Morphisms between indecomposable complexes in the bounded derived category of a gentle algebra

In this article we provide a simple combinatorial description of morphisms between indecomposable complexes in the bounded derived category of a gentle algebra.


Introduction
Derived categories have become useful tools in many areas of mathematics since their introduction in the Grothendieck school of algebraic geometry in the 1960s. In representation theory, for instance, their usefulness is manifest from Rickard's celebrated theorem on derived Morita theory [22], which explains that they are the proper setting for tilting theory. Moreover, they permit the flow of ideas and insight between different areas of mathematics, a trend which was begun in Beilinson's [7] famous equivalence of the derived categories of coherent sheaves on projective spaces and certain finite-dimensional algebras. However, despite their widespread utility certain drawbacks remain: their construction is abstract, and explicit computation is often difficult. In this paper, we aim to make some progress toward alleviating some of these drawbacks.
In the representation theory of finite-dimensional algebras, the structure of the derived categories of hereditary algebras is well-understood. However, this is due to a particularly nice homological property which forces the indecomposable objects of the derived category to be shifted copies of the indecomposable objects of the module category. This allows one to make computations by reducing to the module category.
In representation theory, some recent papers have made derived categories of a certain class of algebras accessible. These are the so-called gentle algebras introduced by Assem and Skowroński [4] in the 1980s. Originally introduced in the context of iterated tilted algebras of type A, they have recently seen a resurgence in the context of cluster theory, occurring as the surface algebras associated with triangulations of certain Riemann surfaces with marked points [3]. They are, therefore, an interesting and topical class of algebras to study in themselves.
However, from the point of view of derived categories they are perhaps of even more interest. The class presents us with a particularly good candidate to begin a systematic 'hands on' study of derived categories of non-hereditary algebras for a number of reasons: • Certain general aspects of their structure are already known: the Avella Alaminos-Geiß invariant describes certain fractionally Calabi-Yau triangulated subcategories whose Auslander-Reiten (AR) components have a boundary [5]. • Vossieck [23] introduced the family of derived-discrete algebras, for which we now understand various non-trivial homological properties [10,11,12,13,14]. These algebras are gentle, and thus they can be used as a template for further study of derived categories of gentle algebras. • Inspired by a classic paper [16] on string algebras, Bekkert and Merklen [8] explicitly described the indecomposable objects of the derived category of a gentle algebra in terms of so-called 'homotopy strings' and 'homotopy bands'. Note that we employ the terminology of [9]. See [15] for related work. • Using the Happel functor F : D b (Λ) ֒→ mod(Λ) [19], whereΛ is the repetitive algebra of Λ, Bobiński [9] gave an algorithm which computes the AR triangules in D b (Λ). Thus, almost all the pieces have been assembled to facilitate effective computation in derived categories of gentle algebras. Indeed, in [2], the authors were able to apply Bobiński's algorithm to describe the AR components of the categories of perfect complexes over cluster-tilted algebras of type A using the parameters involved in Bastian's [6] derived equivalence classification. See [1] for similar results for cluster-tilted algebras of type A.
However, one basic tool is missing: it would be useful to understand all the maps between indecomposable complexes, thereby extending Bobiński's description of irreducible maps. In this paper, we compute a canonical basis for the space of homomorphisms between two indecomposable complexes in D b (Λ): see Theorem 3.15, Theorem 5.10 and Proposition 5. 16. We note that the module category analogue of this was done by Crawley-Boevey and Krause in a series of papers [17,18,21]. In the final two sections, we apply our main result to recover Bobiński's description [9] of the irreducible maps, and to recover a universal Hom dimension bound for discrete derived categories, cf. [13].
We would like to stress here that our results are completely combinatorial descriptions that take place entirely in the homotopy/derived category. One does not need to employ the Happel functor, and therefore, the often unpleasant computations this entails can be avoided. It is thus hoped that these results complete the basic description of derived categories of gentle algebras, and provide tools to gain further understanding of their structure -and thus more concrete intuition regarding triangulated and derived categories. Immediate applications we would have in mind would be determination of tilting and silting objects in such categories.
(1) for each vertex x of Γ, there are at most two arrows starting at x and at most two arrows ending at x; (2) for any arrow a in Γ there is at most one arrow b in Γ such that ab / ∈ I and at most one arrow c in Γ such that ca / ∈ I; (3) for any arrow a in Γ there is at most one arrow b in Γ such that ab ∈ I and at most one arrow c in Γ such that ca ∈ I; (4) the ideal I is generated by paths of length 2. Let P (x) be the indecomposable projective left Λ-module corresponding to x ∈ Γ 0 . We recall the following useful property of gentle algebras; see, for instance [8,Section 3]. Convention 1.2. From now on, by abuse of notation, we shall identify a path p : x ❀ y with its corresponding basis element in Hom Λ (P (y), P (x)).
Throughout this article, we shall fix a gentle algebra Λ = kΓ/I over an algebraically closed field k. Algebraic closure of k is not strictly necessary, but it significantly simplifies the presentation of the combinatorics.
All modules in this paper will be left modules. We shall be interested in three categories: • C := C −,b (proj(Λ)): the category of right bounded complexes of finitely generated projective Λ-modules whose cohomology is bounded; • K := K b (proj(Λ)): the homotopy category of bounded complexes of finitely generated projective Λ-modules -the so-called perfect complexes; • D := D b (Λ) = D b (mod(Λ)): the bounded derived category of finitely generated Λ-modules. Throughout the paper, we shall identify the bounded derived category D with the triangle equivalent category K −,b (proj(Λ)), consisting of right bounded complexes of finitely generated projective Λ-modules whose cohomology is bounded. This identification allows us to use the combinatorics of homotopy strings and bands, which will be described in the next section, throughout the paper. We direct the reader to consult [19] for background on derived and homotopy categories.

Indecomposable objects in D
In this section we give an overview of Bekkert and Merklen's description of the indecomposable objects in D. Their crucial observation is that it is enough to consider complexes where the differential is given by matrices whose entries are either zero or a path (cf. Convention 1.2). The indecomposable objects are obtained by unravelling the differential into 'homotopy strings and bands' corresponding to perfect complexes, and 'infinite homotopy strings' for the unbounded complexes. The reader is encouraged to have the following example in mind when reading this section: Running Example. Let Λ = kΓ/I be given by the quiver The following complex, with the leftmost non-zero term in cohomological degree 0, is an indecomposable object of K: Observe that the criterion d 2 = 0 is obtained by either passing through a relation, or by having 0 in the differential. We notice that this complex 'unfolds' as where the cohomological degrees are written above each module. Moreover, the modules appearing are uniquely determined by the endpoints of the maps, so all information in this complex is communicated by the diagram This is what we will later define as a 'homotopy string'. Another way of encoding this object is as a 'string with degrees' (e, 2, 1)(f, 1, 0)(c, 0, 1)(b, 1, 2)(af, 2, 3).
Remark 2.1. By using transposed matrices, matrix multiplication (i.e. composing maps) fits with composition of paths. For instance; ( c f ) ( b 0 0 e ) = ( cb f e ) = 0. 2.1. Homotopy strings. A homotopy letter is a triple (p, i, j) where p is a path in Γ with no subpath in I, and i, j are integers such that |i − j| ≤ 1. The homotopy letter (p, i, j) is called direct if i < j and inverse if i > j. If i = j then p must be a stationary path and is called a trivial homotopy letter. We set (p, i, j) −1 := (p, j, i). The starting and ending vertices of a homotopy letter are defined as Convention 2.2. We shall often write p as shorthand for (p, i, j). However, the degrees i, j should always be considered to be implicitly present.
A homotopy string is a sequence of pairwise composable homotopy letters such that the following holds: (1) whenever (w r , i, i + 1)(w r−1 , i + 1, i + 2) occurs, w r w r−1 has a subpath in I; (2) whenever (w r , i, i − 1)(w r−1 , i − 1, i − 2) occurs, w r−1 w r has a subpath in I; (3) whenever (w r , i, i + 1)(w r−1 , i + 1, i) occurs, w r and w r−1 do not start with the same arrow; (4) whenever (w r , i, i − 1)(w r−1 , i − 1, i) occurs, w r and w r−1 do not end with the same arrow.
Another way of expressing a homotopy string w = 1 r=n (w r , i r , j r ) is by a diagram i n j n j 2 j 1 where the line labelled w r is an arrow pointing to the right if w r is direct, and an arrow pointing to the left if w r is inverse. Note that each vertex wr • w r−1 corresponds to a unique indecomposable projective Λ-module, namely P (s(w r )) = P (t(w r−1 )).
Let w = 1 r=n (w r , i r , j r ) be a homotopy string. The string complex P w corresponding to w is constructed as follows. We define indexing sets Sitting in degree i, the corresponding complex has the object P i w given by r∈I i P (ϕ w (r)), where ϕ w (r) = t(w r ) for r > 0 and ϕ w (0) = s(w 1 ).

