well known folklore. A proof can be found in [CC,Section 2.2]. (a) Z is generically τ -reduced.
(b) Z contains a rational curve C such that the points of C are pairwise non-isomorphic bricks.
(c) Z contains infinitely many pairwise non-isomorphic bricks. In this case, Proof. Part (i) follows directly from the definitions. Thus assume c A (Z) = 1. By Lemma 3.1, Z contains a rational curve C such that the points of C are pairwise non-isomorphic indecomposable modules with For an arbitrary finite-dimensional K-algebra A, each generically τ -reduced component Z ∈ Irr(A) is a direct sum of indecomposable generically τ -reduced components. This is explained in Section 9.5.

ρ-block decomposition and tangent spaces
Let A = KQ/I, where KQ is a path algebra and I is an admissible ideal generated by a set ρ = {ρ 1 , . . . , ρ m } of relations.

For each
be the smallest subquiver of Q containing the paths p i . Of course, these subquivers might overlap for different relations.
For arrows a, b ∈ Q 1 write a ∼ b if there is some k with a, b ∈ Q(ρ k ). Let ∼ be the smallest equivalence relation on Q 1 respecting this rule. In particular, each a ∈ Q 1 which is not contained in any of the Q(ρ k ) forms its own equivalence class.
Each equivalence class in Q 1 with respect to ∼ gives rise to a subquiver of Q and also to a subalgebra of A. These subalgebras are the ρ-blocks of A. Each vertex i ∈ Q 0 , which has no arrow attached to it yields a 1-dimensional subalgebra (with basis e i ). Such subalgebras are also called ρ-blocks of A.
Not under this name and for a different purpose (tameness proofs), ρ-blocks appear already in [B], see also [AB]. We thank Thomas Brüstle for pointing this out.
Let us remark that each arrow of Q belongs to exactly one ρ-block, whereas a standard idempotent e i can belong to several ρ-blocks. For an arrow a which does not appear in any of the relations in ρ, the path algebra of the quiver is a ρ-block. For example, let Q be the quiver 1 a 1 G G 2 G G · · · a n−1 G G n and let A = KQ. (In this trivial example, we have I = 0 and ρ = ∅.) For n ≥ 2 the ρ-blocks of A are the path algebras of the subquivers where 1 ≤ i ≤ n − 1. For n = 1 there is only one ρ-block, namely A itself.
As another example, let Q be the quiver Then KQ/I is a gentle Jacobian algebra, and there are two ρ-blocks with three vertices and one ρ-block with two vertices. (This algebra arises from a torus with one boundary component and one marked point on the boundary.) Our ρ-blocks are in general very different from the classically defined blocks of an algebra. However, on the geometric level there is at least some resemblance. This will be explained at the end of this subsection. Now let A 1 , . . . , A t be the ρ-blocks of A. For each dimension vector d ∈ N n and 1 ≤ i ≤ t let π i (d) denote the corresponding dimension vector for A i . Each M ∈ mod(A, d) induces via restriction modules π i (M ) ∈ mod(A i , π i (d)) for 1 ≤ i ≤ t in the obvious way.
Proposition 4.1. Let A and A 1 , . . . , A t be defined as above. For M ∈ mod(A, d) the following hold: Proof. (i): Obvious.
(ii): For a ring R let nil(R) be its ideal of nilpotent elements. For R commutative and finitely generated, let Spec(R) be as usual its prime ideal spectrum, which is an affine scheme.
We have an isomorphism of affine schemes Let R i be the coordinate algebra of mod(A i , π i (d)) for 1 ≤ i ≤ t. We get an isomorphism of affine schemes Let B and C be finitely generated commutative K-algebras. Then one easily shows that This yields Applying this via induction to the situation above, we get We get which implies (ii).
Proposition 4.1 allows us to study the tangent spaces of mod (A, d) in terms of the often easier to compute tangent spaces of mod(A i , π i (d)).
Corollary 4.2. Let M ∈ mod (A, d). Then the following are equivalent: Corollary 4.3. Let M ∈ mod (A, d). Then the following are equivalent: Corollary 4.4. Let Z ∈ Irr (A). Then the following are equivalent: For the basic algebra A = KQ/I, let Q(1), . . . , Q(m) be the connected components of the quiver Q. For 1 ≤ i ≤ m let I(i) := I ∩ KQ(i). Then I(i) is generated by a subset ρ(i) of ρ. With B i := KQ(i)/I(i) we get an algebra isomorphism The B i are indecomposable algebras, i.e. they are not isomorphic to the product of two algebras of smaller dimension. In other words, the B i are the classical blocks of A. For a dimension vector d ∈ N n let d(i) be the corresponding dimension vector for B i . We get an isomorphism of affine schemes. The ρ-blocks of A are the disjoint union of the ρ(i)-blocks of the B i .

Modules over gentle algebras
Throughout this section, let A = KQ/I be a gentle algebra with Q = (Q 0 , Q 1 , s, t).
It is straightforward to see that such maps σ and ε exist. We fix σ and ε for the rest of this section.
Later on we will define 1-sided and 2-sided standard homomorphisms. To make this anambiguous, we need the functions σ and τ . 5.2. Strings. For each arrow a ∈ Q 1 we introduce a formal inverse a − . We extend the maps s, t by defining s(a − ) := t(a) and t(a − ) := s(a). We also set (a − ) − = a. Let Q − 1 = {a − | a ∈ Q 1 } be the set of inverse arrows. Now a string C of length l(C) := m ≥ 1 is an m-tuple C = (c 1 , . . . , c m ) such that the following hold: We often just write C = c 1 · · · c m instead of C = (c 1 , . . . , c m ). Let C − := (c − m , . . . , c − 1 ) be the inverse of C, which is obviously again a string.
Sometimes we will just write 1 i instead of 1 (i,t) .
We extend the maps σ and ε to strings as follows: For strings C = (c 1 , . . . , c p ) and D = (d 1 , . . . , d q ) of length p, q ≥ 1, the composition of C and D is defined, provided (c 1 , . . . , c p , d 1 , . . . , d q ) is again a string. We write then CD = c 1 · · · c p d 1 · · · d q . Now let C be any string. The composition of 1 (u,t) and C is defined if t(C) = i and ε(C) = t. In this case, we write 1 (i,t) C = C. The composition of C and 1 (i,t) is defined if s(C) = i and σ(C) = −t. In this case we write C1 (i,t) = C.
If C and D are arbitrary strings such that the composition CD is defined, then For a string C we write C ∼ C − . This defines an equivalence relation on the set of all strings. Let S denote a set of representatives of all equivalence classes of strings for A.
When visualizing a string we draw an arrow a ∈ Q 1 often pointing from northeast to southwest: • Note that in this picture the arrow a − carries just the label a.

Example. Let again
is a string, which looks as follows: . . , c m ) be a string of length m ≥ 1. We define a string module M (C) as follows: The module M (C) has a standard basis (b 1 , . . . , b m+1 ). The generators of the algebra A act on this basis as follows: For i ∈ Q 0 and 1 ≤ j ≤ m + 1 we have and for a ∈ Q 1 and 1 ≤ j ≤ m + 1 we have

Bands. A band for
A is a string B such that the following hold: • l(B) ≥ 2; • B t is a string for all t ≥ 1; • B is not of the form C s for some string C and some s ≥ 2.
Let B be a band, and let C and D be strings such that B = CD. Then DC is a rotation of B. Obviously, any rotation of B is again a band. We write This yields an equivalence relation on the set of all bands for A. Let B be a set of representatives of all equivalence classes of bands for A.
As an example, let A = KQ/I as in Section 5.3. Then 5.6. Band modules. Now let B = (c 1 , . . . , c m ) be a band, and let λ ∈ K * . We define a band module M (B, λ, 1) as follows: The module M (B, λ, 1) has a standard basis (b 1 , . . . , b m ). The generators of the algebra A act on this basis as follows: For i ∈ Q 0 and 1 ≤ j ≤ m we have and for a ∈ Q 1 and 1 ≤ j ≤ m we have For q ≥ 2 and λ ∈ K * there are also band modules M (B, λ, q). They do not play a major role in this article, so we omit their definition. Let us just mention that they form Auslander-Reiten sequences for q ≥ 2. For q ≥ 1, we say that M (B, λ, q) has quasi-length q.

