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A GEOMETRIC MODEL FOR THE DERIVED CATEGORY OF GENTLE ALGEBRAS SEBASTIAN OPPER, PIERRE-GUY PLAMONDON, AND SIBYLLE SCHROLL

Abstract. In this paper we construct a geometric model for the bounded derived category of a gentle algebra. The construction is based on the ribbon graph associated to a gentle algebra in [55]. This ribbon graph gives rise to an oriented surface with boundary and marked points in the boundary. We show that the homotopy classes of curves connecting marked points and of closed curves are in bijection with the isomorphism classes of indecomposable objects in the bounded derived category of the gentle algebra up to the action of the shift functor. Intersections of curves correspond to morphisms and resolving the crossings of curves gives rise to mapping cones. The AuslanderReiten translate corresponds to rotating endpoints of curves along the boundary. Furthermore, we show that the surface encodes the derived invariant of Avella-Alaminos and Geiss.

Contents Introduction Conventions 1. Surfaces with boundaries for gentle algebras 1.1. Ribbon graphs and ribbon surfaces 1.2. Marked ribbon graphs 1.3. The marked ribbon graph of a gentle algebra 1.4. A lamination on the surface of a gentle algebra 1.5. Recovering the gentle algebra from its lamination 2. Indecomposable objects in the derived category of a gentle algebra 2.1. Homotopy strings and bands 2.2. Main result on indecomposable objects of the derived category 3. Homomorphisms in the derived category of a gentle algebra 3.1. Bases for morphism spaces in the derived category 3.2. Morphisms as Intersections 4. Mapping Cones in the derived category of a gentle algebra 5. Auslander-Reiten triangles 5.1. Auslander-Reiten triangles in bounded homotopy categories 5.2. Geometric description of Auslander-Reiten triangles 6. Avella-Alaminos–Geiss invariants in the surface 6.1. The Avella-Alaminos–Geiss invariants Acknowledgements

2 4 4 4 5 6 9 12 12 12 14 18 18 19 25 27 28 29 31 31 33

Date: February 7, 2018. The first author is supported by the DFG grants BU 1866/4-1 and CRC/TRR 191. The second author is supported by the French ANR grant SC3A (ANR-15-CE40-0004-01). The third author is supported by the EPSRC through an Early Career Fellowship EP/P016294/1. 1

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SEBASTIAN OPPER, PIERRE-GUY PLAMONDON, AND SIBYLLE SCHROLL

References

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Introduction Giving a description of the (bounded) derived category of a finite dimensional algebra is a difficult undertaking in general. This problem is best approached by restricting to special classes of examples; for instance, in [41] the notion of derivedtameness of a bounded derived category of finite dimensional modules over a finite dimensional algebra was introduced and in [14] it was shown that for a gentle algebra its bounded derived category is derived-tame (a result which also follows from [54] for gentle algebras of finite global dimension). Gentle algebras first appeared in the from of iterated tilted algebras of type A [6, 7] and type A˜ [8]. It has transpired since that they naturally appear in many different contexts, these include dimer models [19, 20], enveloping algebras of Lie algebras [47], and cluster theory, where they appear as m-cluster tilted and mCalabi Yau tilted algebras as well as Jacobian algebras associated to surfaces with marked points in the boundary [5, 36, 49]. Of particular interest in relation to our construction is the appearance of the derived category of graded gentle algebras in the context of partially wrapped Fukaya categories [43]. The bounded derived categories of gentle algebras have been extensively studied. Their indecomposable objects were completely classified in terms of homotopy strings and bands in [14] and using different matrix reduction techniques in [24, 26, 25]. A basis for their morphism spaces was given in [4], and the cones of these morphisms were studied in [30]. The almost-split triangles of these categories were described in [16], see also [4]. The introduction of a combinatorial derived invariant for gentle algebras in [10] has also sparked a lot of research on the question of when two gentle algebras are derived equivalent, see for instance [18, 9, 34, 33, 1, 2, 46, 17, 51] (other invariants had also been introduced in [15]). This study was also extended to unbounded homotopy categories in [28]. The derived category of related classes of algebras have also been studied in some of the references mentioned above, see also [13, 12, 27, 11] In this paper, given a gentle algebra A, we construct a geometric model of Db (A − mod ) in form of a lamination of an oriented surface with boundary and marked points in the boundary. In [55], see also [56], for every gentle algebra, a ribbon graph was given. Our model is based on the embedding of the ribbon graph into its ribbon surface where the marked points correspond to the vertices of the ribbon graph embedded in the boundary of the surface. The lamination then corresponds to a form of dual of the ribbon graph within the surface. We give an explicit description of the correspondence of homotopy classes of (infinite) curves in the surface with the indecomposable objects (up to shift) in the derived category of a gentle algebra based on the homotopy strings and bands of [14] (Theorem 2.5). Based on the basis of homomorphism in Db (A − mod ) given in [4], we show that these basis elements correspond to crossings of curves (Theorem 3.4). Using the graphical mapping cone calculus given in [30], we show that the mapping cone of a map corresponding to a crossing of curves is given by the resolution of the crossing (Theorem 4.3). The Auslander-Reiten translate of a perfect object in Db (A−mod ) then corresponds to the rotation of the endpoints along the boundary

A GEOMETRIC MODEL FOR THE DERIVED CATEGORY OF GENTLE ALGEBRAS

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of the corresponding curve in the surface (Corollary 5.2). Finally, we show that the surface encodes the derived invariant of Avella-Alaminos and Geiss [10] in terms of the number of boundary components, the number of marked points on each boundary component and the number of laminates starting and ending on each boundary component (Theorem 6.1). As we were finalising our paper, an independent construction of a geometric model of the bounded derived category of homologically smooth gentle algebras has come to our attenion [51]. It is establishing, together with the results in [43], an equivalence with the partially wrapped Fukaya category associated to the surface. While it would seem that our geometric models coincide, our focus is on the explicit description of the connection of the surface combinatorics and the representation theory related to Db (A − mod ). In the context of a classification of thick subcategories of discrete derived categories, a geometric model was given in [21]. Discrete derived categories were classified in [59], where it is shown that they correspond to bounded derived categories of a class of gentle algebras. The geometric model constructed in [21] coincides with our model for the class of discrete derived algebras. Jacobian algebras of (ideal) triangulations of marked surfaces with all marked points in the boundary are gentle algebras [49, 5]. We note that the ribbon graph of such a gentle algebra corresponds exactly to the triangulation of the surface. In this context, the indecomposable objects of the associated cluster category were classified in [23] in terms of arcs and closed curves on the surface, and the AuslanderReiten translation was described in [22]. Bases for the extension spaces were described in terms of crossings of arcs in [31]. These results were then extended to the case where the surface has punctures (that is, marked points in its interior) to objects associated to arcs in [53], and a complete description of indecomposable objects using arcs and closed curves was given in [3]. Furthermore, for gentle algebras associated to triangulations of surfaces with marked points in the boundary, the geometric description of the Auslander-Reiten translation is the same in both the associated module category [23], the cluster category [23] and we show in this paper, that it is the same again in the bounded derived category. We also note that the triangulated marked bounded surface S corresponding to a gentle Jacobian algebra A and the marked bounded surface underlying our geometric model of the bounded derived category Db (A − mod ) correspond if we add a boundary component with no marked points in the interior of each internal triangle in S. However, we also note that the sets of marked points do not necessarily coincide. The layout of the paper is as follows. In Section 1 we construct the marked bounded surface SA of a gentle algebra A from its ribbon graph as well as a lamination of SA . The correspondence of homotopy classes of curves with the objects in the bounded derived category Db (A − mod ) (up to shift) is given in Section 2. In Section 3 we establish a correspondence between the basis of homomorphism in Db (A − mod ) given in [4] and the crossing of curves (in minimal position) in SA . The mapping cones of the basis of homomorphism in terms of resolutions of crossings is given in Section 4 where it is also shown that the Auslander-Reiten translate corresponds to a rotation of both endpoints of the homotopy class of curves corresponding to an indecomposable object in Db (A − mod ). Finally, in Section 6 a description of the derived invariant of Avella-Alaminos and Geiss in terms of the surface is given.

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Conventions In this paper, all algebras will be assumed to be finite-dimensional over a base field k. All modules over such algebras will be assumed to be finite-dimensional. Arrows in a quiver are composed from left to right. 1. Surfaces with boundaries for gentle algebras In this section, we recall the construction of a surface with boundary associated to a gentle algebra. Our main references in this section are [56] and [50]. 1.1. Ribbon graphs and ribbon surfaces. A ribbon graph is an unoriented graph with a cyclic ordering of the edges around each vertex. In order to give a precise definition, it is useful to define a graph as a collection of vertices and halfedges, each of which is attached to a vertex and another half-edge. More precisely: Definition 1.1. A graph is a quadruple Γ = (V, E, s, ι), where • • • •

V is a finite set, whose elements are called vertices; E is a finite set, whose elements are called half-edges; s : E → V is a function; ι : E → E is an involution without fixed points.