Homotopy bands.
A non-trivial homotopy string w = 1 r=n (w r , i r , j r ) is a homotopy band if s(w) = t(w), i n = j 1 , one of {w n , w 1 } is direct and the other inverse, and w is not a proper power of another homotopy string.
We now describe how to construct one-dimensional band complexes. The higher dimensional band complexes will be studied in Section 5, and the definition is given there. Fix a homotopy band w and an element λ ∈ k * . Again we define indexing sets and B i w,λ,1 is defined as for string complexes, that is, For band complexes with n ≥ 2, the components of the differential are determined as for string complexes, except for that corresponding to w 1 , which acquires the scalar λ: P (ϕ w (1)) λw 1 −→ P (ϕ w (n)) for w 1 direct, and P (ϕ w (n)) λw 1 −→ P (ϕ w (1)) for w 1 inverse. When n = 2, the only non-zero component of the differential of B w,λ,1 is λw 1 + w 2 . Note that B w,λ,1 ∼ = B w −1 , 1 λ ,1 . The diagram notation of a homotopy band is an infinite repeating diagram, with the scalar λ attached to w 1 as in the complex: We write this as the diagram Convention 2.4. From now on, unless explicitly needed for emphasis, we shall omit the degrees from unfolded diagrams. However, all homotopy letters in this article carry degrees, therefore, the reader should be aware that they are implicitly always present.
2.3. Infinite homotopy strings. These indecomposable complexes only occur when Λ has infinite global dimension: when the global dimension of Λ is finite all the complexes in D are isomorphic to perfect complexes. If Λ = kΓ/I has infinite global dimension, then Γ contains oriented cycles with 'full relations'. Let C(Λ) denote the collection of arrows a ∈ Γ 1 such that there exists a repetition-free cyclic path a n · · · a 2 a 1 in Γ such that a i+1 a i ∈ I for 1 ≤ i ≤ n and a 1 a n ∈ I, where a 1 = a. We need the following definition from [9].
Definition 2.5. A direct (resp. inverse) antipath is a homotopy string where all homotopy letters are direct (resp. inverse) and arrows in the quiver.
Definition 2.6. Let w = 1 k=n (w k , i k , j k ) be a homotopy string. We say w is (1) left resolvable if (w n , i n , j n ) is direct (i.e. j n = i n + 1), i n ≤ i k , j k for all 1 ≤ k < n, and there exists a ∈ C(Λ) such that (a, i n − 1, i n )w is a homotopy string. We shall say that w is left resolvable by a. (2) primitive left resolvable if there is no direct antipath t k=n (w k , i k , j k ) such that 1 k=t−1 (w k , i k , j k ) is left resolvable. One can write down the obvious dual definitions of (primitive) right resolvable. A homotopy string will be call (primitive) two-sided resolvable if it is both (primitive) left resolvable and (primitive) right resolvable.
If w is left (resp. right) resolvable by a, then gentleness of Λ ensures that this is the unique arrow in Γ 1 by which w is left (resp. right) resolvable. If w is left resolvable then w −1 is right resolvable, and vice versa. We shall call a left or right resolvable homotopy string that is not two-sided resolvable one-sided resolvable. There is then the obvious notion of primitive one-sided resolvable.
Suppose w = 1 k=n (w k , i k , j k ) is left resolvable by a 1 ∈ C(Λ), which sits in the repetition-free cyclic path a m . . . a 1 in Γ. We form the left infinite homotopy string ∞ w by concatenating infinitely many appropriately shifted copies of the cycle on the left of w, i.e. the unfolded diagram of ∞ w has the form Analogously, we form the right infinite homotopy string w ∞ and the two-sided infinite homotopy strings ∞ w ∞ from a right resolvable homtopy string or two-sided resolvable homotopy string, respectively.
If w is one-sided left resolvable we obtain the corresponding infinite string complex from the left infinite homotopy string ∞ w, or equivalently the right infinite homotopy string (w −1 ) ∞ . Dually for w right resolvable. For w two-sided resolvable we represent it by ∞ w ∞ and ∞ (w −1 ) ∞ .
2.4. The indecomposable objects of D. We first set up some notation. A cyclic rotation of the homotopy band w is a homotopy band of the form Consider the equivalence relation ∼ −1 generated by identifying a homotopy string with its inverse, and the equivalence relation ∼ r generated by identifying a homotopy band with its cyclic rotations and their inverses. The following will denote complete sets of representatives of the specified objects under the given equivalence relations: We mention here that a homotopy band can always be considered as a homotopy string, thus we must take the disjoint union: a band gives rise both to a string complex and a family of band complexes.

Morphisms between indecomposable objects of D
In this section, we shall describe a canonical basis for the set of homomorphisms between (finite or infinite) string and/or one-dimensional band complexes; higher dimensional bands are dealt with in Section 5. The proof that the maps described here do indeed form a basis is contained in Section 4.
We first set up some notation and define three canonical classes of maps. Fix w ∈ St⊔Ba and λ w ∈ k * . Define We have abused notation here by allowing the scalar λ w to disappear when dealing with Q w in the case that w is a homotopy band; it should be treated as implicitly present.
For f ∈ Hom C (Q v , Q w ) and a degree t, the map f t : Q t v → Q t w can be written as a matrix between the finitely-many indecomposable summands of Q t v and Q t w . Each entry of this matrix is a linear combination of paths (see Proposition 1.1). We refer to a single term in this sum as a component of f t . Moreover, a component of f is taken to be a component of f t for some degree t. Throughout this section we fix two homotopy strings or bands v and w and the corresponding complexes Q v and Q w . We consider 'maps' between the unfolded diagrams of Q v and Q w which, at each projective, will look like: If f is at the leftmost end of the unfolded string complex Q v we say that v L is zero; likewise for v R , w L , and w R . Again, f is a linear combination of paths and we will use the term component to refer to a single summand of f . In the next sections we shall define maps occurring in a canonical basis of Hom C (Q v , Q w ).