5.7.
Classification of modules. The following classification theorem was first proved by Wald and Waschbüsch [WW] using covering techniques. There is an alternative proof by Butler and Ringel [BR] using functorial filtrations. Both articles [BR] and [WW] also contain a combinatorial description of all Auslander-Reiten sequences for string algebras. Recall that all gentle algebras are string algebras.
Theorem 5.1. Let A = KQ/I be a gentle algebra. The modules M (C) and M (B, λ, q) with C ∈ S, B ∈ B, λ ∈ K * and q ≥ 1 are a complete set of pairwise non-isomorphic representatives of isomorphism classes of indecomposable modules in mod (A).
5.8. Homomorphisms. For a string C we define S(C) as the set of triples (D, E, F ) such that the following hold: (ii) Either l(D) = 0, or D = D a − for some a ∈ Q 1 and some string D ; (iii) Either l(F ) = 0, or F = bF for some b ∈ Q 1 and some string F . Following our convention for displaying strings, a triple (D, E, F ) ∈ S(C) with l(D), l(F ) ≥ 1 yields the following picture, where the left (resp. right) hand red line stands for the string D (resp. F ), and the blue line stands for E.
We clearly see that M (C) has a submodule isomorphic to M (E) and that the corresponding factor module is isomorphic to M (D ) ⊕ M (F ). Let be the obvious canonical inclusion.
Dually, for a string C we define F(C) as the set of triples (D, E, F ) such that the following hold: (i) C = DEF ; (ii) Either l(D) = 0, or D = D a for some a ∈ Q 1 and some string D ; For such a (D, E, F ) ∈ F(C) with l(D), l(F ) ≥ 1 we get the following picture, where the left (resp. right) hand green line stands for the string D (resp. F ), and the blue line stands for E. For a pair (C 1 , C 2 ) of strings we call a pair ) is 2-sided. Let h be admissible as above, and let The following picture describes f h for the case E 1 = E 2 and l( Depending if some of the four strings D 1 , F 1 , D 2 , F 2 are of length 0 or not, there are 16 different types of oriented standard homomorphisms.  [BR].) Therefore we can restrict our attention to band modules of quasi-length 1.
Krause [K] extended Theorem 5.2 to homomorphisms also involving band modules. We just recall a special case here, where we only consider band modules of quasi-length 1.

For a band B let
Let B 1 and B 2 be bands, and let C be a string. Let , respectively. All of these are 2-sided. This involves of course a choice of scalars λ 1 and/or λ 2 , in case we deal with B 1 and/or B 2 . For a band module M (B, λ, 1), the identity is also called a standard homomorphism. Similarly as before, we call f h oriented if E 1 = E 2 . For further details we refer to [K].
Theorem 5.3 ( [K]). For M and N string modules or band modules of quasi-length 1, the set of standard homomorphisms M → N is a basis of Hom A (M, N ). 5.9. Auslander-Reiten translation of string modules. Let A be a gentle algebra, and let M ∈ mod(A) be a non-projective string module. It follows that τ A (M ) is also a string module, and that we are in one of the five situations displayed in Figure 2, see [BR,Section 3]. (We use here the same way of illustrating strings and string modules as in [Sch,Section 3].) The subfactor of M and τ A (M ) defined by the string between the two red points is called the core of M . (In the 5th case, the core is just the 0-module.) The core of M does not change under the Auslander-Reiten translation.
The strings E i in Figure 2 are left-bounded direct strings, and the strings F i are right-bounded direct strings. The strings E 1 a − 1 and a 2 E − 2 are hooks in the sense of [BR], and the strings F − 1 b 1 and b − 2 F 2 are cohooks in the sense of [BR]. For each arrow a = a 1 = b 2 ∈ Q 1 there is exactly one Auslander-Reiten sequence of type 5. In this case, there is a string which yields the middle term of an Auslander-Reiten sequence All other Auslander-Reiten sequences involving string modules are of types 1, . . . , 4, and their middle terms are a direct sum of two indecomposable string modules. For details we refer to [BR].
5.10. Auslander-Reiten formulas. The following is a well known statement from Auslander-Reiten theory, see for example [ARS, ASS, R].
Theorem 5.4 (Auslander,Reiten). Let A be a finite-dimensional basic algebra. For M, N ∈ mod(A) the following hold: . Lemma 5.5. Let A be a gentle algebra. For any band module M ∈ mod(A) the following hold: Proof. (i): This is well known, see for example [BS,Corollary 3.6].
(ii): This is proved for example in [BR,Section 3].
Note that part (ii) of the above lemma holds also for all string algebras A.
Corollary 5.6. Let A be a gentle algebra, and let M, N ∈ mod (A). If M is a band module, then 5.11. Rank functions for gentle algebras. Let A = KQ/I be a gentle algebra, and let d ∈ N n be a dimension vector. A map r : Q 1 → N is a rank function for (A, d) if the following hold: (Using a slightly different wording, this definition appears in [CC,Section 5].) For M ∈ mod(A) the rank function of M is defined by One easily checks that r M is a rank function for (A, d) where d = dim(M ). Furthermore, each rank function for (A, d) is obtained in this way.
The following lemma is well known and follows directly from the definitions of string and band modules. Since each A-module is isomorphic to a direct sum of string modules and band modules, the claim follows.
Let r and r be rank functions for (A, d). We write r ≤ r if r(a) ≤ r (a) for all a ∈ Q 1 . This defines a partial order on the set of rank functions for (A, d).
For a rank function r for (A, d) let This is a non-empty closed subset of mod (A, d).