We think of s as a function sending each half-edge to the vertex it is attached to, and of ι as sending each half-edge to the other half-edge it is glued to. This definition is equivalent to the usual definition of a graph, and in practice we will draw graphs in the usual way. Definition 1.2. A ribbon graph is a graph Γ endowed with a permutation σ : E → E whose orbits correspond to the sets s−1 (v), for all v ∈ V . A ribbon graph is a graph Γ endowed with a permutation σ : E → E such that the cycles of σ correspond to the sets s−1 (v), for all v ∈ V . In other words, a ribbon graph is a graph endowed with a cyclic ordering of the half-edges attached to each vertex. Any ribbon graph can be embedded in the interior of a canonical oriented surface with boundary, called the ribbon surface, in such a way that the orientation of the surface is induced by the cyclic orderings of the ribbon graph. Whenever we deal with oriented surfaces in this paper, we will call clockwise orientation the orientation of the surface, and anti-clockwise orientation the opposite orientation. When drawing surfaces or graphs in the plane, we will do so that locally, the orientation of the surface or graph becomes the clockwise orientation of the plane. Definition 1.3. Let Γ be a connected ribbon graph. The ribbon surface SΓ is constructed by gluing polygons as follows. • For any vertex v ∈ V with valency d(v) ≥ 1, let Pv be an oriented 2d(v)-gon. • Following the cyclic orientation, label every other side of Pv with the halfedges e ∈ E such that s(e) = v. • For any half-edge e of Γ, identify the side of Pv labelled e with the side of the polygon Ps(ι(e)) labelled ι(e), respecting the orientations of the polygons. In this definition, we exclude the degenerate case where Γ has only one vertex and no half-edges.

A GEOMETRIC MODEL FOR THE DERIVED CATEGORY OF GENTLE ALGEBRAS

P1

f1 g1

e2

f1

f2

P2

e2

e1 1

e1

5

f2

2 g2

g1

g2

Figure 1. Example of a ribbon graph Γ with orientation given by the planar embedding and with half edge labelling on the left and on the right the associated ribbon surface SΓ obtained by gluing the two polygons P1 and P2 corresponding to vertices 1 and 2 of the ribbon graph. Note that SΓ is oriented, and that we can embed Γ in SΓ as follows: the vertices of Γ are the centers of the polygons Pv , and the half edges of Γ are arcs joining the center of each Pv to the middle of the side with the same label. By [50, Corollary 2.2.11], SΓ is, up to homeomorphism, the only oriented surface S in which we can embed Γ, preserving the cyclic ordering around each vertex, and such that the complement of the embedding of Γ in S is a disjoint union of discs (we say that Γ is filling for S). Moreover, by [50, Proposition 2.2.7], the number of boundary components of SΓ is equal to the number of faces of Γ, according to the following definition. Definition 1.4. Let Γ be a ribbon graph. A face of Γ is an equivalence class, up to cyclic orientation, of tuples of half-edges (e1 , . . . , en ) such that ( ι(ep ) if s(ep ) = s(ep−1 ), • ep+1 = where the indices are taken modulo n; σ(ep ) otherwise, • the tuple is non-repeating, in the sense that if p 6= q and ep = eq , then ep+1 6= eq+1 . 1.2. Marked ribbon graphs. When we study gentle algebras in Section 1.3, we will obtain ribbon graphs endowed with one additional piece of information. We will call these marked ribbon graph, and we define them as follows. Definition 1.5. A marked ribbon graph is a ribbon graph Γ together with a map m : V → E such that for every vertex v ∈ V , m(v) ∈ s−1 (v). In other words, a marked ribbon graph is a ribbon graph in which we have chosen one half-edge m(v) around each vertex v. If Γ is a marked ribbon graph, we can construct its ribbon surface SΓ like in Definition 1.3. Moreover, with the additional information given by the map m, we can do the following: Proposition 1.6. There is an orientation-preserving embedding of Γ in SΓ which sends all vertices of Γ to boundary components of SΓ such that for each vertex

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v ∈ V , the boundary component lies between m(v) and σ(m(v)) in the clockwise orientation. This embedding is unique up to homotopy relative to ∂SΓ . Proof. With the notations of Definition 1.3, to prove the existence of the embeding, it suffices to move v to the unlabelled side of Pv that lies between the sides labeled with m(v) and σ(m(v)). Uniqueness follows from the fact that there is precisely one boundary component inside every face of Γ.  We call an embedding as in Proposition 1.6 a marked embedding of Γ in SΓ . 1.3. The marked ribbon graph of a gentle algebra. Here, we follow [55], see also [56, Section 3]. Gentle algebras are finite-dimensional algebras having a particularly nice description in terms of generators and relations. Their representation theory is well understood and their study goes back to [42, 35, 60, 29]. Let us recall their definition: Definition 1.7. An algebra A is gentle if it is isomorphic to an algebra of the form kQ/I, where (1) Q is a finite quiver; (2) I is an admissible ideal of Q (that is, if R is the ideal generated by the arrows of Q, then there exists an integer m ≥ 2 such that Rm ⊂ I ⊂ R2 ); (3) I is generated by paths of length 2; (4) for every arrow α of Q, there is at most one arrow β such that αβ ∈ I; at most one arrow γ such that γα ∈ I; at most one arrow β 0 such that αβ 0 ∈ / I; and at most one arrow γ 0 such that γ 0 α ∈ / I. Definition 1.8. For a gentle algebra A = kQ/I, let • M be the set of maximal paths in (Q, I), that is, paths w ∈ / I such that for any arrow α, αw ∈ I and wα ∈ I; • M0 be the set of trivial paths ev such that either v is the source or target of only one arrow, or v is the target of exactly one arrow α and the source of exactly one arrow β, and αβ ∈ / I; • M = M ∪ M0 . We call M the augmented set of maximal paths of A. Then the marked ribbon graph ΓA of A is defined as follows. (1) The set of vertices of ΓA is M. (2) For every vertex of ΓA corresponding to a path ω, there is a half-edge attached to ω and labeled by i for every vertex i of Q through which ω passes. Note that this includes the vertices at which ω starts and ends. Furthermore, if ω passes through i multiple times (at most 2), then there is one half-edge labeled by i for every such passage. (3) For every vertex i of Q, there are exactly two half-edges labeled with i. The involution ι sends each one to the other. (4) For each vertex ω of ΓA , the vertices through wich the path ω passes are ordered from starting point to ending point. The permutation σ sends each vertex in this ordering to the next, with the additional property that it sends the ending point of ω to its starting point. (5) The map m takes every ω to the half-edge labeled by its ending point. Remark 1.9. In a dual construction, the marked ribbon graph of a gentle algebra can also be defined using instead of M, the augmented set of all paths in Q such

A GEOMETRIC MODEL FOR THE DERIVED CATEGORY OF GENTLE ALGEBRAS

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that any subpaths of length 2 is in I. This is the set of forbidden threads as defined in [10]. Using Section 1.2, we can now define a surface with boundary and marked points for every gentle algebra. Definition 1.10. Let A = kQ/I be a gentle algebra. Its ribbon surface SA is the ribbon surface of ΓA . It has marked points on the boundary and collection of arcs joining them given by the embedding of ΓA as in Proposition 1.6. Thus the marked points of SA are in bijection with the vertices of ΓA , and the arcs joining them are in bijection with the vertices of Q. Example 1.11.

(1) Let A be the algebra defined by the quiver α

α

α

1 2 3 1 −→ 2 −→ 3 −→ 4

with no relations. The ribbon graph ΓA of this algebra is e2 e3 e1

α1 α2 α3

and its ribbon surface SA is a disc. •

e4 •





• (2) Let A be the algebra defined by the quiver α

α

α

1 2 3 1 −→ 2 −→ 3 −→ 4

with relations α1 α2 and α2 α3 . The ribbon graph ΓA of this algebra is e1 α1 α2 α3 e4 and its ribbon surface SA is, again, a disc. • •







For any gentle algebra A, the edges of ΓA cut SA into polygons as follows. Proposition 1.12. Let A = kQ/I be a gentle algebra, and let ΓA and SA be as in Definitions 1.8 and 1.10. Then SA is divided into two types of pieces glued together by their edges:

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(1) polygons whose edges are edges of ΓA , except for exactly one boundary edge, and whose interior contains no boundary component of SA ; (2) polygons whose edges are edges of ΓA and whose interior contains exactly one boundary component of SA with no marked points. Proof. Take any point X in the interior of SA which does not belong to any edge of ΓA . Then this point belongs to a polygon Pv as in Definition 1.3. This polygon has 2d sides (for a certain integer d) and contains exactly one marked point on its boundary, from which emanate d edges of ΓA . Below is the local picture if Pv is an octogon:

We see that X belongs to a region of Pv (grayed on the picture) that is partly bounded by a segment of a boundary component B of SA . Around this boundary component are other polygons Pv1 , Pv2 , . . . , Pvr , each containing exactly one marked point on its boundary. The picture around this boundary component B is as follows (in the example, B is a square).