3.1.
Single and double maps. Suppose we are in the following situation: where f is some non-stationary path in the quiver.
Definition 3.1. A map in Hom C (Q v , Q w ) will be called a single map if it has only one non-zero component whose unfolded diagram is as above and satisfies the following conditions: Then we have the following: which gives rise to a single map P v → P w : Suppose we have the following situation: such that ( * ) commutes for non-stationary paths f L , f R . Definition 3.3. If (L1) and (L2) hold for f L and (R1) and (R2) hold for f R , then the diagram above induces a map Q v → Q w with two non-zero components in consecutive degrees given by f L and f R . We call such a map a double map; write D v,w for the set of double maps Q v → Q w .

Remark 3.4.
To obtain all double maps Q v → Q w in terms of diagrams as above, it is necessary to consider overlaps between unfolded diagrams of Q v and any Q w ′ such that Q w ∼ = Q w ′ ; see Section 2.4 for the relevant equivalence relations on homotopy strings and bands.
The unfolded diagram gives rise to the double map: Notation 3.6. Let f ∈ S v,w and suppose the unique non-zero component of f corresponds to the path p : x ❀ y in Γ. Then we write f = (p). Similarly, for f ∈ D v,w whose two nonzero components correspond to the paths p : x ❀ y and q : x ′ ❀ y ′ , we write f = (p, q). The notation should be suggestive of an infinite vector, in which all entries are zero apart from those which are written.
We next highlight two important classes of single and double maps.
Definition 3.7. Recall the setup in (1). A map f ∈ S v,w will be called a singleton single map when its non-zero component is given by a path p and we are in the situation of the following unfolded diagrams (up to inverting one of the homotopy strings): where r ∈ {1, n}, s ∈ {1, m}, either v r is inverse (or zero) or v r p = 0, and either w s is inverse (or zero) or pw s = 0. We denote the set of all singleton single maps Definition 3.8. Recall the setup in (2). A map f = (f L , f R ) ∈ D v,w will be called a singleton double map if it satisfies the following condition (or its dual).
We denote the set of all singleton double maps Condition (D) means that to find singleton double maps, it is sufficient to look for homotopy letters v i of v and w j of w such that v i and w j sit in the same degrees, with the same orientation, and a 'proper right substring' of v i is a 'proper left substring' of w j .

3.2.
Graph maps and quasi-graph maps. Suppose the unfolded diagrams of Q v and Q w overlap as follows: The double lines represent isomorphisms and all of the squares with solid lines commute as paths in the quiver. Consider the following left endpoint conditions. ( LG1) The arrows v L and w L are either both direct or both inverse and there exists some (scalar multiple of a) non-stationary path f L such that the square ( * ) commutes. (LG2) The arrows v L and w L are neither both direct nor both inverse. In this case, if v L is non-zero then it is inverse and if w L is non-zero then it is direct.
There are dual right endpoint conditions, which we call (RG1) and (RG2), respectively.
Definition 3.9. If one of (LG1) or (LG2) hold and one of (RG1) or (RG2) hold, the diagram induces a map Q v → Q w , whose non-zero components are exactly those described by the diagram. Such maps are called graph maps; write G v,w for the set of graph maps Q v → Q w . The unfolded diagram on the left gives rise to the graph map on the right.
Definition 3.11. If none of the conditions (LG1), (LG2), (RG1) nor (RG2) hold, then the diagram no longer induces a map, however we shall say that there is a quasi-graph map from Q v to Q w .
Then there exist p single maps Q v → Q w given by the paths u p , u p−1 , ..., u 1 in the appropriate degrees. There are also two (single or double) maps with non-zero components given by v L , w L , v R or w R in the appropriate degree. For example, if v L is direct and w L is inverse than there is a double map (v L , w L ). We call these maps the associated quasigraph map representatives. Let Q v,w be a fixed set of quasi-graph map representatives The following diagram describes a quasi-graph map Q v → Σ −1 Q w : Remarks 3.14. We highlight the following.
(1) (Quasi-)graph maps extend to infinite homotopy strings in the obvious way; if a left (respectively right) endpoint condition holds and the substrings to the right (respectively left) are equal and infinite then we can define a (quasi-)graph map with infinitely many components which are isomorphisms. Cf. [18] for infinite graph maps between modules. (2) To standardise what we mean by a (quasi-)graph map we take the following conventions. If v and w are both homotopy strings, then each isomorphism is an identity. If one of v and w is a homotopy band, then the leftmost isomorphism in the diagram is an identity and the remaining isomorphisms will be determined by this. Note that any other choice of isomorphisms will result in a scalar multiple of such a standardised map. (3) It is necessary to replace w by an equivalent homotopy string/band to obtain all of the possible (quasi)-graph maps Q v → Q w ; cf. Remark 3.4. (4) There is one pathological example arising from the identity map B w,λ,1 → B w,λ,1 . This is a graph map since it 'travels' the whole way around the band. In particular, there are no real endpoint conditions. As such it also defines a quasi-graph map B w,λ,1 → Σ −1 B w,λ,1 . This situation will be treated in more detail in Section 5.
3.3. The main theorem. We have now assembled all the maps that occur in a canonical basis of Hom D (Q v , Q w ) and can state our main result succinctly as the following.
The next section concerns the proof of this result.
The set B D v,w has two elements. The first is a singleton single map (af ) in degree 2 from the right end-point of v to the left end-point of w. The second is given by the following quasi-graph map Q v → Σ −1 Q w , where the associated quasi-graph map representatives Q v → Q w are indicated by dashed arrows (single maps) and dotted arrows (double map):

Proof of the main theorem
Let v, w ∈ St ⊔ Ba and Q v , Q w be as in the previous section. We shall split up the proof of Theorem 3.15 into two parts. The first part establishes a canonical basis for Hom C (Q v , Q w ). In the second part, we identify which elements of this basis are homotopic or null-homotopic.

4.1.
A basis at the level of complexes. In this section, we establish the following: . The proof of Proposition 4.1 is inspired by [18,Section 1.4]. We first need two technical lemmas from which we will deduce that B C v,w is a linearly independent set. Lemma 4.2. Suppose we have the following situation: where f C is a non-stationary path and v L is direct if and only w L is direct; similarly for v R and w R . If the squares ( * ) and ( * * ) commute, then at most one of ( * ) and ( * * ) has a non-zero commutativity relation.
Proof. We analyse the case when all four homotopy letters v L , v R , w L and w R are direct; the remaining cases are analogous. Suppose ( * ) has a non-zero commutativity relation. Then It follows that the paths f C and w L start with in the same arrow. By the definitions of homotopy strings and bands, w L w R = 0 and, by condition (4) of gentleness, we must also have that f C w R = 0. That is, f C w R = v R f R = 0 and ( * * ) has a zero commutativity relation.
Dually, whenever ( * * ) has a non-zero commutativity relation, ( * ) has a zero commutativity relation. Proof. We give a proof for the case where we have a non-zero component of a graph map. The arguments for single and double maps are analogous.
We make use of the diagram from Lemma 4.2. Suppose that f C is a non-zero component of a graph map. We show that, if it exists, f L is completely determined. Note that if v L and w L have different orientations then the arrow f L does not make sense since the degrees will not match. So suppose v L and w L are either both direct or both inverse.
Suppose f C is an isomorphism and v L and w L are both direct. Then the square ( * ) must commute so f L is the unique path such that Suppose f C is given by a path, then according to Lemma 4.2 we must be at the left or right endpoint of the diagram. If v L is direct and v L f C = 0, then we must be at the left end of the diagram and f L = 0. If v L f C = 0, then we are are at the right end of the diagram and f L is an isomorphism. Similarly, when w L is inverse and f C w L = 0, then f L = 0. If f C w L = 0, then f L is an isomorphism.
We can apply dual arguments to conclude that the diagam is also completely determined to the right (i.e. f C determines f R ).
We show that b cannot be written as a linear combination of Then some b i , 1 ≤ i ≤ n, must also have this non-zero component because the algebra is gentle and so only has zero relations. By Lemma 4.3, b = b i and so k i = 1 and i−1 ). By the shape of homotopy strings and bands, P (ϕ v (a)) and P (ϕ w (b)) are each connected to at most two non-zero components of the differential. We must therefore consider the unfolded diagrams (as in Lemma 4.2 but with h t ab in place of f C ). Without loss of generality, assume that h t ab is an isomorphism or a scalar multiple of a path. By Lemma 4.3, there is a unique (scalar multiple of an) element of B C v,w with this component and, by Lemma 4.2, this must be a summand of h. If this is not the whole of h, then we choose another non-zero component of h and continue until we have found a complete decomposition of h. Thus h ∈ Span B C v,w . Proposition 4.1 gives us canonical bases for the Hom spaces between indecomposable complexes of D considered as objects of C. We next turn our attention to homotopy classes of these maps. The following section highlights the strategy of our approach.