Schemes of complexes
As already mentioned in the introduction, the study of schemes of modules over gentle algebras can (to a large extent) be reduced to schemes of complexes. This section deals with all necessary results on schemes of complexes.
6.1. Definition of schemes of complexes. For n ≥ 1 let and I is the ideal generated by all paths of length 2. (For n = 1, Q has just one vertex and no arrows. For n = 1, 2, we set I = 0.) For n ≥ 1 let C n := KQ/I, where Q is the quiver 1 a 1 G G 2 a 2 G G · · · a n−2 G G n − 1 a n−1 G G n an g g and I is the ideal generated by all paths of length 2. For C n we adopt the convention that all indices are meant modulo n.
Let A be one of the algebras C n or C n . By scheme of complexes we mean the affine schemes mod (A, d) with d ∈ N n . This definition is a bit more general than the one used by De Concini and Strickland [DS], who consider only the case C n .
The representation theory of A is extremely well understood. Obviously, A is a representation-finite gentle algebra. So all its indecomposable modules are string modules. For each vertex i ∈ Q 0 there is a simple module S i and an indecomposable projective modules P i , and these are all indecomposable A-modules up to isomorphism. The modules S 1 , . . . , S n , P 1 , . . . , P n are pairwise non-isomorphic, with the exception of P n being equal to S n in case A = C n . Using the usual notation for string modules, for each i ∈ Q 0 we have S i = M (e i ) and It is straightforward to compute homomorphism spaces and extension groups between A-modules. All this can be proved in an elementary fashion using mainly Linear Algebra. The next two lemmas contain all the homological data we need.
where i ∈ Q 0 and a ∈ Q 1 . In these cases, we have dim Hom A (X, Y ) = 1 with only one exception for A = C 1 , where we have dim Hom A (P 1 , P 1 ) = 2.
where a ∈ Q 1 . In these cases, we have dim Ext 1 A (X, Y ) = 1.
Let A be one of the algebras C n or C n . Let d = (d 1 , . . . , d n ) ∈ N n be a dimension vector, and let r be a rank function for (A, d). Then there exists a unique (up to if i = 1 and A = C n , r(a n−1 ) if i = n and A = C n , r(a n ) + r(a n−1 ) if i = n and A = C n . Proposition 6.3. Let A be one of the algebras C n or C n , and let d ∈ N n . For each rank function r for (A, d) we have Now the claim follows from [Z, Theorem 1 and its Corollary].
Corollary 6.4. Let A be one of the algebras C n or C n , and let d ∈ N n . For each M = M d,r the following are equivalent: (ii) The rank function r is maximal.
Corollary 6.5. Let A be one of the algebras C n or C n , and let d ∈ N n . Then Furthermore, this becomes an equality if and only if M does not have a simple direct summand.
Proof. We have The claim follows.
Proposition 6.7 (Rigid modules). Let A be one of the algebras C n or C n , and let d ∈ N n . For M ∈ mod(A, d) the following are equivalent: For A = C 1 we assume now additionally that d = (d 1 ) with d 1 even. Then the two conditions above are equivalent to the following: Proof. The equivalence (i) ⇐⇒ (ii) follows from Lemma 6.2. The implication (i) =⇒ (iii) is true in general and follows from Voigt's Lemma 2.2.
(iii) =⇒ (ii): Assume that (ii) does not hold. Thus there is an arrow a such that S a is isomorphic to a direct summand of M .
Thus M is properly contained in the orbit closure of For A = C 1 and d = (d 1 ) with d 1 even, we get that M has a direct summand isomorphic to S s(a) ⊕ S s(a) . (Here we used that d 1 is even.) We get a non-split short exact sequence Thus M is properly contained in the orbit closure of ).
In both case, this shows that O M is not open.
The module S a in Proposition 6.7(ii) is a critical summand of type I of M . In Proposition 6.7(ii) we have Recall that a τ -rigid module is automatically rigid. Thus to get a decription of all τ -rigid modules, it suffices to look at rigid modules.
Proposition 6.8 (τ -rigid modules). Let A be one of the algebras C n or C n , and let d ∈ N n . For a rigid M ∈ mod(A, d) the following are equivalent: then the rigid module M has a direct summand isomorphic to S a , a contradiction to Proposition 6.7. If X ∼ = P t(a) , then X ⊕ Y ∼ = P a . This proves the claim.
The module P a in Proposition 6.8(ii) is a critical summand of type II of M . 6.3. Generic reducedness and singular locus. Proposition 6.9. Let A = KQ/I be one of the algebras C n or C n , and let d = (d 1 , . . . , d n ) ∈ N n . For Z ∈ Irr(A, d) the following are equivalent: (i) Z is not generically reduced.
(ii) A = C 1 and d 1 is odd.
Proof. (i) =⇒ (ii): Suppose that (ii) does not hold. Then it follows from Proposition 6.7 that Z contains a rigid module M . Then Z = O M and Z is generically reduced by Corollary 2.5.
(ii) =⇒ (i): Assume that (ii) holds. Then In particular, M is not rigid and therefore Z is not generically reduced, again by Corollary 2.5. Proposition 6.9 is not really original. Using very different methods, it is shown in [DS,Theorem 1.7] that mod (C n , d) is reduced for all d. Reducedness is in general a much stronger and harder to prove property than being generically reduced. Also the schemes mod( C n , d) should be reduced provided n ≥ 2. A proof for n = 2 is in [St,Proposition 1.3]. Proof. Let r be a rank function for (A, d) and let For A = C n we adopt the convention that P j = S j = 0 and r j = q j = 0 for all j / ∈ Q 0 , and for A = C n we use all indices modulo n.
Case 1: A = C 1 or A = C 2 . In this case, mod (A, d) is always an affine space. Therefore all modules M are smooth. On the other hand condition (ii) is never satisfied. This proves (i) ⇐⇒ (ii).
Thus M is singular.
This shows that M is singular if and only if q 1 , q 2 ≥ 1. But this condition is equivalent to (ii).
Case 4: n ≥ 3. Let Case 4(a): Assume that q i , q i+1 , q i+2 ≥ 1 and q i + q i+2 > q i+1 for some i. Similarly as in Case 3 one shows that M is contained in at least two different irreducible components of mod (A, d). Thus M is singular.
Case 4(b): Assume that for all i with q i , q i+1 , q i+2 ≥ 1 we have q i + q i+2 ≤ q i+1 . It follows immediately that q i−1 = q i+3 = 0 for all such i. In other words, we have i ∈ H 3 .
We get that M is contained in exactly one irreducible component Furthermore, for each i ∈ H 3 replace The module N is rigid and therefore smooth.
Thus M is smooth if and only if H 3 = ∅. This finishes the proof.
In Proposition 6.10(ii) we have Consequently, we have The singularities of the closures of the GL d (K)-orbits of the schemes mod(C n , d) have been described by Lakshmibai [L] for n = 3 and by Gonciulea [Go] for arbitrary n. Note the difference to Proposition 6.10, where we look at the singularities of the whole scheme.
6.4. ρ-blocks of gentle Jacobian algebras. Let A = KQ/I be a gentle Jacobian algebra. It follows from the definitions that the ρ-blocks of A are isomorphic to C 1 , C 2 or C 3 . We call them 1-blocks, 2-blocks or 3-blocks, respectively.
A 1-block can only occur if A = C 1 . Here we used that gentle Jacobian algebras are by definition connected. Now let A s be a 1-block or 2-block. Then the schemes mod(A s , d) are obviously just affine spaces. In particular, they are irreducible, and all modules M ∈ mod(A s , d) are smooth and reduced. Furthermore, mod(A s , d) contains a unique τ -rigid module. In particular, mod(A s , d) is generically τ -reduced.
Next, let A s be a 3-block of A. For convenience, we assume that A = C 3 = KQ/I, where Q is the quiver 1 and I is generated by the paths a 2 a 1 , a 3 a 2 and a 1 a 3 .
Lemma 6.11. Let A be a 1-block, 2-block or 3-block as above. For τ -rigid A-modules M and N the following are equivalent: Proof. By the discussion above, the statement is clear for 1-blocks and 2-block. Thus assume A is a 3-block as above.
(ii) =⇒ (i): By Proposition 6.8 there are four types of τ -rigid A-modules: where r i ≥ 0 and s i ≥ 1 for all i.
First, let M be of type 0 with dim(M ) = d = (d 1 , d 2 , d 3 ). It follows that For a fixed d, this system of linear equations has exactly one solution. This proves (ii) =⇒ (i) for modules of type 0.
Next, let M be of type i for some 1 ≤ i ≤ 3 with dim(M ) = d = (d 1 , d 2 , d 3 ). It follows that For a fixed d, this system of linear equations has exactly one solution. This proves (ii) =⇒ (i) for modules of type i.
Finally, we observe that modules of different types have always different dimension vectors. This finishes the proof.
Lemma 6.12. Let A be a 1-block, 2-block or 3-block as above. For M ∈ mod(A, d) the following are equivalent: (i) M is singular; (ii) M is contained in at least two different irreducible components of mod (A, d).
Proof. By the discussion above, the statement is clear for 1-blocks and 2-block. Thus assume A is a 3-block as above.
It follows that q 2 + q 3 > q 1 . Now one proceeds as in the proof of Proposition 6.10 to show that M is contained in at least two different irreducible components.
(ii) =⇒ (i): This holds for arbitrary finite-dimensional K-algebras, see Proposition 2.9. 7. Irreducible components for gentle algebras 7.1. Irreducible components. Finding the irreducible components of schemes of modules over gentle algebras is rather easy, since each of these schemes is isomorphic to a product of schemes of complexes.
Let A = KQ/I be a gentle algebra, and let A 1 , . . . , A t be its ρ-blocks. For each ρ-block A s there is a unique such that there exists an algebra homomorphism f s : A s → A s with the following properties: (i) f s sends vertices to vertices and arrows to arrows.
(ii) f s is bijective on the sets of arrows.
(In (i) we think of the vertices as standard idempotents.) This follows directly from the definition of a gentle algebra and from the definition of a ρ-block. We say that A s is of type A s . Let n s (resp. n s ) be the number of vertices of A s (resp. For example, let A = KQ, where Q is the quiver So here we have I = 0 and ρ = ∅. There are two ρ-blocks A 1 and A 2 of type C 2 , i.e. A 1 = A 2 = C 2 . Define f 1 : A 1 → A 1 by 1 → 1, 2 → 2, a 1 → a, and define f 2 : A 2 → A 2 by 1 → 2, 2 → 3 and a 1 → b. For s = 1, 2 and a dimension vector d for A s we have d = d.