Two cases arise. Case 1: There is at least one marked point on B. In this case, the point X belongs to a polygon cut out by edges of ΓA and by exactly one boundary edge on B, as illustrated in the following picture.

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Case 2: There are no marked points on B. In this case, the point X belongs to a polygon on SA cut out by edges of ΓA and which contains the boundary component B, as illustrated below.

This finishes the proof.



Remark 1.13. Let A be a gentle algebra with ribbon graph ΓA and associated ribbon surface SA . Suppose that ΓA admits v vertices, 2e half-edges and f faces. 1) The complement of SA is a disjoint union of open discs. 2) The Euler characteristic χ(ΓA ) = v − e + f of the ribbon graph Γ is equal to c the Euler characteristic of Sc A , where SA is the surface without boundary obtained from SA by gluing an open disc to each of the boundary components of SA . 3) The genus of SA (as well as the genus of Sc A ) is equal to 1 − χ(ΓA )/2 that is the genus of SA is (e − v − f + 2)/2. 1.4. A lamination on the surface of a gentle algebra. On any surface with boundary and marked points on the boundary, the notion of lamination is defined in [38, Definiton 12.1]. We will need to modify the definition slightly for what follows. Definition 1.14. Let S be a surface with boundary and a finite set M of marked points on its boundary. A lamination on S is a finite collection of non-selfintersecting and pairwise non-intersecting curves on S, considered up to isotopy relative to M . Each of these curves is one of the following: • a closed curve not homotopic to a point; or

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• a curve from one non-marked point to another non-marked point, both on the boundary of S. We exclude such curves that are isotopic to a part of the boundary of S containing no marked points. A curve that is part of a lamination is called a laminate. Remark 1.15. In [38, Definition 12.1], the case of a curve from a non-marked point to another on the boundary that is isotopic to a part of the boundary containing exactly one marked point is also excluded. For our purposes, we need to allow such curves in our laminations. Let A = kQ/I be a gentle algebra, and let SA be its ribbon surface as in Definition 1.10. We will now define a canonical lamination of the ribbon surface of a gentle algebra. Proposition 1.16. Let A = kQ/I be a gentle algebra, and let SA be its ribbon surface as in Definition 1.10. There exists a unique lamination L of SA such that (1) L contains no closed loops; (2) for every vertex i of Q (that is, every edge of ΓA ), there is a unique curve γi ∈ L such that γi crosses the edge labeled by i of the embedding of ΓA once, and crosses no other edges; (3) L contains no other curves than those described in (2). Proof. Every edge E of ΓA is part of two (not necessarily distinct) faces, in the sense of Definition 1.4, and each of these faces encloses a boundary component in SA . Therefore, if a curve γ in a lamination crosses E, then either it starts and ends on these two boundary components, or it has to cross at least another edge. Moreover, there is a unique curve starting on one of these two boundary components and ending on the other that crosses E once and no other edges of ΓA .  Example 1.17. We give the laminations for the two gentle algebras in Example 1.11. (1)

α1 1

α2 2

α3 3

4

Figure 2. On the right side is the ribbon graph embedded in the ribbon surface as well as the lamination of the hereditary gentle algebra on the left.

(2)

A GEOMETRIC MODEL FOR THE DERIVED CATEGORY OF GENTLE ALGEBRAS

α 1

β 2

11

γ 3

4

Figure 3. On the right side is the ribbon graph embedded in the ribbon surface as well as the lamination of the gentle algebra on the left with relations αβ and βγ.

Definition 1.18. Let A = kQ/I be a gentle algebra. Then we denote by LA the lamination described in Proposition 1.16, and we call it the lamination of A. Definition 1.19. Let A be a gentle algebra with marked ribbon graph ΓA and associated ribbon surface SA together with a marked embedding of ΓA into SA . Denote by M the set of marked points (these are the vertices of ΓA ) in SA . An arc γ in SA is a homotopy class of non-contractible curves whose endpoints coincide with marked points on the boundary. A closed curve in SA is a free homotopy class of non-contractible curves whose starting points and ending points are equal and lie in the interior of SA . A closed curve is primitive if it is not a non-trivial power of a different closed curve in the fundamental group of SA . In addition to the arcs with endpoints in the marked points, we also consider particular classes of rays and lines in SA . Recall that a ray is a map r : (0, 1] → SA or a map r : [0, 1) → SA and that a line is a map l : (0, 1) → SA . In what follows all rays will be such they start or end in a marked point and such that the other end wraps infinitely many times around a single unmarked boundary component with no marked points. All lines will be such that on each end they wrap infinitely many times around a single boundary component with no marked points. An infinite arc is given by homotopy classes associated to rays or lines in SA as follows: Let B and B 0 be boundary components in SA such that B ∩ M = B 0 ∩ M = ∅. We call such boundary components unmarked. We say two rays r : (0, 1] → SA and r0 : (0, 1] → SA wrapping infinitely many times around the same unmarked boundary component B are equivalent if r(1) = r0 (1) ∈ M and if for every closed neighbourhood N of B the induced maps r, r0 : [0, 1] → SA /N are homotopy equivalent relative to their endpoints. Similarly, we say two lines l : (0, 1) → SA and l0 : (0, 1) → SA are equivalent if they wrap infinitely many times around the same unmarked boundary components B and B 0 on each end and if for every closed neighbourhood N of B and N 0 of B 0 the induced maps l, l0 : [0, 1] → SA /(N ∪ N 0 ) are homotopy equivalent relative to their endpoints. Remark 1.20. It will sometimes be useful to think of boundary components with no marked points as punctures in the surface. Infinite arcs wrapping around such a boundary component can then be viewed as arcs going to the puncture.

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1.5. Recovering the gentle algebra from its lamination. The surface SA and lamination LA of a gentle algebra A contain, by construction, enough information to recover the algebra A. We record the procedure in the following proposition. Proposition 1.21. Let A = kQ/I be a gentle algebra, and let LA be the associated lamination (see Definition 1.18). Define a quiver QL as follows: • its vertices correspond to curves in LA ; • whenever two curves i and j in LA both have an endpoint on the same boundary segment of SA , so that no other curve has an endpoint in between, then there is an arrow from i to j if the endpoint of j follows that of i on the boundary in the clockwise order. Let IL be the ideal of kQL defined by the following relations: whenever there are curves i, j and k in LA that have an endpoint on the same boundary segment of SA , so that the endpoint of k follows that of j, which itself follows that of i, and if α : i → j and β : j → k are the corresponding arrows, then βα is a relation. Then A∼ = kQL /IL . 2. Indecomposable objects in the derived category of a gentle algebra Throughout this section let A = KQ/I be a gentle algebra. In this section, we prove that the indecomposable objects of the bounded derived category Db (A − mod ) are in bijection, up to shift, with certain curves on the surface SA . 2.1. Homotopy strings and bands. There are several approaches to the description of indecomposable objects in the bounded derived category of a gentle algebra. One approach makes use of combinatorial objects called homotopy strings and bands [14, 13] and this is the approach that we will use in this paper. In this section, we briefly recall the classification of the indecomposable objects in the bounded derived category of a gentle algebra in terms of homotopy string and band complexes [14]. Throughout this section let A = KQ/I be a gentle algebra. We recall that there is a triangle equivalence Db (A − mod ) ' K −,b (A − proj), where A − proj is the full subcategory of A − mod given by the finitely generated projective A-modules, K −,b (A − proj) is the homotopy category of complexes of objects in A − proj which are bounded on the right and have bounded homology, and Db (A − mod ) is the bounded derived category of A − mod . The definition of the homotopy string and band complexes that we will use is that introduced in [14]. For every a ∈ Q1 , we define a formal inverse a where s(a) = t(a) and t(a) = s(a). We denote by Q1 the set of formal inverses of the elements in Q1 , and we extend the operation (−) to an involution of Q1 ∪ Q1 by setting a = a. A walk is a sequence w1 . . . wn , where wi ∈ Q1 ∪ Q1 is such that s(wi+1 ) = t(wi ). We also allow trivial walks eu for every vertex u of Q. A string is a walk w such that wi+1 6= wi and such that for all substrings w0 = wi wi+1 · · · wj of w with the wi , . . . , wj all in Q1 (or all in Q1 ), we have that w0 ∈ / I (or w0 ∈ / I, respectively). We say that w = w1 . . . wn is a direct (resp. inverse) string if for all 1 ≤ i ≤ n, we have wi ∈ Q1 (resp. wi ∈ Q1 ). A generalized walk is a sequence σ1 . . . σm such that each σi is a string such that s(σi+1 ) = t(σi ).