4.2.
The strategy for constructing homotopy classes. We first recall the general definition; we direct the reader to a standard textbook on homological algebra, for example [20,24] for more information regarding homotopies.
where, as before, the maps are composed from left to right, see Remark 2.1). The family of maps {h i } is called a homotopy from f to g. If g = 0 then f is called null-homotopic.
Consider a map f ∈ B C v,w and let p be a component of f . We can write down an unfolded diagram representation of this component as follows.
Thus, the corresponding component of a homotopic map must be given by for some scalars α, β, γ, δ ∈ k and paths a, b, c, d in the quiver. If the composition of paths does not make sense we take the corresponding scalar to be zero. For instance, if v L is direct then α = 0. Therefore, in order to construct homotopies between maps in B C v,w it is enough to look at ways to construct the path p by 'completing' differential components.
Definition 4.6. We denote by H(f ) the set of maps f ′ such that f ≃ f ′ and f ′ = λg for some g ∈ B C v,w and λ ∈ k * . Remark 4.7. The set H(f ) is not the same as the homotopy class of f , however, if g ≃ f then the decomposition of g into a linear combination of elements of B C v,w will consist of elements of H(f ) ∪ H(0) only. Hence, it suffices to determine the sets H(f ). If f is non-zero and H(f ) is a singleton set then we will say that f belongs to a singleton homotopy class.

4.3.
Basic maps f such that H(f ) is not singleton. Here we start with f ∈ S v,w ∪ D v,w ; we shall see in Section 4.4 that we do not need to consider f ∈ G v,w . The reader may find it helpful to recall Definitions 3.9 and 3.11 and the corresponding endpoint conditions. Proof. For simplicity, we consider only the case f ∈ S v,w . The case f ∈ D v,w is similar. The setup is the following, where by abuse of notation we have denoted the map and its unique non-zero component by f : As explained in Section 4.2, the components corresponding to f in any homotopic map can be constructed only from four possible paths v ′ −1 ,v ′ 0 , w ′ −1 or w ′ 0 illustrated in the following diagrams: Observe that f can be immediately seen to be null-homotopic in the following cases: Suppose we are not in any of the cases (N1)-(N3). Since H(f ) = {f }, at least one, but possibly both, of the following must hold: • f is built from the source differential, i.e. f = v −1 or f = v 0 (but not both); or • f is built from the target differential, i.e. f = w −1 or f = w 0 (but not both). Suppose f can be built from the source differential; the argument when f can be built from the target differential is dual. Without loss of generality, we assume f = v 0 ; if f = v −1 , invert the homotopy string v and re-label v −1 as v 0 . The differential v 0 will be called a used differential, because it has already been used to construct one of the single or double maps, namely f in this instance. There are two cases.
Case: There are no arrows out of ⋆ w . We must be in the situation that w −1 is direct, w 0 is inverse, and v 1 exists and is inverse -for otherwise we would be in case (N2) above. The following diagram describes the situation.
In particular, f = (v 0 ) is homotopic to the single map −h = −(v 1 ). We can now see part of the quasi-graph map constructed: namely, the equality written diagonally. We now wish to continue by building h from a differential. The differential v 1 has already been used, so in order to continue, we must see whether h = w 0 or h = w −1 , and then use the dual argument for maps built from the target differential.
Case: There is an arrow out of ⋆ w . Without loss of generality, assume w 0 = 0 is direct. Note that only one of w 0 or w −1 may be an arrow out of ⋆ w -otherwise f cannot be a well-defined single map. Then there exists The used differentials are v 0 and w 0 in the first case; in the second case v 1 is additionally a used differential. The situation is illustrated below: This gives −g ∈ H(f ) or −(h, g) ∈ H(f ). If the map we obtain at this step is a single map then we carry on, using the dual argument if necessary, to obtain further elements of H(f ). The algorithm terminates when we reach one of the following three cases.
• We reach a single map g for which one of the conditions (N1), (N2) or (N3) is satisfied. In this case f is null-homotopic. • We reach a single map g which is not equal to any of the unused differentials with respect to the already constructed elements of H(f ). This places us in case (3) of the proposition. • We reach a double map; here there are insufficiently many unused differentials to continue to use to construct a homotopy. This places us in case (1) or (2) of the proposition.
If f can also be built from the target differential, we must now return to f and carry out the dual algorithm.
We start by observing that graph maps belong to singleton homotopy classes, and thus are never null-homotopic.
Proof. Suppose f ≃ g : Q v → Q w . Since f is a graph map, there is an unfolded diagram: Without loss of generality, assume that f i = 1 for 0 ≤ i ≤ p. Consider the component f i .
Denote the corresponding component of the map g by g i , which may be zero. Existence of a homotopy between f and g means that the difference between f i and g i is a linear combination, , where the α, β, γ, δ are scalars and the a, b, c, d are paths in the quiver corresponding to the homotopy maps. If the composition does not make sense, we take the corresponding scalar to be zero. Since components of the differential are never zero, the compositions u i a, u i+1 b, cu i and du i+1 are either zero or non-stationary paths in the quiver. It follows that α = β = γ = δ = 0 and g i = 1. Lemma 4.3 gives f = g, whence H(f ) = {f }. If i = p then this argument should be adjusted in the obvious way.
Next we consider singleton homotopy classes of single and double maps. It is clear that a single map f ∈ S v,w is in a singleton homotopy class exactly when its unfolded diagram corresponds to one of (i) − (iv) in Definition 3.7. Therefore, we need only examine when a double map occurs in a singleton homotopy class.
Recall the setup from Section 3.1(2) on page 8. We say that f = (f L , f R ) has no common substring if v 0 = f L f ′ and w 0 = f ′ f R , and has common substring s if f L = v 0 s and f R = sw 0 . Note that these are the only ways in which the commutative square ( * ) in diagram (2) can decompose. Furthermore, s may be a stationary path, in which case f has trivial common substring.
The following lemma shows that any double map in a singleton homotopy class satisfies condition (D) of Definition 3.8. This then completes the proof of Theorem 3.15. (1) If H(f ) = {f }, then f has a common substring.
(2) If f has a non-trivial common substring, then f is null-homotopic.
Proof. Suppose H(f ) is not a singleton set. Then one of the following must hold: then 16 since f L w 0 = v 0 f R it follows that v 0 and v L start with the same arrow, a contradiction. If w ′ L w L = f L , then since f L w 0 = 0, we must have that w L w 0 = 0, a contradiction. Thus v 0 v ′ 0 = f L and v ′ 0 is a non-zero component in the homotopy. But then v ′ 0 w 0 = f R and so f has a common substring v 0 .
Let s be a non-trivial common substring for f , then we can take the required family of maps to be zero everywhere except for the component P (s(v 0 )) → P (t(w 0 )) which is taken to be s.

