The Differential Graded Stable Category of a Self-Injective Algebra

Let A be a finite-dimensional, self-injective algebra, graded in non-positive degree. We define A-dgstab, the differential graded stable category of A, to be the quotient of the bounded derived category of dg-modules by the thick subcategory of perfect dg-modules. We express A-dgstab as the triangulated hull of the orbit category A-grstab/$\Omega$(1). This result allows computations in the dg-stable category to be performed by reducing to the graded stable category. We provide a sufficient condition for the orbit category to be equivalent to A-dgstab and show this condition is satisfied by Nakayama algebras and Brauer tree algebras. We also provide a detailed description of the dg-stable category of the Brauer tree algebra corresponding to the star with n edges.


Introduction
If A is a self-injective k-algebra, then A -stab, the stable module category of A, admits the structure of a triangulated category. This category has two equivalent descriptions. The original description is as an additive quotient: One begins with the category of A-modules and sets all morphisms factoring through projective modules to zero. Phrasing this in categorical terms, we define A -stab to be the quotient of additive categories A -mod /A -proj. The second description, due to Rickard [8], describes A -stab as a quotient of triangulated categories. Rickard obtains A -stab as the quotient of the bounded derived category of A by the thick subcategory of perfect complexes (i.e., complexes quasi-isomorphic to a bounded complex of projective modules). Once this result is known, the triangulated structure on A -stab is an immediate consequence of the theory of triangulated categories. When translated back into the additive description, the homological shift functor [−1] inherited from D b (A -mod) becomes identified with the Heller loop functor Ω, which maps each module to the kernel of a projective cover. The triangulated description provides a well-behaved technical framework for transferring information between A -stab and the derived category, while the additive description allows computations of morphisms to be performed in A -mod rather than D b (A -mod). If A is made into a graded algebra, analogous constructions produce two equivalent descriptions of the graded stable category A -grstab.
If A is a dg-algebra, we use the triangulated description to define the differential graded stable category A -dgstab. More precisely, A -dgstab is defined to be the quotient of the derived category D b dg (A) of dg-modules by the thick subcategory of perfect dg-modules.
The most immediately interesting feature of the dg-stable category is the presence of non-trivial interactions between the grading data and the triangulated structure. In D b dg (A), the grading shift functor coincides with the homological shift functor, and so in A -dgstab the grading shift functor (−1) can be identified with Ω. This phenomenon does not occur in the graded stable category, since the grading shift and homological shift functors in D b (A -grmod) are distinct.
However, working with dg-modules introduces new complications. The presence of dg-modules which do not arise from complexes of graded modules is an obstacle to obtaining a simple additive definition of A -dgstab, without which computation of morphisms becomes much harder, as it must be done in the triangulated setting. In this paper we consider the problem of finding a simple additive description of A -dgstab.
The dg-stable category has been studied by Keller [5], using the machinery of orbit categories; our approach is motivated by his work. In Section 3, we consider the case where A is a non-positively graded, finite-dimensional, selfinjective algebra, viewed as a dg-algebra with zero differential. There is a natural functor A -grstab → A -dgstab which is faithful but not full. This is due to the fact that X ∼ = ΩX(1) for all X ∈ A -dgstab; the corresponding isomorphism almost never holds in A -grstab. To recover the missing morphisms, we turn to the orbit category C(A) := A -grstab /Ω(1). The objects of C(A) are those of A -grstab, and the morphisms X → Y are finite formal sums of morphisms X → Ω n Y (n) in A -grstab. Orbit categories need not be triangulated, but Keller proves they can always be included inside a "triangulated hull". We shall construct a fully faithful functor F A : C(A) → A -dgstab whose image generates A -dgstab as a triangulated category; in other words, A -dgstab is the triangulated hull of C(A).
F A is an equivalence of categories precisely when it identifies C(A) with a triangulated subcategory of A -dgstab. This is in general not the case, as there is no natural way to take the cone of a formal sum of morphisms with different codomains. In Section 4, we provide a sufficient condition for F A to be an equivalence and show that this condition is satisfied by self-injective Nakayama algebras. An example for which F A is not an equivalence is also provided.
In the second half of this paper, we investigate the dg-stable category of nonpositively graded Brauer tree algebras. A Brauer tree is a data of a tree, a cyclic ordering of the edges around each vertex, a marked vertex-called the exceptional vertex-and a positive integer multiplicity associated to the exceptional vertex. The data of a Brauer tree determines, up to Morita equivalence, an algebra whose composition factors reflect the combinatorial data of the tree. We refer to Schroll [10] for a detailed introduction to the theory of Brauer tree algebras and their appearance in group theory, geometry, and homological algebra, but we mention here one application which is of particular relevance. Khovanov and Seidel [6] link the category D b dg (A), where A is a graded Brauer tree algebra on the the line with n vertices, to the triangulated subcategory of the Fukaya category generated by a chain of knotted Lagrangian spheres. The braid group acts on D b dg (A) by automorphisms, and the category A -dgstab can be viewed as the quotient of D b dg (A) by this action.
In Section 5, we show that C(A) is equivalent to A -dgstab for any Brauer tree algebra, and in Section 6 we provide a detailed description of A -dgstab when A corresponds to the star with n edges and multiplicity one.