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Definition 2.1. Let A = KQ/I be a gentle algebra. A finite homotopy string σ = w1 . . . wn , where wi ∈ Q1 ∪ Q1 , is a (possibly trivial) walk in (Q, I) consisting of subwalks σ1 , . . . , σr with σ = σ1 . . . σr and such that (1) σk is a direct or inverse string; (2) if σk , σk−1 are both direct strings then σk−1 σk ∈ I (resp. if both σk , σk−1 are inverse strings then σk−1 σk ∈ I). If σk is a direct string it is called a direct homotopy letter, otherwise it is called an inverse homotopy letter. A homotopy band is a finite homotopy string σ = σ1 . . . σr with an equal number of direct and inverse homotopy letters σi such that t(σr ) = s(σ1 ) and σ1 6= σr and σ 6= τ m for some homotopy string τ and m > 1. A homotopy string or band σ = σ1 . . . σr is reduced if σi 6= σi+1 for all i ∈ {1, . . . , r − 1}. A generalized walk is called a direct (resp. inverse) antipath if each homotopy letter is a direct (resp. inverse) homotopy letter. Definition 2.2. A left (resp. right) infinite generalized walk σ = . . . σ−2 σ−1 (resp. σ = σ1 σ2 . . .) is called a left (resp. right) infinite homotopy string if there exists k ≥ 1 such that . . . σk σk+1 (resp. σ−k−1 σ−k . . .) is a direct (resp. inverse) antipath which is eventually periodic and eventually involves only homotopy letters of length 1. A two-sided infinite generalized walk σ = . . . σ−1 σ0 σ1 . . . is called a two-sided infinite homotopy string if . . . σ−1 σ0 is a left infinite homotopy string and σ0 σ1 . . . is a right infinite homotopy string. To each homotopy string and homotopy band σ as described above is associated a (possibly infinite) complex of projective modules Pσ• [14]. We now recall this construction. Definition 2.3 ([14]). (1) Let σ = σ1 · · · σr be a finite reduced homotopy string. Define v0 = s(σ1 ) and ( vi = t(σi ) for all i ∈ {1, . . . , r}. Define µi + 1 if σi is a direct homotopy letter; further µ0 = 0 and µi+1 = µi − 1 if σi is an inverse homotopy letter, and let µ(σ) := mini∈{0,1,...,r} (µi ). Then the complex d−1

d0

Pσ• = . . . → − P −1 −−→ P 0 −→ P 1 → − ... is given by • for all j ∈ Z, Pj =

M

Pv i ;

0≤i≤r µi =j

• each direct (resp. inverse) homotopy letter σi defines a morphism σi σi Pvi−1 −→ Pvi (resp. Pvi −→ Pvi−1 ). These form the components of the differentials dj in the natural way. We call Pσ• a string object. (2) The definition of Pσ• when σ is an infinite reduced homotopy string is similar, and we again call Pσ• a string object. (3) Let σ = σ1 · · · σr be a reduced homotopy band. Let M be a finite-dimensional indecomposable K[X]-module, and let m = dimK M . Let F be the matrix

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of the multiplication by X for a given basis of M . Define v0 , . . . , vr , µ0 , . . . , µr as for homotopy strings. • Then the complex Pσ,F is defined by • for all j ∈ Z, M Pv⊕m ; Pj = i 0≤i≤r−1 µi =j

• for all i ∈ {1, . . . , r − 1}, the direct (resp. inverse) homotopy letter σ Id

σ Id

−−i−−m → Pv⊕m (resp. Pv⊕m σi defines a morphism Pv⊕m −−i−−m → Pv⊕m ), i i i−1 i−1 where Idm is the m × m identity matrix. These form the components of the differentials dj in the natural way. • The homotopy letter σr defines a final component of the differential. σi F If it is a direct letter, then the morphism used is Pv⊕m −− → Pv⊕m ; r−1 0 σ F

i otherwise, the morphism is Pv⊕m −− → Pv⊕m . 0 r−1 • In this case, Pσ,F is called a band object.

Furthermore, it is shown in [14] that the isomorphism classes of indecomposable objects in Db (A − mod ) up to shift are in bijection with homotopy strings and bands up to inverse. More precisely, the equivalence is modulo the equivalence relation σ ∼ σ for a homotopy string σ, infinite homotopy strings up to inverse, and pairs consisting of a homotopy band up to inverse and up to permutation and of an isomorphism class of indecomposable K[X]-modules. This bijection is the one described in Definition 2.3. Remark 2.4. If the field K is algebraically closed, then the matrix F of Definition 2.3 (3) can always be chosen to be a Jordan block Jm (λ) of size m corresponding • • to a scalar λ ∈ K. Note that for m = 1, we have Pσ,J = Pσ,λ . In the text, if the 1 (λ) result or proof does not depend on the scalar λ, we will sometimes omit it in our • . notation and we will write Pσ• instead of Pσ,λ 2.2. Main result on indecomposable objects of the derived category. We are now ready to prove our classification of indecomposable objects (up to shift) in Db (A − mod ) using curves on the surface SA . Before stating our main result, we recall, see for example [32, 48], the definition of the orbit category Db (A − mod )/[1] where [1] is the shift functor. Namely the objects of Db (A − mod )/[1] are the same as the objects of Db (A − mod ) and HomDb (A−mod )/[1] (X, Y ) = ⊕n∈Z HomDb (A−mod ) (X, Y [n]). Theorem 2.5. Let A = KQ/I be a gentle algebra with marked ribbon graph ΓA and a marked embedding in the associated ribbon surface SA . Let [1] be the shift functor in Db (A − mod ). Then (1) the isomorphism classes of the indecomposable string objects in Db (A − mod )/[1] are in bijection with the finite arcs on SA and infinite arcs on SA whose infinite rays circle around a boundary component in counter-clockwise orientation; (2) the isomorphism classes of the indecomposable band objects in Db (A − mod )/[1] are in bijection with the pairs (γ, M ), where γ is a closed curve on SA satisfying condition (3) of Lemma 2.10 below and M is an isomorphism class of indecomposable K[X]-modules.

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More precisely, arcs correspond to homotopy string complexes, infinite arcs correspond to infinite homotopy string complexes and closed arcs correspond to homotopy bands. Before proving the theorem, we need some results on the geometry of the lamination. Lemma 2.6. (1) The lamination LA subdivides SA into polygons whose sides are laminates and boundary segments. The laminates of LA can be chosen to be the “glued edges” of Definition 1.3. (2) Each polygon contains exactly one marked point. (3) Every boundary segment of SA contains the endpoint of at least one laminate of LA . Proof. It suffices to observe that the “glued edges” of Definition 1.3 cut the surface SA into the polygons Pv of Definition 1.3, which contain exactly one marked point each by definition. Moreover, every boundary segment of these polygons is adjacent to at least one laminate. 

Once we know that a surface is cut into polygons, then any arc is determined by the order in which it crosses the edges of the polygons. Note that the edges crossed correspond exactly to the laminates. Lemma 2.7. Let γ be a possibly infinite arc or a closed curve on SA , and assume that every laminate of LA that γ crosses, it crosses transversally (we can assume this, up to homotopy). (1) If γ is an arc, then it is completely determined by the (possibly infinite) sequence of the laminates that it crosses. (2) If γ is a closed curve, then it is completely determined by the sequence of the laminates that it crosses, up to cyclic ordering. The order in which an arc or a closed curve crosses the laminates gives rise to a homotopy string or band, as we will see in Lemma 2.10. Definition 2.8. Let Pv be a polygon on the surface SA , as per Lemma 2.6, and let Mv be the unique marked point in Pv . Let δ be a curve in Pv starting and ending on edges `1 and `2 of Pv which are laminates. • If Mv lies between `2 and `1 in the clockwise order, then let w1 , . . . , wr be the laminates between `1 = w1 and `2 = wr in clockwise order. By Proposition 1.21, these correspond to vertices of the quiver Q of A which are joined by arrows α1 , . . . , αr−1 . Then define σ(δ) := α1 · · · αr−1 . • If Mv lies between `1 and `2 in the clockwise order, then let w1 , . . . , wr be the laminates between `2 = w1 and `1 = wr in clockwise order. By Proposition 1.21, these correspond to vertices of the quiver Q of A which are joined by arrows α1 , . . . , αr−1 . Then define σ(δ) := (α1 · · · αr−1 )−1 .