Higher-dimensional band complexes
Each pair (w, λ) ∈ Ba × k * determines a homogeneous tube in K ⊂ D: where we refer the reader to Section 5.1 for a precise definition of the higher dimensional band B w,λ,r for r > 1. We shall show that the dimensions of the Hom spaces involving a higher dimensional band complex can be determined using the dimension of the Hom space of the corresponding one-dimensional band occurring at the mouth of the tube.
We start by making these definitions precise and describing unfolded diagrams for higher dimensional tubes. For simplicity, in this section we shall assume that k is an algebraically closed field. 5.1. Definition, example and unfolded diagrams. Let (w, λ, r) ∈ Ba × k * × N and recall that B w,λ,1 := (B i w,λ,1 , D i ), with D i = (d i jk ), denotes the one-dimensional band complex. The r-dimensional band complex is defined as follows: The unfolded diagram of B w,λ,r consists of r aligned copies of the unfolded diagram of B w,λ,1 arranged from top to bottom of the page called layers, which are connected by downwards arrows corresponding to the non-zero entries of A i , called links.
Example 5.1. Let z be the homotopy band from the Running Example. The twodimensional band complex B z,λ,2 is The corresponding unfolded diagram is:

5.2.
Passing through the link.
Definition 5.2. Let 1 ≤ m ≤ r and 1 ≤ n ≤ s. We say a map f ∈ Hom D (B v,λ,r , B w,µ,s ) is lifted from a map f ′ ∈ Hom D (B v,λ,1 , B w,µ,1 ) to the pair (m, n) if the components of f ′ from layer m of B v,λ,r to layer n of B w,µ,s are exactly the same as the components of f ′ and f is the minimal such map in terms of number of non-zero components.
For f ′ ∈ Hom D (B v,λ,1 , B w,µ,1 ), we shall count the number of (homotopy classes of) maps in Hom D (B v,λ,r , B w,µ,s ) which are lifted from f ′ . The idea is to put a copy of f ′ between the pair (m, n) of layers and see if a map arises; such a map will be called a candidate map. If a component of f ′ composes non-trivially with a link arrow, we say that the (candidate) map passes through the link. This will cause lifted maps to have non-zero components between more layers than just the pair (m, n). λ,1 , B w,µ,1 ) such that f is lifted from f ′ to a pair (m, n).
Proof. As with one-dimensional maps, all maps in Hom D (B v,λ,r , B w,µ,s ) are completely determined by any of their non-zero components. Ignoring the link arrows between layers, we simply have r copies of the band in B v,λ,r and s copies in B w,µ,s . It follows that if there is a map f : B v,λ,r → B w,µ,s with a non-zero component from layer m ′ of B v,λ,r to layer n ′ of B w,µ,s , then there is a map f ′ ∈ B v,λ,1 → B w,µ,1 with the same non-zero component and f is lifted from f ′ .
The following lemma is straightforward. λ,1 , B w,µ,1 ) to the pair of layers (m, n). Then: (1) if f ′ is a single map, then f does not pass through any link; (2) if m = r, then f does not pass through a link in B v,λ,r ; (3) if n = 1, then f does not pass through a link in B w,µ,s .
Recall the notation in Notation 3.6. To take care of homotopies for higher-dimensional homotopy bands, we need to modify Definition 4.6 slightly. Recall that the link in B v,λ,r is given by the homotopy letter v 1 .
Definition 5.5. Let f : B v,λ,r → B w,µ,s be a map lifted from a map f ′ to the pair (m, n) as in Definition 5.2. We denote by H (m,n) (f ) the set of k-linear combinations and ( v 1 ), ( w 1 ) are the single maps (v 1 ), (w 1 ) lifted to pairs (m + 1, n) and (m, n − 1), respectively. When m and n are understood, we simply write H(f ) for H (m,n) (f ).
Note that ρ 2 = 0 if and only if the homotopy map passes through the link in B v,λ,r ; similarly for ρ 3 . Thus, the homotopy class of f can be determined by H(f ) and H(f ′ ).

5.3.
A worked example. Before discussing the general behaviour of maps involving higher-dimensional homotopy bands, it is useful to examine an example in detail. This example will exhibit all possibilities regarding lifting of maps and homotopy classes and clarify the strategy in the proofs of the general results.
Throughout this worked example, Λ will be given by the following bound quiver. We consider the homotopy bands v = (e, 0, 1)(c, 1, 0)(a, 0, 1)(bb ′ , 1, 2)(f, 2, 1)(g, 1, 0) and where the degree i will be specified by the diagrams occurring in each example in the context of the particular map or homotopy class we are interested in.
Our first example indicates the typical situation of lifting a singleton homotopy class.
Example 5.6. The candidate map is a graph map h ′ ∈ Hom(B w,λ,1 , B v,µ,1 ) lifted to the pair (1, 1) of layers in Hom(B w,λ,2 , B v,µ,1 ). The components of the lifted map h which are forced by passing through the link are drawn as broken lines (they are all identities).
Note that h ′ also lifts to a map which includes a copy of h ′ from layer 2 of B w,λ,2 to the unique layer of B v,µ,1 . Thus, the graph map h ′ lifts to two maps in Hom D (B w,λ,2 , B v,µ,1 ).
Our next example examines the case of lifting a non-singleton homotopy class. Recall the notation for homotopy equivalent basis maps in Definition 5.5.
We shall now lift H((b)) to Hom D (B v,λ,1 , B w,µ,2 ). From Lemma 5.4(1) it is clear that each map in H((b)) gives rise to two maps in Hom C (B v,λ,1 , B w,µ,2 ) without adding any extra components. For convenience, decorate maps to the first layer of B w,µ,2 with a tilde, i.e.b and so on, and maps to the second layer with a hat, i.e.b and so on.
We use the following diagram to determine H((b)).
The homotopy passes through the link, so we have It is easy to see that (â) ≃ (−ĉ) ≃ (ê,d) ≃ (−µb). Therefore, we determine H((b)) and H((â)) as follows: Now, we have three different homotopy classes, but each of them is a k-linear combination of the other two. This shows that such a homotopy class lifts to two homotopy classes of maps in Hom C (B v,λ,1 , B w,µ,2 ), thus giving rise to two maps in Hom D (B v,λ,1 , B w,µ,2 ).
In the next example we look at what happens when one tries to lift an isomorphism. The following example shows that one cannot lift an identity morphism on a one-dimensional homotopy band to every pair of layers (m, n).
Example 5.8. Consider the homotopy band w above and let λ ∈ k * . Taking a copy of the identity from layer 1 of B w,λ,2 to B w,λ,1 , we are forced to take the dashed components as before. However, once we reach the end of the band we must add dashed arrows to the right-hand side of the link as well: but then the square involving the link does not commute: we have 2b + λb = b + λb. Therefore, we cannot lift a copy of the identity to layer 1 in B w,λ,1 and get a well-defined map of homotopy band complexes.
In the final example of this section, we consider the homotopy class arising from the identity map on a one-dimensional band complex.
Example 5.9. Consider the complex B w,λ,1 . The identity map on B w,λ,1 gives rise to a homotopy class in Hom D (B w,λ,1 , ΣB w,λ,1 ) which is non-zero: where we have written the (b) twice to emphasise the following key point: this gives a non-trivial way in which to obtain the tautologous homotopy equivalence b − b ≃ 0. Note that there can be other (homotopy classes of) maps in Hom(B w,λ,1 , ΣB w,λ,1 ); these behave as in Examples 5.6 and 5.7.