As a less trivial example, let
y y and I is generated by the paths {a i+1 a i | 1 ≤ i ≤ 6}. Then A has only one ρ-block, The following result follows almost immediately from [DS], see also [CW,Propositions 3.4 and 5.2]. Note that Carroll and Weyman [CW] only consider the class of gentle algebras admitting a colouring. However, the result holds in general. Proof. Let A 1 , . . . , A t be the ρ-blocks of A. Recall that for each d we have an isomorphism which yields a bijection Irr(A, d) → Irr(A 1 , π 1 (d)) × · · · × Irr(A t , π t (d)).
Now the isomorphisms f s,πs(d) : mod(A s , π s (d)) → mod(A s , π s (d) ) and the description of irreducible components of varieties of complexes (see Corollary 6.5) yield the result.

String and band components and generic decompositions. Let A = KQ/I be a gentle algebra. An indecomposable irreducible component Z of mod(A, d)
is a string component provided there is a string C such that the orbit O M (C) is dense in Z. In this case, C is (up to equivalence of strings) uniquely determined by Z, and we write Z = Z (C).

An indecomposable component
with pairwise different λ 1 , . . . , λ q ∈ K * . Lemma 7.2. Let A be a gentle algebra. For Z ∈ Irr(A, d) let be the canonical decomposition of Z. Then c A (Z) = q.

Generically reduced components.
Theorem 7.4. Let A be a gentle algebra, and let A 1 , . . . , A t be its ρ-blocks. For d = (d 1 , . . . , d n ) ∈ N n and Z ∈ Irr(A, d) the following are equivalent: (i) Z is generically reduced; (ii) For each loop a ∈ Q 1 , the number d s(a) is even.
Proof. We know from Corollary 4.4 that Z is generically reduced if and only if π i (Z) is generically reduced for all 1 ≤ i ≤ t. Now the result follows from Proposition 6.9.
Corollary 7.5. Let A be a gentle algebra without loops. Then each Z ∈ Irr(A) is generically reduced.
Note that Corollary 7.5 is exactly the statement of Theorem 1.2. 7.4. Singular locus. The following theorem describes the singular locus of schemes of modules over gentle algebras. It turns out that the rank function of a module determines completely if this module is singular or not.
Theorem 7.6. Let A = KQ/I be a gentle algebra. Let M ∈ mod (A, d), and let r = r M : Q 1 → Q 0 be the rank function of M . The following are equivalent: (i) M is singular; (ii) There exist a, b ∈ Q 1 with s(a) = t(b) and ab ∈ I such that the following hold: (1) r(a) < d t(a) , r(b) < d s(b) and r(a) + r(b) < d s(a) .
(2) If a ∈ Q 1 with s(a ) = t(a) and a a ∈ I, then r(a ) + r(a) < d t(a) . ( of affine schemes. In particular, π i (M ) is singular if and only if f s,π i (d) (π i (M )) is singular.
By Proposition 6.10 we know all singular points of mod(A i , π i (d) ). The conditions Theorem 7.6(ii) and Proposition 6.10(ii) are equivalent. More precisely, let A i be the ρ-block containing the arrows a and b. Then f i,π i (d) (π i (M )) has a direct summand isomorphic to S ab if and only if condition Theorem 7.6(ii) holds. This finishes the proof.
Theorem 7.7. Let A be a gentle Jacobian algebra. For M ∈ mod(A, d) the following are equivalent: (i) M is singular; (ii) M is contained in at least two different irreducible components of mod (A, d).
Proof. Let A 1 , . . . , A t be the ρ-blocks of A. We know that M is singular if and only if π i (M ) is singular for some 1 ≤ i ≤ t.
We also know that M is contained in two different components if and only if π i (M ) is contained in two different components. Now the claim follows from Lemma 6.12. In other words, h A (Z) = 1 and M is a brick. It follows that Z is a brick component.
Note that Proposition 7.11 yields Theorem 1.5.
Corollary 7.12. Let A be a gentle algebra, and let Z ∈ Irr(A) be a direct sum of band components. Then Z is generically τ -reduced. In other words, c A (Z) = h A (Z), thus Z is generically τ -reduced.

By Theorem 2.11 we get ext
Theorem 7.13. Let A be a gentle algebra. For Z ∈ Irr(A, d) the following are equivalent:

Proof. (i) =⇒ (ii): Let
be a direct sum of string and band components. For a generic M ∈ Z we get c A (Z) = q, see Lemma 7.2. In other words, Clearly dim End A (M ) ≥ p + q. So dim(Z) = dim(GL d (K)) implies p = 0. In other words, Z is a direct sum of band components. This finishes the proof. Combining Corollary 7.12 and Theorem 7.13 proves Theorem 1.6.
Theorem 7.14 ( [CC]). Let A be an acyclic gentle algebra. Then the following hold: and I is generated by {ab}. Then A is a gentle algebra, which does not admit a colouring in the sense of [CC].

Let
and I is generated by {a 2 , e 2 , c 1 c 2 , c 2 c 1 }. This is a gentle algebra admitting a colouring. For d = (2, 2, 2, 2), the affine scheme mod (A, d) has 3 irreducible components, and all of these are band components.