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α3 δ α2 α1

Lemma 2.9. Let Pv and δ be as in Definition 2.8. Then σ(δ) is a homotopy letter. Proof. By Proposition 1.21, the compositions of the arrows of σ(δ) are not in the ideal of relations of A.  Lemma 2.10. (1) Let γ be a finite arc on SA . Let `1 , `2 , . . . , `r be the laminates crossed (in that order) by γ, as per Lemma 2.7. For every i ∈ {1, 2, . . . , r − 1}, let γi be the part of γ between its crossing of `i and of `i+1 . Let (Q r−1 i=1 σ(γi ) if r > 1; σ(γ) := e `1 if r=1. Then σ(γ) is a homotopy string. (2) Let γ be an infinite arc. Assume that on any infinite end of γ, the arc cycles infinitely many times around a boundary component in counter-clockwise orientation. Let (`i ) be the sequence of laminates crossed by γ (this sequence can be infinite on either side). For every i, let γi be the part of γ between its crossing of `i and of `i+1 . Let Y σ(γ) := σ(γi ). i

Then σ(γ) is an infinite homotopy string. (3) Let γ be a primitive closed curve on SA . Let `1 , `2 , . . . , `r be the laminates crossed (in that order) by γ. For every i ∈ {1, 2, . . . , r − 1}, let γi be the part of γ between its crossing of `i and of `i+1 , and let γr be the part of γ between its crossing of `r and of `1 . Let r Y σ(γ) := σ(γi ). i=1

If there is an equal number of inverse and direct homotopy letters among the σ(γi ), then σ(γ) is a homotopy band. Proof. In all three cases, for any index i, if σ(γi ) and σ(γi+1 ) are both direct homotopy letters, then by Proposition 1.21, composition of the last arrow of σ(γi ) and of the first of σ(γi+1 ) form a relation. The argument for consecutive inverse homotopy letters is similar. This proves (1). To prove (2), assume that γ is an infinite arc. Then γ eventually wraps around one of the boundary components without marked points. By Lemma 2.6, there is at least one laminate with one endpoint on this boundary component. Thus, by Proposition 1.21, every full turn of γ around the boundary component induces a

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subword of σ(γ) of the form α1 · · · αr , where the αi form an oriented cycle of Q such that every composition is a relation. Thus σ(γ) is eventually periodic, with homotopy letters of length one. Since the infinite ends of γ cycle around a boundary component in counter-clockwise direction, we get that the start (or the end) of σ(γ), if infinite, is a direct (resp. inverse) antipath. This proves (2). Qr To prove (3), assume that γ is a primitive closed curve. Write σ(γ) := i=1 σ(γi ) as in the statement of the Lemma. Clearly, s(σ(γ1 )) = t(σ(γr )) and σ(γ1 ) 6= σ(γr ). The condition on the number of inverse and direct homotopy letters among the σ(γi ) ensures that σ(γ) is a homotopy band.  Remark 2.11. In Lemma 2.10 (3), it should be stressed that the condition on the number of inverse and direct homotopy letters among the σ(γi ) is not satisfied by all closed curves. It would be interesting to find a geometric characterisation of the closed curves that do satisfy the condition. Conversely, any homotopy string or band defines an arc or a closed curve on SA . Lemma 2.12. (1) For any finite homotopy string τ , there exists a unique finite arc γ on SA (up to homotopy) such that τ = σ(γ). (2) For any infinite homotopy string τ , there exists a unique infinite arc γ on SA (up to homotopy) such that τ = σ(γ). (3) For any homotopy band b, there exists a unique closed curve γ on SA (up to homotopy) such that b = σ(γ). Proof. We only prove (1); the proofs of (2) and (3) are similar. Write τ = τ1 · · · τr , where each τi is a homotopy letter. Write τi = αi1 · · · αisi , where the αij are either all arrows or all inverse arrows. By Proposition 1.21, since there are no relations in the (possibly inverse) path αi1 · · · αisi , then there are laminates `1i , . . . , `si i +1 inside a unique polygon Pv such that `ji and `j+1 have an endpoint on the same boundary i j segment of Pv and `j+1 follows ` in the clockwise order if τi is a direct homotopy i i letter, and counter-clockwise order if τi is an inverse homotopy letter. Define γi to be a segment in Pv going from `1i to `si i +1 if τi is a direct homotopy letter, or the other way around if τi is an inverse homotopy letter. We can assume that the endpoint of γi is the starting point of γi+1 . If we define γ(τ ) to be the concatenation of γ1 , . . . , γr , then σ(γ(τ )) = τ1 · · · τr = τ . This proves the existence result. To prove uniqueness, assume that γ and γ 0 are such that σ(γ) = σ(γ 0 ). Let τ be the (unique) reduced expression of the homotopy string σ(γ) = σ(γ 0 ). Then γ(τ ) is homotopic to γ and γ 0 . Indeed, if σ(γ) is reduced, then τ = σ(γ) and we are done. Otherwise, it means that in the expression σ(γ)1 · · · σ(γ)r of σ(γ) as a product of homotopy letters, there are two adjacent letters σ(γ)i and σ(γ)i+1 that are inverse to each other. Then the corresponding segments in the polygon Pv described above are the same path going in opposite directions; their concatenation is thus homotopic to a trivial path. Thus if we cancel the two inverse homotopy  letters, we get that γ σ(γ)1 · · · σ(γ)i−1 σ(γ)i+2 · · · σ(γ)r is homotopic to γ σ(γ) . By induction on the number of reduction steps to get from σ(γ) to τ , we get that  γ σ(γ) = γ(τ ). The same applies if we replace γ by γ 0 . This proves the uniqueness, and finishes the proof of the Lemma. 

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With this, we can prove Theorem 2.5. Proof of Theorem 2.5. It follows from the results of [14] that indecomposable objects of Db (A − mod) are in bijection with homotopy strings (finite and infinite) and homotopy bands paired with an isomorphism class of indecomposable K[X]modules. By Lemma 2.10, we can associate a homotopy string or band to each of the curves listed in the statement of Theorem 2.5. Then Lemma 2.12 ensures that this defines the desired bijections.  3. Homomorphisms in the derived category of a gentle algebra The morphism spaces in the bounded derived category Db (A − mod ) of a gentle algebra A were completely described in [4]. Our aim in this section is to describe a basis of the morphism spaces in the orbit category Db (A − mod )/[1] in terms of curves on the surface SA that was associated to A in Section 1. 3.1. Bases for morphism spaces in the derived category. We now briefly recall the results of [4]. These results are proved in the case where the base field K is algebraically closed; for the rest of this section, we will assume that we are in this situation. Also, their results deal with morphisms in Db (A − mod ), but we will immediately translate them to the setting of the orbit category Db (A − mod )/[1]. This actually makes some combinatorial conditions simpler since, for instance, we do not need to keep track of the position of subwords in a given homotopy string or band. Let σ and τ be two homotopy strings or bands. Let λ, ν ∈ K ∗ . Let Pσ• and Pτ• be the associated indecomposable objects in Db (A − mod )/[1] (we write Pσ• instead • of Pσ,λ if σ is a homotopy band). In all that follows, we consider σ and τ only up to the action of the inverse operation ?; this means that whenever we are comparing σ and τ , we also need to compare σ and τ in order to get all morphisms. 3.1.1. Graph maps. Assume that σ and τ have a maximal subword in common, say σi σi+1 · · · σj and τi τi+1 · · · τj , with each σ` equal to τ` . We as well allow this subword to be a trivial homotopy string. Consider the following conditions. LG1: Either the homotopy letters σi−1 and τi−1 are both direct and there exists a path p in Q such that pτi−1 = σi−1 , or they are both inverse letters and there exists a path p in Q such that τi−1 = σi−1 p. LG2: The homotopy letter σi−1 is either zero or inverse, and τi−1 is either zero or direct. RG1: Dual of (LG1). RG2: Dual of (LG2). If one of (LG1) and (LG2) holds, and one of (RG1) and (RG2) holds, then one can construct a morphism from Pσ• to Pτ• called a graph map. Note that if σ and τ are infinite homotopy strings, then the definition above extends to the case where the strings have an infinite subword in common: if this subword is on the left, then one simply drops conditions (LG1) and (LG2), for instance.

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3.1.2. Quasi-graph maps. Keep the above notations. If none of the conditions (LG1), (LG2), (RG1) and (RG2) hold, then one can construct a morphism in Db (A − mod )/[1] from Pσ• to Pτ• , called a quasi-graph map. Again, this definition extends to infinite homotopy strings in the natural way. Note that a quasi-graph map gives rise to a homotopy class of single and double maps, defined in the next section. In fact, all single and double maps that are not singleton maps arise in this way, see [4]. 3.1.3. Single maps. Assume that there are direct homotopy letters σi and τj and a non-trivial path p such that s(p) = t(σi ) and t(p) = t(τj ). (What follows also works if σi and τj are both inverse letters by working with σ and τ instead). Consider the following conditions: L1: If σi is direct, then σi p = 0. L2: If τi is inverse, then pτ i = 0. R1: If σi+1 is inverse, then σ i+1 p = 0. R2: If τi+1 is direct, then pτi+1 = 0. If conditions (L1), (L2), (R1) and (R2) are satisfied, then p induces a morphism of complexes from Pσ• to Pτ• called a single map. Assume, moreover, that 0 0 • σi+1 is zero or is a direct homotopy letter of the form pσi+1 , where σi+1 is a direct homotopy letter; • τi is zero or is a direct homotopy letter of the form τi0 p, where τi0 is a direct homotopy letter. If that is the case, then p induces a morphism from Pσ• to Pτ• in Db (A−mod )/[1] called a singleton single map. 3.1.4. Double maps. Keeping the above notations, assume now that there are nontrivial paths p and q such that s(p) = s(σi ), t(p) = s(τj ), s(q) = t(σi ) and t(q) = t(τj ), and such that σi q = pτi . If conditions (L1) and (L2) above are satisfied for p and conditions (R1) and (R2) are satisfied for q, then p and q induce a morphism of complexes from Pσ• to (a shift of) Pτ• called a double map. If, moreover, there exists a non-trivial path r such that σi = σi0 r and τi = rτi0 , with σi0 and τi0 direct homotopy letters, then p and q induce a morphism from Pσ• to Pτ• in Db (A − mod )/[1] called a singleton double map. 3.1.5. The basis. We can now state the main result of [4]. Theorem 3.1 (Theorem 3.15 of [4]). A basis of the space of morphisms from Pσ• to Pτ• in Db (A − mod )/[1] is given by all graph maps, quasi-graph maps, singleton single maps and singleton double maps in Db (A − mod )/[1]. Definition 3.2. The basis described in Theorem 3.1 will be called the ALP basis. 3.2. Morphisms as Intersections. As before, let A = kQ/I denote a fixed gentle algebra throughout this section and SA denote its surface (see Section 1). The purpose of this section is to show that for indecomposables X, Y ∈ Db (A−mod )/[1] and any two representing curves γX , γY on SA (see Theorem 2.5) with a minimal possible number of crossings, the set of intersections (satisfying additional conditions) give rise to a subset of any ALP basis of HomDb (A−mod )/[1] (X, Y ) as described in