5.4.
Maps which are not self-extensions. In this section we shall make a number of statements regarding dimensions of Hom spaces. It is useful to first set up some notation. Let Rad D (P • , Q • ) denote the space of non-isomorphisms P • → Q • . Following standard notation in algebraic geometry, we write We now state the main result of this section.
We now prove the first assertion of Theorem 5.10 in a sequence of lemmas; the second assertion is proved similarly. From now on assume that v, w ∈ Ba, λ, µ ∈ k * and B v,λ,1 ≇ ΣB w,µ,1 . λ,1 , B w,µ,1 ). If H(f ′ ) = {f ′ } and f ′ is not an isomorphism, then f ′ can be lifted to any pair of layers in B v,λ,r and B w,µ,s .
Proof. This follows directly from Lemma 5.4, noting that the resulting maps may acquire extra components if they pass through a link.
The following lemmas deal with the generic case of Theorem 5.10; the example to bear in mind is Example 5.7.
Lemma 5.12. Let v and w be homotopy bands, and λ, µ ∈ k * such that B v,λ,1 = ΣB w,µ,1 and B v,λ,1 = B w,µ,1 . Let f (m,n) : B v,λ,1 → B w,µ,1 be a lift of f ∈ S v,w ∪ D v,w , which is not null-homotopic, to the layers (m, n). Consider the lift g (m,n) of any map g ∈ H(f ). Then g (m,n) is homotopic to linear combination of representatives from only H (i,j) (f ) for i ≥ m and j ≤ n. 5.5. Self-extensions of bands. As in Lemma 5.14, we must be careful when considering the homotopy class in Hom C (B v,λ,1 , ΣB v,λ,1 ) corresponding to the identity in Hom C (B v,λ,1 , B v,λ,1 ). Here, the example to keep in mind is Example 5.9.
Lemma 5.15. Let 1 ≤ i ≤ r and 1 ≤ j ≤ s. Denote the lift of (v 1 ) to layers (i, j) by Proof. The single map (v 1 ) : B v,λ,1 → ΣB v,λ,1 is a representative of the quasi-graph map corresponding to the identity B v,λ,1 → B v,λ,1 . Moreover, this quasi-graph map determines a non-trivial homotopy from (v 1 ) to (v 1 ). Lifting this non-trivial homotopy to the pair of layers (i, j) gives ,j) , which outputs the homotopy equivalences as claimed.
Proposition 5.16. Let r, s ∈ N, v ∈ Ba and λ ∈ k * . Then Proof. Observe that every basis element of Hom D (B v,λ,1 , ΣB v,λ,1 ) passes through the link at most once except for the quasi-graph map corresponding to the shifted identity. As before, each of these basis elements can be lifted to (m, n) for any m, n and so hom D (B v,λ,r , ΣB v,λ,s ) is at least rs · (hom D (B v,λ,1 , ΣB v,λ,1 ) − 1). It remains to determine how many linearly independent homotopy classes are lifted from this remaining homotopy class.
Repeated application of Lemma 5.15 yields the following homotopy equivalences: . . .
Therefore, if r ≥ s, then none of these homotopy equivalences includes a zero map, and therefore none of the homotopy classes above are null. In particular, there are s homotopy classes. If r < s, the the first s − r homotopy classes are actually null-homotopic, which leaves s − (s − r) = r non-null homotopy classes. To see that all other lifts of (v 1 ) are null-homotopic, we again apply Lemma 5.15. In this case, it yields that 0 All that remains to show is that lifts of all other single maps occurring as representatives of the quasi-graph map corresponding to id B v,λ,1 are linear combinations of representatives of the homotopy classes described above. This follows directly from Lemma 5.12.
Many tubes occurring in representation theory are hereditary. It is natural to ask whether the same is true for the homogeneous tubes that arise from homotopy bands. This can be seen not to be the case using the following straightforward example.