Generically τ -reduced components for gentle Jacobian algebras
In this section, we concentrate on the description of generically τ -reduced components for gentle Jacobian algebras. Some of this can be generalized to arbitrary gentle algebras. We leave this endeavor to the reader. 8.1. Simple summands of restrictions. Let A = KQ/I be a gentle Jacobian algebra and let A 1 , . . . , A t be its ρ-blocks. For a ∈ Q 0 ∪ Q 1 and 1 ≤ s ≤ t let δ a,As := 1 if a belongs to A s , 0 otherwise.
Lemma 8.1. Let A = KQ/I be a gentle Jacobian algebra and let A 1 , . . . , A t be its ρ-blocks. For a string module M = M (C) ∈ mod(A) and any ρ-block A s , the A s -module π s (M ) has a simple direct summand if and only if one of the following hold: Proof. For C = 1 i the claim is clear.
This follows directly from the definition of a string module. The claim follows.
Lemma 8.2. Let A = KQ/I be a gentle Jacobian algebra, and let A 1 , . . . , A t be its ρ-blocks. For a band module M ∈ mod(A) and any ρ-block A s , the module π s (M ) has no simple direct summand. In particular, π s (M ) is a projective A s -module.
Proof. Let M = M (B, λ, q) be a band module where B = (c 1 , . . . , c r ). For each 1 ≤ i ≤ r we have c i = a ± i for some a i ∈ Q 1 . We get This follows directly from the definition of a band module. The claim follows.
8.2. Non-vanishing of Hom A (M, τ A (M )). Let A be a gentle Jacobian algebra, and let A 1 , . . . , A t be its ρ-blocks. Recall from Section 6.2 the definition of critical summands of type I or II for modules over C n or C n . We say that M ∈ mod(A, d) has a critical summand of type I (resp. type II) if there exists some 1 ≤ i ≤ t such that π i (M ) has a critical summand of type I (resp. of type II).
Lemma 8.3. Let A be a gentle Jacobian algebra, and let A 1 , . . . , A t be its ρ-blocks. For M ∈ mod(A) the following are equivalent: (i) M does not have a critical summand of type I or II.
Proof. This follows from Propositions 6.7 and 6.8 Lemma 8.4. Let A be a gentle Jacobian algebra, and let A 1 , . . . , A t be its ρ-blocks. Let M 1 , M 2 ∈ mod(A) such that the following hold: There exists a ρ-block A i containing an arrow a ∈ Q 1 such that S s(a) is (up to isomorphism) a direct summand of π i (M 1 ), and S t(a) is (up to isomorphism) a direct summand of π i (M 2 ). Then Ext 1 A (M 1 , M 2 ) = 0.
Proof. We can assume that M 1 and M 2 are both indecomposable. By Lemma 8.2 we know that M 1 = M (C 1 ) and M 2 = M (C 2 ) are both string modules. By Lemma 8.1 we can assume without loss of generality that s(C 1 ) = s(a) and t(C 2 ) = t(a) and that C 1 a −1 C 2 is a string. We obtain a non-split short exact sequence Thus Ext 1 A (M 1 , M 2 ) = 0. Corollary 8.5. Let A be a gentle Jacobian algebra, and let A 1 , . . . , A t be its ρ-blocks.
Lemma 8.6. Let A be a gentle Jacobian algebra, and let A 1 , . . . , A t be its ρ-blocks. Let M 1 , M 2 ∈ mod(A) such that the following hold: There exists a 3-block A s containing an arrow a ∈ Q 1 such that S s(a) is (up to isomorphism) a direct summand of π s (M 1 ), and P t(a) is (up to isomorphism) a direct summand of π s (M 2 ). Then dim Hom A (M 2 , τ A (M 1 )) = 0.
Proof. We can assume that M 1 and M 2 are both indecomposable. We know that M 1 = M (C 1 ) for some string C 1 (see Lemma 8.2) and M 2 = M (C 2 ) or M 2 = M (C 2 , λ, q) for some string or band C 2 , respectively.
If M 2 is a band module, then there is a surjective homomorphism M (C 1 , λ, q) → M (C 1 , λ, 1). Thus in this case we can assume without loss of generality that q = 1.
We can assume that the 3-block A s is of the form with a 2 a 1 , a 3 a 2 , a 1 a 3 ∈ I and a = a 1 .
Without loss of generality we can assume that s(C 1 ) = 1 and that either l(C 1 ) = 0 or C 1 = (c 1 , . . . , c m ) such that c m / ∈ A 2 . We can also assume that C 2 = C a 2 C for some strings C and C and we can assume that C = (c 1 , . . . , c r ) with c 1 ∈ Q −1 1 . We want to construct a non-zero homomorphism Let E be a path of maximal length such that a −1 1 E is a string. It follows that τ A (M (C 1 )) = M (E E) for some string E , where E is either of length 0 or of the form E = E a −1 1 for some string E , compare Section 5.9. Let F be a path of maximal length such that F F = C . Thus C 2 = C a 2 F F . It follows that F is of length 0 or of the form F = b −1 F for some b ∈ Q 1 and some string F . This yields a surjective homomorphism Furthermore, we have E = F G for some direct string G . We get a standard homomorphism Thus is the desired non-zero homomorphism. This finishes the proof.
Corollary 8.8. Let A be a gentle Jacobian algebra, and assume that M ∈ mod(A) has a critical summand of type I or II. Then dim Hom A (M, τ A (M )) = 0. 8.3. Proof of Theorem 1.4. Let A = KQ/I be a gentle Jacobian algebra, and let A 1 , . . . , A t be its ρ-blocks. Let Z ∈ Irr (A). We want to show that the following are equivalent: For 1 ≤ i ≤ p let N i := M (C i ), and for 1 ≤ j ≤ q let N p+j := M (B j , λ j , 1).