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Section 3.1 Before we start, let us set up some notation and clarify our assumptions for the objects involved. Throughout this section, we will replace all boundary components of SA without marked points by punctures, and consider infinite arcs wrapping around such a boundary component as a finite arc going to the puncture (see ˜ A the Remark 1.20). Denote π : S˜A → SA a fixed universal covering map and L ˜ A is a lamination of S˜A and set of all lifts of laminates L ∈ LA . In particular, L π induces a universal covering map (defined in the obvious way) from the ribbon ˜ A dual to L ˜ A to the ribbon graph dual to LA , which is ΓA (see Definition graph Γ ˜ ˜ A along edges. 1.8). As such, SA is glued from polygons indexed by the vertices of Γ −1 ˜ ˜ M := π (M) ⊆ ∂SA . Given a subset J ⊂ LA , we write |J| for the union of all laminates in J considered as a subset of S˜A . Assumption 3.3. For the entire section, we fix two objects X1 , X2 ∈ Db (A − mod )/[1] associated to curves γ1 , γ2 in SA . We identify a closed curve with its corresponding map S 1 → SA . We make the following assumptions on γ1 and γ2 : 1) If γi intersects the boundary, then γi is an arc and the intersection is one of its endpoints. 2) The set of curves {γ1 , γ2 } ∪ LA is in minimal position, that is, the number of intersections of each pair of (not necessarily distinct) curves in this set is minimal in their respective homotopy class. As pointed out in [58], it follows from [39] and [52] that these assumptions do not impose any restrictions on the homotopy classes of γ1 and γ2 . In particular, it follows from there that for any given γ1 , γ2 in minimal position, we can deform the laminates in such a way that {γ1 , γ2 } ∪ LA is in minimal position. We write γ1 ∩ γ2 for the set {(s1 , s2 ) ∈ [0, 1]2 | γ1 (s1 ) = γ2 (s2 )} of intersections. If it causes no ambiguity, we identify (s1 , s2 ) ∈ γ1 ∩ γ2 with its image γ1 (s1 ) = γ2 (s2 ) ∈ SA . With the relation to morphisms in mind, which have a domain and a codomain, we → − consider the set γ1 ∩ γ2 consisting of all interior intersections and those boundary intersections of γ1 and γ2 , such that locally around the intersection, γ1 ’lies before’ γ2 in the counter-clockwise orientation as shown in the following picture. γ1

∂SA γ2

Let us state the main result of this section. Theorem 3.4. Let X1 , X2 ∈ Db (A − mod )/[1] be string or band objects sitting at the mouth of a homogeneous tube and for each i ∈ {1, 2}, let γi be a representative of Xi , such that the number of intersections of γ1 and γ2 is minimal in their respective

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homotopy classes. Let B be the basis of HomDb (A−mod )/[1] (X1 , X2 ) described in Theorem 3.1. There exists an explicit injection → − B : γ1 ∩ γ2 ,→ B. Moreover, the following hold true. i) If X1 and X2 are not isomorphic band objects, then B is a bijection. ii) If X1 and X2 are isomorphic band objects, B is not surjective and the missing elements in its image are an invertible graph map and its dual quasi map. Remark 3.5. If X := X1 = X2 in Theorem 3.4, then the quasi map mentioned ∼ X[1] in an Auslanderin ii) corresponds to a connecting morphism X → τ X[1] = Reiten triangle in Kb (A − proj) ending in X. The proof of Theorem 3.4 occupies the rest of this section. The underlying idea is that if two curves intersect, they arrive from distinct directions and stay close to each other in a neighbourhood until they eventually depart towards different directions resulting in the effect that certain subcurves share a sequence of intersections with the laminates. We will make this more precise now. The Walk of an Intersection. We will be lifting arcs on SA to arcs on S˜A and closed curves on SA to infinite lines on S˜A . More precisely, let Ωi denote the universal covering space of the domain of γi and ρi the corresponding covering map. For convenience, we may assume ρi is given by either the identity or the exponential map exp : R → S 1 . By a lift of γi we mean an equivalence class of maps γ˜ : Ωi → S˜A , such that π ◦ γ˜ = γi ◦ ρi , where two such maps γ˜ , γ˜ 0 are considered equivalent if and only if there exists a Deck transformation T : Ωi → Ωi , such that γ˜ = γ˜ 0 ◦ T . → − Let p = (s1 , s2 ) ∈ γ1 ∩ γ2 and let q˜ be any preimage of q := γ1 (s1 ) = γ2 (s2 ) under the covering map π. For each i ∈ {1, 2} and each preimage s˜i ∈ ρ−1 i ({si }), there exists a unique lift γ˜i : Ωi → S˜A of γi , such that γ˜i (˜ si ) = q˜. Denote J the set of all ˜ of L ˜ A , such that L ˜ ∩ γ˜i = ∅ for some i ∈ {1, 2}. Denote Sq˜ the closure of the arcs L connected component of S˜A \ |J| which contains q˜. In other words, Sq˜ is the region of S˜A containing the laminates intersected by both γ˜1 and γ˜2 . This region is a disc, as we now show. Somethings missing. Before this we need the result that lifts of γi are simple in order to use the bigon criterion Lemma 3.6. For each i ∈ {1, 2}, let γ˜i be a lift of γi . The following hold true. • Either the curves γ˜1 and γ˜2 are homotopic, or they have only one intersection point. • The surface Sq˜ is homoeomorphic to a closed disc. Proof. It suffices to show that Sq˜ contains only a finite number of laminates. As there is nothing to show if γ1 or γ2 is an arc, we assume that γ1 and γ2 are closed curves and suppose Sp contains an infinite number of laminates. Consequently, σ(γ1 ) = σ(γ2 ) and hence γ1 ' γ2 by primitivity. If q˜ = (s1 , s2 ), let ti ∈ R, such that si ∈ [ti , ti+1 ] and such that γ˜1 (t1 ) and γ˜2 (t2 ) lie on a single laminate L. The order of γ˜1 (t1 ) and γ˜2 (t2 ) on L is the same as the order of γ˜1 (t1 +1) and γ˜2 (t2 +1) on the respective laminate. It follows that γ˜1 |[t1 ,t1 +1] and γ˜2 |[t2 ,t1 +2] intersect at least twice. It follows from [57] that γ˜1 and γ˜2 are simple. By the bigon criterion (see [37], Proposition 1.7), we can find a homotopy from γ˜1 |t1 ,t1 +1] which descends to a