Application: Irreducible morphisms between string complexes
In this section, we recover Bobiński's description [9] of the irreducible maps in K and extend it to D. In [9], Bobiński employed the Happel functor F : D ֒→ mod(Λ) to determine the AR structure of K ֒→ D. We present an algorithm intrinsic to K −,b (proj(Λ)), which allows one to compute all irreducible maps in D. (Note that only the full subcategory of perfect complexes K ֒→ D admits AR triangles.) By [16], it is known that for any P ∈ ind(K), there are at most two irreducible maps starting and ending at P .
• We must consider formal inverses of trivial homotopy strings, say (1 x , i, i) −1 = (1 x , i, i), as distinct homotopy strings; of course they give rise to the same complex. • For each homotopy string w, there are unique trivial homotopy strings 1 x and 1 y such that the compositions w1 x and 1 y w are defined.
We set up some notation and terminology.
The support of f is the homotopy substring supp(f ) := l k=n (w k , i k , j k ) of w. For l ≤ i ≤ n we will also say that f is supported at P (ϕ w (i)). If f has only one non-zero component, say f 0 : P (ϕ w (k)) → P (ϕ v (l)), we say that f is supported at P (ϕ w (k)).
Recall the definition of antipath from Definition 2.5. A direct (resp. inverse) antipath θ is called maximal if there is no a ∈ Γ 1 such that θ(a, i, i + 1) is a direct antipath (resp. (a, i, i − 1)θ is an inverse antipath). Remark 6.2. (Maximal) antipaths do not have to be finite; however, we cannot compose a homotopy string with an infinite antipath if this takes us outside D. In particular, if θ = θ 1 θ 2 · · · is an infinite direct antipath and w is a finite homotopy string such that wθ 1 is defined, then the composition wθ is not in D. Similarly, if θ = · · · θ 2 θ 1 is an infinite inverse antipath and θ 1 w is defined, then the composition θw is not in D. Note that infinite antipaths correspond to oriented cycles in the quiver with 'full relations'. 6.1. Algorithm for determining irreducible maps. The algorithm is stated here for K. We extend it to infinite strings in Section 6.2. Let w = 1 k=n (w k , i k , j k ) ∈ St. The strategy of the algorithm is as follows: consider the identity map on P w : We alter the map 'from the left' in a minimal way to get a new map. This gives a new string w + and a new map f + : P w → P w + , which may each be zero. However, when they are nonzero, the resulting map will turn out to be irreducible. This deals with one of the two possible irreducible maps. To get the other, we alter the map in a minimal way 'from the right'. This gives a new string w + and a new map f + : P w → P w + , which again may each be zero. At least one of f + or f + will be nonzero. We explicitly give the algorithm which alters the map in a minimal way 'from the left'. The algorithm doing this 'from the right' is dual.
Algorithm 6.3. The algorithm proceeds as follows, where we carry out each step in sequence unless instructed otherwise. The map f ′ is defined in each step in Figure 1.
Step 1: If there is a direct homotopy letter u such that uw is a homotopy string and u is a maximal path, we set w ′ = uw, and go to Step 8.
Step 2: Remove the longest direct antipath which is a left substring of w. Write ψ w := w n · · · w r+1 for this antipath.
Step 3: If (w r , i r , j r ) is inverse and there exists a ∈ Γ 1 such that w r a = 0, then set w ′ = (w r a, i r , j r ) 1 k=r−1 (w k , i k , j k ), and go to Step 8.
Step 4: If (w r , i r , j r ) is inverse but there is no a ∈ Γ 1 such that w r a = 0, then set w ′ = 1 k=r−1 (w k , i k , j k ), and go to Step 9.
Step 5: If (w r , i r , j r ) is direct, then decompose w r = aa ′ with a ∈ Γ 1 and set w ′ = (a ′ , i r , j r ) 1 k=r−1 (w k , i k , j k ), and go to Step 8.
Step 6: If w is a direct antipath and there exists a ∈ Γ 1 with t(a) = ϕ w (0) with w 1 a = 0, then set w ′ to be the trivial homotopy string such that ww ′ is defined, and go to Step 8.
Step 7: Set f ′ = f + to be the zero map P w → 0 and terminate the algorithm.
Step 8: If there is a maximal inverse antipath θ = 1 k=m (θ k , i ′ k , j ′ k ) such that θw ′ is defined as composition of homotopy strings, we set w + = θw ′ and let f + have the same components as f ′ and terminate the algorithm.
Step 9: Set f + = f ′ and w + = w ′ . Notation 6.4. We shall refer to the the dual algorithm that alters a homotopy string in a minimal way 'from the right' as Algorithm 6.3 ′ ; a prime will also be affixed to denote the dual of each of the corresponding steps, i.e. the dual of Step i is Step i ′ .
Note that Step 7 only occurs for homotopy strings that are either trivial or antipaths. Remark 6.5. We highlight the following: (1) The maps f + and f + , when non-zero, are either graph maps or single maps.
(2) If both f + and f + are non-zero, then the two maps are different at the level of chain maps. If w or w ′ are trivial, then this is taken care of by the functions S and T . (3) Whenever there is an a ∈ Γ 1 with s(a) = x, then Step 1 or its 'right' dual will occur when considering the trivial homotopy string (1 x , i, i). If there are two arrows a and b with s(a) = x = s(b) then both Step 1 and its dual will occur.
Step 3 Step 5 Step 6 Figure 1. Diagrams defining the homotopy strings w ′ and maps f ′ produced in each step of Algorithm 6.3, with the maximal inverse antipath θ of Step 8 also added where appropriate.
(4) A case analysis shows that if a maximal antipath θ is added in Step 8, then this antipath is finite.
Theorem 6.6. Let w be a homotopy string, and let f + and f + be the outputs produced by Algorithm 6.3 and its dual, respectively. If f + (respectively f + ) is non-zero, then it is irreducible.
In the remainder of this section, we verify Theorem 6.6 in the case that f + = 0. The case f + = 0 is dual. 6.2. Irreducible maps involving infinite homotopy strings. Let w ∈ St 1 ∪ St 2 . We extend the above algorithm by possibly extending graph maps to infinite graph maps, and by adding the following step between Step 8 and Step 9: Step 8.5: If there is a maximal, infinite inverse antipath θ = · · · (θ 1 , i ′ 1 , j ′ 1 ) such that (θ 1 , i ′ 1 , j ′ 1 )w ′ is defined as composition of homotopy strings, we set f + to be the zero map P w → 0 and terminate the algorithm.
As commented above, the condition of this step is impossible if w is a finite homotopy string. Moreover, this step ensures that we never go outside D. Proof. The 'if' direction is already established. Consider v ∈ St 1 such that w := ∞ v is left infinite. When we run the algorithm, we perform Step 2 and remove the left infinite part of w, which corresponds to a cyclic path ρ. We will then be able to add an arrow a in one of Step 3, Step 5 or Step 6 (take a to be the arrow preceding w r−1 in ρ). Next Step 8.5 will occur, since we can always add infinitely many copies of ρ, but this time as inverse homotopy letters.
If v ∈ St 1 and w := v ∞ , we will never remove the infinite part from w as it consists of inverse homotopy letters. Hence, if the output w + is non-zero, it is also in St 1 . Corollary 6.8. If w is a two-sided infinite homotopy string, then there are no irreducible maps in D with P w as source.
In the remainder of the section, we consider only irreducible maps in K, but it is clear that the irreducible maps involving one-sided homotopy strings will behave in the same way.
6.3. Non-irreducible maps. Before verifying Theorem 6.6, we eliminate some maps which can be easily seen to be non-irreducible. Proposition 6.9. Let v, w ∈ St and g ∈ Hom K (P v , P w ). Then g is not irreducible if: (1) g is a double map; (2) g ∈ G v,w is such that both endpoints are non-isomorphisms; (3) g = (g s , . . . , g 0 ) is such that g s = ab with a, b non-stationary paths in Γ; (4) g = (g s , . . . , g 0 ) is such that g 0 = ab with a, b non-stationary paths in Γ. Proof.
(1) Write g = (c, d) where c and d are paths in the quiver. Then g is, up to inverting v and w, given by the diagram to the left, and the factorisation of g is given by the diagram to the right.
Cases (2), (3) and (4) are similar and can be verified by drawing the corresponding diagrams, noting in (3) and (4) one needs to check only single and graph maps.
As a consequence, no single map in the homotopy class of a double map is irreducible. We now examine single maps more closely, giving a necessary condition for irreducibility, which is then used to eliminate further possibilities for irreducible single maps. Proposition 6.10. If g : P v → P w is an irreducible single map, then g is of the form o o up to inverting w or v, where a ∈ Γ 1 , and v 0 and/or w 1 may be zero. Moreover, g is in a singleton homotopy class; see Definition 3.7.
Proof. Suppose g ∈ S v,w with non-zero support, by abuse of notation, denoted by g : P → P ′ with P, P ′ ∈ ind(proj(Λ)). This situation is indicated below.
• One can easily check that in the following cases, g is not irreducible: • v 0 and v 1 are both non-zero, or precisely one is non-zero and it has P as source.
• w 0 and w 2 are both non-zero, or precisely one is non-zero and it has P ′ as target. Indeed, if both conditions hold g factors as two graph maps and one single map (with g as its non-zero support), and if only one holds then g factors as one graph map and one single map. Corollary 6.11. Suppose g ∈ S v,w has unfolded diagram taking the form (3) in Proposition 6.10. Then g is not irreducible if: (1) v (or w) is not a uniformly oriented homotopy string; (2) any homotopy letter of v and w is a path of length longer than 1.
Proof. The following diagrams indicate the factorisations: Proof of Theorem 6.6. We start by setting up the notation for the section. Throughout w ∈ St, w ′ and map f ′ : P w → P w ′ will be the homotopy string and map output at Steps 1-7, and w + and f + : P w → P w + the homotopy string and map output at Steps 7-9. Note that f ′ differs from f + if and only if w + = θw ′ , where θ = (θ m , j m , i m ) · · · (θ 1 , j 1 , i 1 ) is an inverse antipath. Write f + = (f k , . . . , f 0 ) where, using the notation from Algorithm 6.3, k ∈ {0, r − 1, r, n}.
Lemma 6.12. We have the following: (1) f + is a well-defined map.
(3) For each v ∈ St such that vw ′ is defined and such that the components of f ′ also determine a map g : P v → P vw ′ , the map g factors through f + .
Proof. This is easily verified. Observe that if f k is the leftmost component of f + , and if an antipath was added in Step 8 such that f k θ 1 occurs in the diagram, then f k θ 1 = 0.
The following lemma shows that if there is a graph map with source P w satisfying certain criteria, then at least one of the maps f + and f + is a graph map. Lemma 6.13. Let w be a non-trivial homotopy string. If there exists g ∈ G w,v starting after the left endpoint of w and stopping with an isomorphism at the right endpoint of w, then f + is a graph map and g factors through f + .
Proof. It follows from the hypotheses of the lemma that w is not an antipath, so neither Steps 6 nor 7 occur. A straighforward case analysis shows that g factors through f + even if v is not a homotopy substring of w + . Lemma 6.14. Suppose f + ∈ G w,w + and g : P w → P u is a map such that supp(g) is a homotopy substring of supp(f + ). If g is not supported on the source of f k , then g factors through f + .
Proof. If g ∈ G w,u , then since f k−1 , . . . , f 0 are isomorphisms, supp(g) is also a homotopy substring of w + . It is straightforward to check that the restriction g ′ : P w + → P u of g is also a graph map, and g = g ′ f + . Similarly for g ∈ S w,u .
There is a dual statement of Lemma 6.14 for f + . Lemma 6.15. Let w ∈ St be such that f + ∈ G w,w + . If g : P w → P u is such that supp(f + ) and supp(g) have no overlapping part, then f + = 0 and g factors through f + .
The map f + is output at Step 3. Here w is non-trivial and g 0 need not be the leftmost non-zero component of g. Case 1: Here g 0 is either the leftmost or rightmost non-zero component of g. In the former case w r g 0 = 0 and g 0 = a (in the notation of Algorithm 6.3). By Proposition 6.9(2), we may assume that g −t is an isomorphism. Now g factors through f + by Lemma 6.12 if t < r. If t = r, f + is a graph map (argue as in Lemma 6.13), and the conclusion follows by Lemma 6.14. Case 2: In this case g does not factor through f + . The components g −1 , . . . , g −t+1 are isomorphisms; g −t may be an arrow. The g s , . . . , g 1 are all isomorphisms if ψ w is nontrivial, and zero otherwise. If f + is a graph map then g factors through f + by Lemma 6.13. If f + is not a graph map, then 1 k=r (w k , i k , j k ) is an inverse antipath, which implies g −t is an isomorphism. It follows that w = u since adding anything to the endpoints forces either f + to be output in Step 1, or a graph map f + in Step 1 ′ , whence g is an isomorphism. Case 3: Apply the same argument as when g 0 was the rightmost non-zero component of g in Case 1.
The map f + is output at Step 4. Here Cases 1 and 3 are straightforward. For Case 1 apply Lemma 6.14. For Case 3 it is immediate that g factors through f + . Case 2: If g 0 is the leftmost non-zero component then by previously given arguments, g factors through f + . So assume not: we must have that g 1 is an isomorphism, for otherwise f + would have been output at Step 3. Applying the argument as in Case 2 above, g s , . . . , g 1 are each isomorphisms. If t < r − 1 then g factors through f + by Lemma 6.14. If t = r − 1, g is either an isomorphism or else Algorithm 6.3 ′ outputs f + at Step 1 ′ , whence g clearly factors through f + .
The map f + is output at Step 5. Here f + cannot be output at Step 6 ′ of Algorithm 6.3 ′ since w r is direct, so f + is a graph map. For Case 2, argue as in Case 2 above. Case 3 is again a straighforward verification. Case 1: If g 0 is the leftmost non-zero component, then g 0 = a (in the notation of Algorithm 6.3) factors through f + . If g 0 is the rightmost non-zero component then use Lemma 6.13.
The map f + is output at Step 6. Note that the direct antipath w may be trivial. Case 1 and 3: If g 0 = a (in the notation of Algorithm 6.3) then in the graph map case g factors through f + by Lemma 6.13. In the single map case, w is a trivial homotopy string and g factors through f + , which is output either at Step 1 ′ or Step 6 ′ of Algorithm 6.3 ′ . Similarly when g 0 = a. Case 2: Here, the isomorphism g 0 is dragged down the entire homotopy string so that v has the form v = · · · cwd · · · for some homotopy letters c and d. The homotopy letter c cannot be direct (f + would be output at Step 1) nor inverse (g would not be a map). Similarly, d is not direct. Hence v = wd · · · . If d is trivial then g is an isomorphism. If d is inverse then f + is output at Step 1 ′ and g clearly factors through f + .
The map f + = 0 is output at Step 7. One now applies the dual arguments for f + to get the required factorisation through f + .