Now it follows from Corollary 8.7 that
Hom As (π s (N i ), τ As (π s (N j ))) = 0 for all i = j, and also for all i = j with 1 ≤ i ≤ p. Since N p+1 , . . . , N p+q are band modules, we get from Lemma 8.2 that also in this case Hom As (π s (N i ), τ As (π s (N i ))) = 0.
This proves that π s (M ) is a τ -rigid A s -module for all s. Thus π s (Z) ∈ Irr τ (A s ).
We have ) for all 1 ≤ j ≤ q and 1 ≤ k ≤ p + q. For the third equality we used Corollary 5.6. Thus Z is generically τ -reduced if and only if Hom A (N k , τ A (N i )) = 0 for all 1 ≤ i ≤ p and 1 ≤ k ≤ p + q. To get a contradiction, assume that for some 1 ≤ i ≤ p and some 1 ≤ k ≤ p + q. On the other hand, we know that We know that f factors through some injective Amodule. Without loss of generality, we can assume that this injective module equals I r for some r ∈ Q 0 . Thus we have f = f 1 • f 2 with f 1 ∈ Hom A (I r , τ A (N i )) and f 2 ∈ Hom A (N k , I r ). Again without loss of generality we can assume that is a standard homomorphism. (Here we use the same notation and terminology as in [Sch].) The module I r is of the form where C and D are direct strings in Q such that Cγ, Dγ ∈ I for all γ ∈ Q 1 . Since τ A (M i ) is not injective, we know that f 1 cannot be a monomorphism. Thus I r is not simple and we can assume without loss of generality that C = α 1 · · · α v and that (C) if k = v, G = 1 s (C) .
Since A is a gentle algebra, we also know that α v γ ∈ I for all γ ∈ Q 1 . This implies E = 1 s(α) . Set a := α k .
The following picture shows I r , where I r / Ker(f 1 ) is given by the string F between the blue vertices. • By the properties of f 1 discussed above, we see that we must be in the 2nd, 4th or 5th case and that F coincides with the subfactor of τ A (M ) marked by the two rightmost blue points. Here we refer to Section 5.9 for the description of τ A (M ).
We get ab ∈ I. Thus there exists a third arrow c ∈ Q 1 with s(c) = t(a) and t(c) = s(b). So the arrows a, b, c form a 3-block, say A s , of A. So we are in the following situation: • c s s (In the 5th case, the red bullet in this picture should be green.) Clearly π s (N k ) contains M (a) as a direct summand, and π s (N i ) contains S s(b) as a direct summand.
It follows that π s (N k ⊕ N i ) has a direct summand isomorphic to S s(b) ⊕ P s(a) . Now Proposition 6.8 implies that π s (N k ⊕ N i ) and therefore also π s (M ) is not τ -rigid in mod (A s ). This finishes the proof. 8.4. Proof of Theorem 1.3. Let A be a gentle Jacobian algebra, and let A 1 , . . . , A t be its ρ-blocks. Let Z 1 , Z 2 ∈ Irr τ (A). We want to show that the following are equivalent: (ii) =⇒ (i): This direction is trivial.
The following beautiful result due to Plamondon shows that the generic g-vectors parametrize the generically τ -reduced decorated components.
Theorem 9.1 ([P1, Theorem 1.2]). Let A be a basic algebra. Then the map is bijective. 9.5. Decomposition of generically τ -reduced components. An irreducible component Z ∈ decIrr (A, (d, v)) is called indecomposable if there exists a dense open subset U ⊆ Z, which contains only indecomposable decorated modules. This is the case if and only if Z = (Z , 0) with Z ∈ Irr(A, d) indecomposable or Z = {S − i } for some i. In particular, if Z ∈ decIrr (A, (d, v)) is indecomposable, then either d = 0 or v = 0.
The Zariski closure Z 1 ⊕ · · · ⊕ Z t of Z 1 ⊕ · · · ⊕ Z t is an irreducible closed subset of decmod (A, (d, v)) and is called the direct sum of Z 1 , . . . , Z t . Note that Z 1 ⊕ · · · ⊕ Z t is in general not an irreducible component.
Theorem 9.2 ([CLFS, Theorem 1.3]). For Z 1 , . . . , Z t ∈ decIrr(A) the following are equivalent: Each Z ∈ decIrr τ (A) is a direct sum of indecomposable generically τ -reduced components, which are uniquely determined up to reordering. of differentiability class C 1 , with derivative vanishing in at most finitely many points of S 1 , such that the following hold:  Let π : S 1 → S 1 be the universal cover of S 1 . For a loop γ : S 1 → S in (S, M) let γ := γ • π : S 1 → S.
We call this the periodic curve associated with γ.
10.3. Laminations and triangulations. By a lamination of (S, M) we mean a pair L = (γ, m), where γ is a (finite) subset of A(S, M) ∪ L(S, M) such that Int(γ i , γ j ) = 0 for all γ i , γ j ∈ γ, and m : γ → Z >0 is a map. Instead of L = (γ, m) we also write L = {(γ 1 , m 1 ), . . . , (γ t , m t )}, where γ = {γ 1 , . . . , γ t } and m i = m(γ i ) for 1 ≤ i ≤ t. By abuse of terminology, we also say that γ is a lamination. Note that each element in γ is a simple curve or a simple loop. We think of m i as the multiplicity of γ i in the lamination L. Let Lam(S, M) be the set of laminations of (S, M). Note that in [MSW2,Definition 3.17], the set Lam(S, M) is denoted by Each boundary component of (S, M) with m marked points has m boundary segments, each connecting two consecutive marked points.
A curve γ in S \ M is simple if additionally the following holds: (A4) γ is injective, i.e. γ does not intersect itself. Let A(S\M) be the set of curves in S\M up to homotopy, such that γ(0) and γ(1) never leave their respective boundary segment, and up to the equivalence γ ∼ γ −1 . More precisely, we consider here homotopies H t (1)) belonging to the same boundary segment as γ(0) (resp. γ(1)) and such that H 0 = γ.
As before, we just write γ for the class of γ in A(S \ M).
By a classical lamination of (S, M) we mean a pair L = (γ, m), where γ is a (finite) subset of A(S \ M) ∪ L(S, M) such that Int(γ i , γ j ) = 0 for all γ i , γ j ∈ γ, and m : γ → Z >0 is a map. Here Int(γ i , γ j ) is defined in the obvious way. Again by abuse of terminology, we also say that γ is a classical lamination. Let Lam(S \ M) be the set of classical laminations of (S, M).
Given a curve γ ∈ A(S, M), let τ 1/2 (γ) ∈ A(S \ M) be the curve obained from γ by rotating its endpoints in clockwise direction to the adjacent boundary segment. This yields a bijection A triangulation T of (S, M) consists of all boundary segments together with a maximal collection T • of curves in (S, M) such that Int(γ i , γ j ) = 0 for all γ i , γ j ∈ T • . In this case, we have where g is the genus of S and b is the number of boundary components of S, see for example [FST,Proposition 2.10].
Note that the classical laminations defined above correspond to the X -laminations in the sense of Fock and Goncharov [FG]. Let T be a triangulation of (S, M), and let A T be the associated gentle Jacobian algebra. We refer to Section 10.5 for a precise definition of A T . To a lamination L of (S, M) we will associate a certain generic decorated A T -module, which is a direct sum of indecomposable τrigid modules, of certain band modules of quasi-length 1, and of negative simples. In Section 11, we will look at the Caldero-Chapoton functions of these modules, which can be thought of as generating functions of Euler characteristics of quiver Grassmannians. In contrast, Allegretti [A] works with certain A-laminations (see [A, FG] for a definition), and he associates A T -modules, which are direct sums of indecomposable τ -rigid modules, of band modules with arbitrary quasi-length, and of negative simples. He then looks at certain generating functions of Euler characteristics of transversal quiver Grassmannians. We assume that γ is minimal in the sense that To γ we associate a sequence (a, τ j 1 , . . . , τ jm , b), where a = γ(0) and b = γ(1), and there exist 0 < t 1 < · · · < t m < 1 such that γ(t i ) ∈ τ j i . We illustrate this in Figure 4. Note that the curves τ i 1 , . . . , τ im do not have to be pairwise different. We do have, however, τ i j = τ i j+1 for all 1 ≤ i ≤ m − 1. The curve γ −1 yields (b, τ jm , . . . , τ j 1 , a). Analogously, with a loop γ : S 1 → S in (S, M) we associate a sequence (a, τ j 1 , τ j 2 , . . . , τ jm , τ j 1 , a), where a = γ(1). Starting in 1 ∈ S 1 in clockwise orientation, we assume that γ first passes through τ j 1 , then through τ j 2 etc. We can assume here that a ∈ τ j 1 . This is illustrated in Figure 5.