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δ1

δ2

˜ A , whereas the blue The surface Sq˜: Dashed curves belong to L and the red curve show δ1 and δ2 . Solid black lines belong to ∂ S˜A . σ(p) has 4 letters in this example. homotopy of γ1 that reduces the number of intersections with γ2 - a contradiction. Next, suppose q˜, q˜0 ∈ γ˜1 ∩ γ˜2 are distinct interior intersections. Since each γ˜i intersects a laminate at most once, it follows Sq˜ = Sq˜0 and the first part implies that there exists a homotopy from a subarc γ˜i |[t,t+1] of γ˜1 which descends to a homotopy from γ1 which reduces the number of intersections with γ2 .  The sequence of those intersections of δi with the laminates of LA which are not an endpoint of δi , define a (possibly empty) homotopy string σi . By definition of δ1 and δ2 , σ1 = σ2 and we write σ(p) := σ1 . If σ(p) is non-empty, it is a subwalk of σ(γi ) for each i ∈ {1, 2} in a canonical way, where in case of a homotopy band w = w1 · · · wn , we mean that σ(p) is a subwalk of the cyclic two-sided infinite walk · · · w1 · · · wn w1 · · · . We emphasize that referring to σ(p) as a subwalk of σ(γi ) refers to a specific occurrence of σi as a subsequence of σ(γi ). If on the other hand σ(p) is empty, it means that π ◦ δ1 and π ◦ δ2 are contained in ˜ A , i.e. the closure of a connected component a single polygon P of the lamination L of SA \ |LA |. Remark 3.7. Since we have replaced boundary components without marked points with punctures, one should note that if γ1 and γ2 meet at a puncture, then they cross infinitely many laminates. In this case, the walk σ(p) is infinite on one side. Lemma 3.8. If σ(p) is empty, then the unique marked point in P as before is contained in M (see Definition 1.8). Proof. Suppose the marked point in P is an element in M0 . Then the only laminate on the boundary of Sq˜ (as constructed in the definition of σ(p)) is a lift of the laminate L associated with a vertex x ∈ Q0 . But by definition of Sq˜, there exists i ∈ {1, 2}, such that γi does not cross L. Thus, γi is contained in P and homotopic to a constant path - a contradiction.  We now explain how an intersection gives rise to a morphism. For j ∈ {1, 2}, denote p1j , . . . , pm j the ordered sequence of intersections of the curve δj with the boundary or the laminates of LA . We may assume that if σ(p) is non-empty, then

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23

for each i ∈ (1, m), pi1 and pi2 lie on the same laminate. Then δ1 and δ2 in Sq˜ are homeomorphic to two arcs crossing in a closed disc; their endpoints alternate on the boundary of Sq˜. We distinguish two cases: a) if σ(p) is non-empty, then either p12 comes immediately before p11 in the counter-clockwise orientation of ∂Sq˜, or vice versa; b) if σ(p) is empty (i.e. m = 2), let w be the unique marked point on the boundary of the polygon containing δ1 and δ2 . Then either w is after an endpoint of δ2 and before an endpoint of δ1 in the counter-clockwise oritentation of ∂Sq˜, or vice versa. The subsequent lemmata prove that δ1 and δ2 encode a basis element in a natural way. Lemma 3.9. Let B be the basis of HomDb (A−mod )/[1] (X1 , X2 ) described in Theo→ − rem 3.1, and let p ∈ γ1 ∩ γ2 . Then p gives rise to an element B(p) ∈ B. Furthermore, i) If σ(p) is non-empty, then B(p) is a graph map if p12 comes immediately before p11 in the counter-clockwise orientation, and a quasi map otherwise. ii) If σ(p) is empty, then p 6∈ ∂SA and B(p) is a singleton single or singleton double map. Proof. Suppose first that σ(p) is non-empty. Let Pv be the polygon in which p11 , p12 , p21 , p22 are found, and let w be the marked point on its boundary. The position of w with repsect to p11 and p12 decides whether one of the conditions (LG1) to (RG2) of Section 3.1 is satisfied, as illustrated below. Note that this includes the case that p is a boundary intersection, i.e. p11 and p12 both coincide with the marked point.

δ1 δ2

δ1 δ2

Figure 5: The endpoint conditions LG1/RG1 (left) and LG2/RG2 (right) m As the analogous statements hold for pm−1 , pm−1 , pm 1 and p2 , it follows that the 1 2 1 intersection point p defines the data of a graph map if p2 comes immediately before p11 in the counter-clockwise orientation and the data of a quasi map otherwise. We note that in the case where δ1 and δ2 meet in one puncture, then σ(p) is infinite on one side, and we need to look at the conditions (LG1) to (RG2) on one side only. This finishes the proof for case i). To prove case ii), suppose that σ(p) is empty and that p 6∈ ∂SA . Consequently, δ1 and δ2 are contained in a single polygon Pv . Depending on the position of the

24

SEBASTIAN OPPER, PIERRE-GUY PLAMONDON, AND SIBYLLE SCHROLL

marked point in Pv , p gives rise to different types of singleton maps. If the marked point lies between pj2 and pj1 in counter-clockwise order for some j ∈ {1, 2}, then p defines the data of a singleton map, which is a single map. In the other situation,

δ1

δ1

δ2

δ2

Figure 6 i.e. the marked point lies between pj1 and pj2 in counter-clockwise order, p gives rise to a singleton double map. In the previous picture this is the situation we obtain by interchanging the labels δ1 and δ2 . Finally, if δ1 or δ2 have a marked endpoint it can be seen that p gives rise to a singleton single map.

δ1

δ2 Figure 7  Remark 3.10. The graph and quasi maps which occur in the previous Lemma as B(p) are associated with a finite subword or an infinite subword which is bounded in one direction. In particular, this excludes the case of two-sided infinite subwords as in the case of invertible graph maps. Remark 3.11. Suppose γ1 ' γ2 and p ∈ ∂SA . Then X1 = X2 [m] for some m ∈ Z and fp is an invertible graph map with all components given by a multiple of the identity. If X1 = X2 the identity can be either seen as a map X1 → X2 or as a map X2 → X1 , the former being represented by the unique boundary intersection in → − γ1 ∩ γ2 and the latter being represented by the other boundary intersection, which → − constitutes an element in γ2 ∩ γ1 .

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25

Lemma 3.12. Let f ∈ HomDb (A−mod )/[1] (X1 , X2 ) be an element of B which is neither a graph map nor a quasi map associated with an two-sided infinite subword → − (see Remark 3.10). Then there exists a unique p ∈ γ1 ∩ γ2 , such that f = B(p). Proof. Let γ˜1 be a lift of γ1 . We distinguish two cases. First, assume that f is a graph or quasi map and denote σ the maximal common subword associated with f as in Section 3.1. Our assumptions imply that σ is finite. ˜ denote a laminate crossed by γ˜1 , which corresponds to s(σ) and denote γ˜2 Let L ˜ As above, when we defined σ(p) of an the unique lift of γ2 , which intersects L. intersection p, define S as the closure of the connected component of S˜A \ |J|, ˜ A , such that L ∩ γ˜i = ∅ for some where J denotes the set of all laminates L ∈ L i ∈ {1, 2}. Denote δi the restriction to γ˜i to S, i.e. δi := γ˜i |γ˜ −1 (S) . We orient δ1 i ˜ A \ J in the same order. and δ2 in such a way that they intersect the laminates L Since σ is finite, S is homeomorphic to a closed disc and does not consist of a single polygon. As we have seen in the proof of Lemma 3.9, the endpoint conditions are equivalent to certain cofigurations of δ1 , δ2 and the marked point in the ’outer polygons’ of S (see Figure 5), which force δ1 and δ2 to intersect in a unique point q. It follows by Lemma 3.6 that p := π(q) is the unique intersection of γ1 and γ2 and by construction, B(p) = f . Next, assume that f is a singleton single or singleton double map. If f is a single map, denote p the non-trivial path which appears in the definition of single maps, see Section 3.1.3. Otherwise, let p denote the non-trivial path, which was denoted by r in the definition of singleton double maps, see Section 3.1.4. There exists a ˜ A , which corresponds to p and is crossed by γ˜1 . We polygon S of the lamination L write γ˜2 for the unique lift of γ2 , which crosses S, and denote δi the restriction of γ˜i to S. The combinatorial conditions in the definition of singleton single and singleton double maps are then equivalent to certain configurations of the marked point in S and the endpoints of δ1 and δ2 as shown in Figure 6 and Figure 7. As → − usually, this proves that δ1 and δ2 intersect in a (again unique) point p ∈ γ1 ∩ γ2 , such that B(p) = f .  The previous lemma finishes the proof of Theorem 3.4. 4. Mapping Cones in the derived category of a gentle algebra In this Section we will show that the mapping cone of a map in Db (A − mod ) is given by the homotopy strings of the two curves resolving the corresponding crossing. Theorem 4.1. [30] Let A be a gentle algebra and let Pσ• and Pτ• be indecomposable objects in Db (A) with homotopy strings or bands σ and τ . Let f • ∈ HomDb (A) (Pσ• , Pτ• ) be an ALP basis element. Then the indecomposable summands of the mapping cone Mf•• are given by the homotopy strings and bands occurring in the green and red boxes resulting from the following graphical calculus. (1) Let σ = . . . σi−2 σi−1 σi . . . σj σj+1 σj+2 . . . and τ = . . . τi−2 τi−1 τi . . . τj τj+1 τj+2 . . . and suppose f • is a graph map with common homotopy substring σi . . . σj = τi . . . τj . Then Mf•• = Pc•1 ⊕ Pc•2 with homotopy strings c1 = . . . σi−2 σi−1 τ i−1 τ i−2 . . . and c2 = . . . τ j+2 τ j+1 τj+1 τj+2 . . .