Application: Discrete derived categories
For the definition and background on discrete derived categories we refer the reader to [11,13,23]. Derived-discrete algebras are derived equivalent to either path algebras of simply-laced Dynkin quivers or the bound path algebra Λ(r, n, m) defined in Figure 2; when we refer to discrete derived categories, we shall always mean D b (Λ(r, n, m)).
In this section, we recover the universal Hom-space dimension bound described in [13] when Λ(r, n, m) is of finite global dimension and extend it to the case Λ(r = n, n, m), which has infinite global dimension. Recall from [8] that Λ(r, n, m) has no homotopy bands. From now on Λ := Λ(r, n, m) and St(Λ) will denote homotopy strings over Λ.
A subword of a homotopy string w is defined in the obvious fashion: the left-and rightmost homotopy letters of the subword may be (incomplete) substrings of the corresponding letters of w. We now describe all the homotopy strings for a discrete derived category. Note that, when r = n, by our labelling convention there are no 'b' arrows.
Lemma 7.1. Consider the following homotopy strings: where v k is the k-fold concatenation of • c n−1 G G · · · c n−r+1 G G • b n−r ...b 0 G G • . Then: (1) if r < n, all homotopy strings are (shifted) copies of subwords of the w k for k ≥ 0; (2) if r = n, all homotopy strings are (shifted) copies of subwords of w and w k for k ≥ 0.
Lemma 7.3. Suppose v, w ∈ St(Λ) and consider the following unfolded diagram: (1) Suppose ( * ) represents a quasi-graph map P v → Σ −1 P w . If v L is non-zero, then one of v ′ L or w ′ L is zero. (2) Suppose ( * ) represents a graph map P v → P w . Then either (i) f L = 0 and one of v ′ L or w ′ L is zero; or (ii) f L = 0 and if v L is non-zero then v L = (a −1 ...a −i ) −1 for some 1 ≤ i ≤ m. Dual statements hold for the right-hand end of the diagram.

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Proof. Recall from Definition 3.11 that if v L is not zero then v L = w L . We note that the only ways that v L and w L can differ is if one of them is zero or if one (or both) of them is a subpath of b n−r . . . b 0 a −1 . . . a −m , and it is clear that in all cases one of v ′ L or w ′ L is zero. This shows (1); (2) is similar.
The upshot of Lemma 7.3 is that any graph map P v → P w or quasi-graph map P v → Σ −1 P w spans every degree where P w and P v are both non-zero.
Proof. For r > 1, observe that Lemma 7.2 combines with Lemmas 4.3 and 7.3 to give hom D b (Λ) (P v , P w ) in the cases that there is a graph map P v → P w or a non-singleton homotopy class P v → P w .
For r ≥ 1, we claim that if there is a single or double map f : P v → P w such that H(f ) is a singleton homotopy class, then hom D (P v , P w ) = 1. If f is a single map, a case analysis reveals that, of the options presented in Definition 3.7, only (i) could arise. Clearly, there can be no other basis maps P v → P w . Similarly, by considering all of the possible cases where f is a double map, we find that either v or w is a homotopy string of length 1 and that in each of these cases there is no other possible basis map.
When r = 1, the only way there can be more than one basis map P v → P w is if there is a graph map and a single map supported in the same degree, both the graph map and the homotopy class containing this single map will span the rest of the string, by  (1, 1, 3), a graph map P v → P w , and a quasi-graph map P v → Σ −1 P w .