10.5. From triangulations to gentle Jacobian algebras. Let T be a triangulation of an unpunctured marked surface (S, M). Assume that T • consists of n curves τ 1 , . . . , τ n . Then Q = Q T is by definition the quiver with vertices 1, . . . , n. The arrows of Q are defined as follows: As displayed in Figure 6, there are three types of triangles defined by T , and two of these yield arrows in Q, as indicated in the picture. Note that the non-labelled sides of the triangles are meant to be boundary segments of (S, M), and note that our arrows point in clockwise direction. Other authors might choose the opposite convention. The algebra associated to T is then A T := KQ/I, where I is generated by the paths a 2 a 1 , a 3 a 2 , a 1 a 3 arising from triangles with all three sides in T • .
a 2 a 3 Figure 6. How triangles yield arrows The algebra A T was first studied by [ABCP] and [LF1], where it was defined as the Jacobian algebra P C (Q T , W T ) of a quiver with potential.
Theorem 10.2 ([ABCP, Section 2]). The Jacobian algebras A T arising from triangulations of unpunctured marked surfaces are exactly the gentle Jacobian algebras.
10.6. From curves and loops to string and band modules. Let (S, M) be an unpunctured marked surface, and let T be a fixed triangulation of (S, M). The band associated with the curve in Figure 5 looks as in the following picture, where the two blue vertices have to be identified: Note that for an arbitrary gentle algebra A there is also a geometric model for the derived category D b (mod (A)) (see [HKK, LP, OPS]), which differs substantially from the one for mod (A) used in this article. Note that the results in [BZ] are formulated in terms of the cluster category associated with (S, M). Theorem 10.4 is a straightforward reformulation in terms of decorated A T -modules.
In Section 10.9 we reprove and generalize Theorem 10.4 by also including band modules. For the following two statements we refer to [BZ,Theorem 3.6]. (Note that the orientation of our Q T is opposite to the one used in [BZ].) We orient each boundary component of S by requiring that when following the orientation, the surface lies to the left. We call this the induced orientation of the boundary component.
If M is non-projective, then γ τ A (M ) = τ (γ), where τ (γ) is obtained from γ by rotating the points a = γ(0) and b = γ(1) of γ to the next marked point on their respective boundary component, following the induced orientation.
Dually, if M is non-injective, then γ τ −1 A (M ) = τ −1 (γ), where τ −1 (γ) is obtained from γ by rotating a and b to the next marked points on their respective boundary component, following the opposite induced orientation.
The proof of these statements uses the combinatorial descriptions of τ A (M ) and τ −1 A (M ) given in [BR] and [WW].
For more details we refer to [BZ,Section 3]. Proof. Since q ≥ 2, we have γ M = γ q for some primitive loop γ. It follows that Int(γ M , γ M ) = 0.
Lemma 10.6. Let S − i be a negative simple decorated A-module, and let M be an indecomposable decorated A-module. Then the following are equivalent: Proof. Suppose that M = S − j is also negative simple. Then the equivalence of (i) and (ii) follows directly from the definitions. Next, assume that M = (M, 0). Let Recall that the notions of ρ-blocks and of the associated restriction maps π i were defined in Section 4.
Lemma 10.7. Let M and N be indecomposable A-modules such that γ M and γ N have a Type I or Type II intersection as shown in Figure 7. Then Proof. Assume we are in Type I: Let A i be the ρ-block of A containing the arrow 2 → 1. Then π i (M ) = S 1 and π i (N ) = S 2 . By Lemma 8.4 we get Ext 1 A (N, M ) = 0, which implies Hom A (M, τ A (N )) = 0.
Next, assume we are in Type II: Let A i be the ρ-block of A containing the arrow 1 → 2. Thus A i also contains the arrows 2 → 3 and 3 → 1. We get π i (M ) = P 2 and π i (N ) = S 1 . By Lemma 8.6 this implies Lemma 10.8. Let A 1 , . . . , A t be the ρ-blocks of A. Let M and N be indecomposable A-modules. Then the following are equivalent: (i) γ M and γ N have an intersection of type I or II.
Proof. This is a direct consequence of Propositions 6.7 and 6.8. Proof. (i) Let N = M (C) be a string module, and let f : M → τ A (N ) be a standard homomorphism. Thus, up to symmetry, f is given by one of the ten pictures in Figures 8 and 9. The green curves in these pictures stand now for γ M and the red curves for γ τ A (N ) . Now τ −1 ( γ τ A (N ) ) = γ N is obtained by a rotation in the direction opposite to the induced orientation. By a straightforward case by case analysis we obtain Int( γ M , γ N ) = 0 in all ten cases.
(ii) Let N be a band module of quasi-length 1. Then (i) =⇒ (ii): Assume that (ii) does not hold. Without loss of generality let Hom A (M, τ A (N )) = 0. If N is a string module, then the result follows from Lemma 10.11(i). Next, suppose N = M (B, λ, q) is a band module. The periodic curve γ N and also the condition Hom A (M, τ A (N )) = 0 are independent of t. So we can assume that q = 1. By assumption we have M ∼ = N . Thus rad A (M, τ A (N )) = 0. Now the result follows from Lemma 10.11(ii).
The following theorem corresponds to Theorem 1.7.
Thus, we assume that t = 1 and set Z B := Z B,1 . Let N = M (B, µ, 1) for some µ ∈ K * with µ = λ. Note that γ M = γ N . By Theorem 10.  10.10. Shear coordinates and g-vectors. Let A = A T as above. As mentioned before, a result by W. Thurston (see [FT,Theorem 12.3]) says that there is a bijection s T : Lam(S, M) → Z n sending a lamination to its shear coordinate. We briefly and informally recall the construction of s T .
Then we are in one of the four cases displayed in Figure 10, where the red line is a segment of the curve τ 1/2 (γ) and the dotted arrows indicate possible arrows of A.
(There is an arrow on the left if and only if τ = τ a , and there is an arrow on the right if and only if τ = τ b .) Next, consider a simple loop γ = (a, τ j 1 , . . . , τ jm , τ j 1 , a) ∈ L(S, M).
For each 1 ≤ k ≤ m, we look at the triple In both cases (i.e. γ ∈ A(S, M) and γ ∈ L(S, M)), the shear coordinate of γ (with respect to T ) is defined as s T (γ) := (s 1 , . . . , s n ), where Here δ j k ,i denotes the Kronecker delta and if (τ , τ j k , τ ) looks as in case (1)  (1) Recall that by Plamondon [P1, Theorem 1.2], there is a bijection g T : decIrr(A) τ → Z n sending a generically τ -reduced decorated component to its g-vector.
The proof of the following result is a bit tedious but straightforward. It follows essentially the ideas from Labardini-Fragoso [LF2,Theorem 10.0.5]. Note that [LF2] deals with a dual situation and only considers curves. The case of loops is however easier than the curve case and uses the same arguments. Note also that [LF2] uses a different (but equivalent) definition of g-vectors.
Proposition 10.14. With A = A T as above, the diagram