26

SEBASTIAN OPPER, PIERRE-GUY PLAMONDON, AND SIBYLLE SCHROLL

is given by: · · · σi−3 σi−2



σi−1

σi





σj





σj+1



p

· · · τi−3 τi−2

σj+2 σj+3 · · ·

q



τi−1





τi





τj

τj+1



τj+2 τj+3 · · ·

(2) Let σ = . . . σi σi+1 . . . and τ = . . . τj τj+1 . . . and suppose f • is a single map. Then Mf•• = Pc•1 ⊕ Pc•2 with homotopy strings c1 = . . . σi−1 σi pτ j τ j−1 . . . and c2 = . . . σ i+2 σ i+1 P τj+1 τj+2 . . . is given by: · · · σi−1 σi−1

σi



σi+1





p

· · · τj−2 τj−1



σ i+1 pτj+1 τj+1



τj

σi+2 σi+3 · · ·



τj+2 τj+3 · · ·

(3) Let σ = . . . σi−2 σi−1 σi σi+1 σi+2 σi+3 . . . and τ = . . . τj−2 τj−1 τj τj+1 τj+2 . . . and suppose f • is a double map. Then Mf•• = Pc•1 ⊕ Pc•2 with homotopy strings c1 = . . . σi−2 σi−1 pτ j−1 τ j−2 . . . and c2 = . . . σ i+2 σ i+1 qτj+1 τj+2 . . . is given by: · · · σi−3 σi−2



σi−1



σi

p

· · · τi−3 τi−2



τj−1

σi+1





σi+2 σi+3 · · ·

q



τi



τj+1



τj+2 τj+3 · · ·

Remark 4.2. Note that quasi-graph maps are implicitly treated in the previous theorem, since they give rise to homotopy classes of single and double maps. Theorem 4.3. Let f • : Pσ• → Pτ• be a map in Db (A−mod ) and let Mf•• = Pc•1 ⊕Pc•2 be its mapping cone (as described in Theorem 4.1). Then the curves γ(c1 ) and γ(c2 ) corresponding to Pc•1 and Pc•2 are given by the following resolution of the crossing of the curves γ(σ) and γ(τ ) corresponding to Pσ• and Pτ•

γ c1 γσ

γτ

γ c2

Figure 8. Curves associated to the mapping cone Mf•• = Pc•1 ⊕Pc•2 of a map f • : Pσ• → Pτ• .

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Proof. We first consider the case when f • is a graph map. Locally in the surface this corresponds to the following configuration.

γ c1

γ:::τi−2 τi−1

γ:::σi−2 σi−1

γσi :::σj = γτi :::τj

γσi+1 σi+2 :::

γτi+1 τi+2 :::

γ c2

Figure 9. Curves associated to the mapping cone Mf•• = Pc•1 ⊕ Pc•2 of a map f • : Pσ• → Pτ• when f • is a graph map. In this picture we have decomposed γ(σ) into segments γ(. . . σi−2 σi−1 ), γ(σi . . . σj ) and γ(σj+1 σj+2 . . .) and γ(τ ) into segments γ(. . . τi−2 τi−2 ), γ(τi . . . τj ) and γ(τj+1 τj+2 . . .). The blue dotted region corresponds to the topological disc Sq˜ of Section ??. Thus we see that the curve γc1 at the top is split into two subcurves, so that γ(c1 ) = γ(. . . σi−2 σi−1 ) γ(. . . τi−2 τi−1 )−1 . This proves that Pc•1 is has the form in the statement of the theorem. A similar argument at the bottom of the picture proves the result for Pc•2 . Next, we treat the case of single maps. In that case, γ(σ) and γ(τ ) meet in a polygon which forms the whole of Sq˜.

γ:::σi−2 σi−1

γτi

γτi+1 τi+2 :::

γ:::τi−2 τi−1

γ c1

γ σi

γ c2

γσi+1σi+2:::

Figure 10 We see that γ(c2 ) is obtained by γ(τi+1 τi+2 . . .)−1 γ(p)−1 γ(σi+1 σi+2 . . .), as in the previous case. We also see that γ(c1 ) is obtained in a similar fashion, by noticing that σi pτ i contains one copy of p, since σi ends in (and τ i starts in) p−1 . The remaining cases of a double map or of a single map arising from an intersection on the boundary of SA are treated in a similar fashion.  5. Auslander-Reiten triangles It is shown in [44] that in the bounded derived category of a module category of a finite dimensional algebras, Auslander-Reiten triangles ending in an indecomposable object X exist if and only if X is perfect. For a gentle algebra A, the perfect

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SEBASTIAN OPPER, PIERRE-GUY PLAMONDON, AND SIBYLLE SCHROLL

objects in Db (A − mod ) are given by the string objects with finite homotopy string and the band objects. The purpose of this section is to show that an AuslanderReiten triangle of a string object X can be read off by means of any arc γX on SA which represents X in the sense of Theorem 2.5. The precise results are the following (the notations of the results will be explained later in the section). Theorem 5.1. Let w be a homotopy string and for each u ∈ {w, w+ ,+ w, τ−1 w}, let γu be an arc on SA associated with u, such that all arcs are in minimal position. There exists an Auslander-Reiten triangle 

Pw•

f+ + f



Pw• + ⊕ P+•w

( g+

+

g)

Pτ•−1 w

h

Pw• [1]

such that the following holds true. + and γ+ w ' + γw . 1) We can orient γX in a way, such that γw+ ' γw • • 2) Every morphism Pu1 → Pu2 in the triangle above is a morphism associated → − with the distinguished intersection γu1 ∩ γu2 . With each boundary component B of SA we associate a certain homeomorphism τB of SA . We will see that Theorem 5.1 implies the following Corollary. Corollary 5.2. Let X ∈ Kb (A − proj)/[1] be indecomposable and Q let γX (resp. γτX ) be a representing curve of X (resp. τX) on SA . Define τ := B (τB ), where B is indexed by the set of connected components of SA . Then, γτX = τ ◦ γX . In Figure 11 we give an example of the geometric realisation of the AuslanderReiten translate of an indecomposable object in Db (A − mod ).

γX

τ γX = γ τ X

Figure 11. Example of the curves associated to an indecomposable object X in Db (A − mod ) and its Auslander-Reiten translate τX. 5.1. Auslander-Reiten triangles in bounded homotopy categories. Since gentle algebras are Gorenstein [40], it follows from [45], see also [44], that the category Kb (A − proj) has Auslander-Reiten triangles, i.e for every indecomposable object Z ∈ Kb (A − proj), there exists a distinguished triangle X

f

Y

g

Z

h

X[1]

in Kb (A − proj), such that the composition h ◦ u vanishes for every non-split morphism u : U → Z. Such a triangle is unique up to (non-unique) isomorphism. Furthermore, X is indecomposable and is denoted by τZ. It defines a bijection τ

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on the set of isomorphism classes of indecomposble objects. The first explicit description of such triangles for gentle algebras was given in [16]. We recollect some important facts about an Auslander-Reiten triangles Pw•

f

Y

g

Z

h

X[1]

in Kb (A − proj). • Y is the direct sum of at most two indecomposable objects. • If Pw is a band complex associated with a m-dimensional K[X]-module, then Z ∼ = Pw• and Y is (isomorphic to) a direct sum of band complexes associated with m ± 1-dimensional K[X]-modules. • If Pw is a string complex, then Z and the direct summands of Y are isomorphic to string complexes Pτ−1 w , Pw• + and P+•w (if defined) associated with homotopy strings τ−1 w, w+ and + w, which we will recall below. • We can identify Y with Pw+ ⊕ P+ w and Z with Pτ−1 w in such a way that h is element of an ALP basis and f (resp. g) is the sum of ALP basis elements f + : Pw• → Pw• + and + f : Pw• → P+•w (resp. g + : Pw• + → Pτ•−1 w and + g : P+•w → Pτ•−1 w ). It can be shown that if Pw is a band complex associated with a 1-dimensional K[X]-module, then h as above is not represented by an intersection. In particular, if the representing closed curve of Pw is simple, there is no chance for any of the maps in an Auslander-Reiten triangle to be represented by an intersection. As a consequence, we restrict ourselves to Auslander-Reiten triangles which involve string complexes. For the rest of this section, let w be a fixed homotopy string. We postpone the description of the homotopy string w+ and f + to the proof of Theorem 5.1. The homotopy strings τ−1 w, + w and the other maps f , g + , + g are implicitly defined  + via the relations + w = w+ and τ−1 w = + (w+ ) = (+ w) . For example, g + is given by the map u+ : P+ w → P(+ w)+ = Pτ−1 w . The definition of τ−1 has to be + understood in the sense that at least one of the expressions + (w+ ) and (+ w) is defined, and, in case both are defined, they agree. 5.2. Geometric description of Auslander-Reiten triangles. In this section we will prove Theorem 5.1 and Corollary 5.2. To describe the action of the Auslander-Reiten translation in terms of curves, we need the following definition. Definition 5.3. Let D be an oriented annulus, B a boundary component of D and I a finite set of marked points on B. Choose an arbitrary orientation preserving homeomorphism φ : D → D≤2 \ D