Bangle functions and generic Caldero-Chapoton functions
We will assume throughout that our surface with marked points (S, M) is connected and has no punctures. We fix a triangulation T with internal edges T • = (τ 1 , τ 2 , . . . , τ n ). 11.1. Strings and Bands. Recall from Section 10.4 that we identify each curve γ ∈ A(S, M) \ T • with a certain sequence (a, τ j 1 , . . . , τ jm , b), where a, b ∈ M and the τ j i are the sequence of arcs of T • which are crossed by γ in a minimal way, up to homotopy. Denote by ∆ i the triangle of T , which contains the arcs τ j i and τ j i+1 , and which contains the segment [γ(t i ), γ(t i+1 )] of γ for 1 ≤ i ≤ m − 1. This sequence can be coded into a (decorated) quiver Q T γ of type A m with vertices {1, 2, . . . , m}. Now, in ∆ i there exists an unique arrow a i of the quiver Q T (see Section 10.5), which goes either from τ j i to τ j i+1 , or from τ j i+1 to τ j i . In the first case we draw an arrow with label a i from i to i + 1. In the second case, we draw an arrow with the same label from i + 1 to i. We call Q T γ the string of γ with respect to the triangulation T . Analogously, we associate with a loop γ = (a, τ j 1 , . . . , τ jm , τ j 1 , a) ∈ L(S, M) a quiver of type A m−1 with vertices {1, 2, . . . , m}. The only difference is that now we have an additional triangle ∆ m , which contains the edges τ jm , τ j 1 , and the segment [γ(t m ), γ(t 1 )] of γ. In this case ∆ m determines the direction of the arrow between a m between 1 and m. We call in this case Q T γ the band of γ with respect to T .
11.2. MSW-functions. In this section we will use the conventions and definitions from [MSW2, Section 3] without further reference.
Musiker, Schiffler and Williams [MSW2] assign to each homotopy class γ ∈ A(S, M) (resp. γ ∈ L(S, M)) a snake graph (resp. band graph) G = G T,γ . We assume that in each tile G 1 , G 2 , . . . G l of G, the diagonal goes from SE to NW, and we always think that G is drawn from SW to NE.
Remark 11.1. The graph G comes with a distinguished good resp. perfect matching P − which consists of the external edges of G which are either vertical and belong to a negatively oriented tile, or are horizontal and belong to a positively oriented tile. On the other hand, the tile G j and the position of its two neighbours record how γ crosses the quadrilateral surrounding τ j i in the neighbourhood of γ(t i ). With these two observations it is an easy exercise to show that where s T (γ) is the shear coordinate vector (see Section 10.10) of γ with respect to T .
In [MSW2,Definition 2.19] the (skew-symmetric) signed adjacency matrix B T ∈ Z n×n of a triangulation T of (S, M) is introduced. With these conventions in place we have Q T = Q(−B T ) for our quiver Q T from Section 10.5. The (coefficientfree) cluster algebra A(B T ) associated with B T is just A (S,M) . Let A • (B T ) be the corresponding cluster algebra with principal coefficients.
Remark 11.3. In [MSW2,Definitions 5.3 and 5.6] the authors associate to their graph G = G T,γ a poset structure Q G on the set {1, 2, . . . , m} by describing its Hasse quiver. We leave it as an exercise that our quiver Q T γ from Section 11.1 is opposite to the Hasse quiver in [MSW2]. Thus, the poset ideals of Q G are precisely the subsets I of vertices of Q T γ which are closed under predecessors. We call such subsets order coideals of Q T γ .
In [MSW2,Definition 3.4] a Laurent polynomial X T γ = 1 cross(T, γ) P x(P )y(P ) ∈ R := Z[x ± i , y i ] i=1,2,...,n (11.2) is defined, where the sum runs over the perfect resp. good matchings of G. We agree that X T γ i = x i for γ i ∈ T • and for L = (ξ, m) ∈ Lam(S, M) one sets The following result is implicit in [MSW2, Sections 5 and 6]: Lemma 11.4. For each γ ∈ A(S, M) ∪ L(S, M) holds where the summation runs over the order coideals I of Q T γ and for j = 1, 2, . . . , n.
Proof. According to [MSW2,Theorem 5.7] the lattice L(G) of good matchings of G is in natural bijection with the distributive lattice of order coideals of Q T γ . More precisely, to a good matching corresponds the coideal I(P ), which consists of the labels of the tiles of G which are enclosed by P P − .
On the other hand, by [MSW2,Proposition 6.2] x γ ∈ R is homogeneous of degree if we agree that deg x j = e j ∈ Z n and deg y j = − n i=1 b ij e i ∈ Z n . Thus in view of (11.1) we have to show that x(P )y(P ) x(P − ) = i∈I(P )ŷ j i for all good matchings P of G. 11.3. Dual CC-functions and MSW-functions. We introduce the anti principal ice quiver Q T , which is obtained from Q T by adding an additional set of frozen vertices {1 , 2 , . . . , n }, and an additional arrow p i : i → i for i = 1, 2, . . . , n. The potential W T mentioned in Section 10.5 can be naturally viewed as a potential for Q T and it is not hard to see that ( Q T , W T ) is a non-degenerate QP with finitedimensional Jacobian algebra A T = P C ( Q T , W T ).
Definition 11.5. The dual Caldero-Chapoton function with respect to A T of a decorated representation M = (M, V ) of A T is the Laurent polynomial where Gr e A T (M ) is the quiver Grassmannian of factor modules with dimension vector e of the A T -module M , and χ is the topological Euler characteristic.
Note that for a decorated representation M of A T we have in fact g A T (M) = (g A T , 0, . . . , 0). This is so, since for each A T -module M with minimal projective presentation P 1 → P 0 → M → 0, the same sequence can be taken as a minimal projective presentation of M viewed as an A T -module, due to the shape of Q A .
Remark 11.6. Obviously, the dual Caldero-Chapoton-function is the same as the usual Caldero-Chapoton-function for the corresponding dual module, more precisely  Moreover the CC(M) for decorated reachable E-rigid A T -modules M are precisely the cluster monomials for the cluster algebra A • (B T ) ⊂ R with principal coefficients, see for example [DWZ2].
Remark 11.7. For a curve γ ∈ A(S, M)\T • let M γ := (M γ , 0) be the corresponding decorated A T -module. For a primitive γ ∈ L(S, M) let M γ := (M γ,λ , 0) for some λ ∈ C * . Note that M γ,λ is a band module of quasi-length 1. In these two cases, the quiver Q T γ is the coefficient quiver of the string module M γ (resp. of the band module M γ,λ ). Moreover, the order coideals of Q T γ can be identified with the coordinate factor modules of M γ , see also [MSW2,Remark 5.8]. Finally, for γ ∈ T • let M γ be the associated negative simple decorated A T -module.
Proposition 11.8. For a curve or primitive loop γ ∈ A(S, M) ∪ L(S, M) we have Proof. We use Lemma 11.4 to compare both expressions. As a consequence of Proposition 10.14, we get s T (γ) = g A T (M γ ) = g A T (M γ ). In view of Remark 11.7 our claim follows now from [Hau,Theorem 1.2].
11.4. Bangle functions are generic. Recall that our set of laminations Lam(S, M) from Section 10.3 is the same as the set of C • (S, M) of C • -compatible collection of arcs and simple (= essential) loops in [MSW2,Def. 3.17].
Since on the other hand, we have by definition it is sufficient to prove X T γ = CC T (η T (γ)) for γ an arc or a simple loop. This is trivial if γ ∈ T • , thus we have to distinguish only two cases: Case 1: γ is an arc which does not belong to T . In this case, the string module M γ is τ -rigid and therefore η T (γ) = O Mγ , compare Theorem 10.13. So our claim follows directly from Proposition 11.8.
Case 2: γ is a simple loop. In this case η T (γ) is the closure of the union of a the orbits of a family of modules, namely η T (γ) = λ∈C * O (M γ,λ ,0) .
In this case we have again by Proposition 11.8 X T γ = CC A T ((M γ,λ , 0)) for all λ ∈ C * , and we are done.
By specializing the coefficients to 1, the equality B T = G T from Theorem 11.9 yields B T = G T . Thus Theorem 1.8 is proved.
12. An example Let (S, M) be the sphere with three disks cut out, and one marked point on each boundary component. In Figure 11 we display a triangulation T of (S, M), where the arcs of T are marked in green, together with a loop σ in (S, M). It is easy to read off the quiver Q T (following our convention) and the signed adjacency matrix B T (following the convention of [MSW2]). Both are shown in Figure 12. Recall that with these convention in place we have Q T = Q(−B T ).
Musiker, Williams and Schiffler [MSW2] associate to each loop σ a band graph G = G T,σ with respect to a triangulation T . In our example, we obtain the band graph G displayed in Figure 13. Note that G has m = 7 tiles, corresponding to x 3 x 1 x 5 x 6 x 5 x 3 x 6 x 3 x 6 x 1 x 5 x 4 x 6 x 2 = x 5 x 3 x 4 x 2 = x s(σ) Figure 13. Band graph G = G T,σ with P − (thick edges) and x s(σ) Recall from Section 11.3 that Musiker, Schiffler and Williams associate to G a Hasse quiver Q G , which is opposite to our coefficient quiver Q T σ of the band module M σ,λ for λ ∈ C * , see Remarks 11.1 and 11.7. We display the coefficient quiver Q T σ in Figure 14. Note that the two encircled vertices have to be identified. Thus the order coideals of Q T σ (i.e. coordinate factor modules of M σ,λ ) are in bijection with the good matchings of G. More precisely, the tiles which are enclosed by the symmetric difference P P − for a good matching P are identified with a basis of the corresponding coordinate factor module. Finally we display in Figure 15, three of the 27 good matchings of G. In each case the edges of the matching P are highlighted in orange, whilst the tiles which are enclosed by P P − are highlighted in yellow. Moreover, we show in each case the contribution of P to X T σ . x(P 1 )y(P 1 ) x(P − ) = x 3 y 4 x 6 x 5 x 2 x 4 y 6 x 3 x 1 x 1 y 2 x 3 x 6 =ŷ 4ŷ6ŷ2 . x(P 2 )y(P 2 ) P − = x 3 y 4 x 5 x 6 x 1 y 2 x 3 x 6 =ŷ 4ŷ2 .  The relation between perfect matchings and coordinate submodules of string modules has been also studied in a more general setup by Canakci and Schroll [CS].