Complexes
If A is any additive category, we write Comp(A) for the category of (cochain) complexes over A. We shall write our complexes as (C • , d • C ), where d n C : C n → C n+1 for all n ∈ Z. We write Ho(A) for the category of complexes and morphisms taken modulo homotopy. If A is an abelian category, we let D(A) denote the derived category of A. On any of these subcategories, we shall use the superscript b (resp., +, −) to denote the full, replete subcategory generated by the bounded (resp., bounded below, bounded above) complexes.
If A = A -mod for some algebra A, we shall write Ho perf (A -mod) and D perf (A -mod) for the thick subcategories of Ho(A -mod) and D(A -mod), respectively, generated by A. The objects are those complexes which are homotopy equivalent or quasi-isomorphic, respectively, to bounded complexes of projective modules; we refer to them as the strictly perfect and perfect complexes, respectively.
On any of the above categories, we let [n] denote the n-th shift functor, de- [1] in Ho(A). Abusing notation, for any morphism f : X → Y in a triangulated category T , we shall write C(f ) to refer to any choice of object completing the triangle . This will cause no confusion. We write τ ≤n , τ ≥n , τ <n , τ >n for the truncation functors on D(A) defined by the canonical t-structure. More explicitly, if X • is a complex, the kth term of τ ≤n X • is X k if k < n, 0 if k > n, and ker(d n X ) if k = n. τ >n X • is defined analogously; here the nth term equal to im(d n X ). We also denote by X ≤n the complex whose kth term is X k for k ≤ n and 0 for k > n. We denote X ≥n , X <n , X >n similarly, and refer to these complexes as the sharp truncations of X • .

Modules and the Stable Category
If A is an algebra over a field k, we let A -mod denote the category of finitely generated right A-modules, and let A -proj denote the full subcategory of finitely generated projective modules. A-Mod and A-Proj will denote the categories of all modules and projective modules, respectively.
Given an A-module X, we define the socle of X, soc(X) to be the sum of all simple submodules X. We define the radical of X, rad(X), to be the intersection of all maximal submodules of X, and we define the head of X to be the quotient hd(X) = X/rad(X). We note that rad(A), where A is viewed as a right module over itself, is equal to the Jacobson radical of A. If X is finitely generated, then rad(X) = Xrad(A). (See for instance, Benson [1], Proposition 1.2.5) We let A -stab denote the stable module category of A. The objects of A -stab are the objects of A -mod, and Hom A -stab (X, Y ) is defined to be the quotient of Hom A -mod (X, Y ) by the subspace of morphisms factoring through projective modules. There is a full, essentially surjective functor A -mod ։ A -stab which is the identity on objects. If A is self-injective, then A -stab admits the structure of a triangulated category. In this case, it has been shown by Rickard [8] that A -stab is equivalent as a triangulated category to D b (A -mod)/D perf (A -mod).

Graded Modules
Let A be a graded algebra over a field k. We denote by A -grmod and A -grproj the categories of finitely generated graded right modules and finitely generated graded projective right modules, respectively. We shall use upper case letters when the modules are not required to be finitely generated, in analogy with Section 2.2.
The stable category of graded modules A -grstab can be defined analogously to A -stab. When A is self-injective, we once again have that A -grstab is triangulated and equivalent to D b (A -grmod)/Ho b (A -grproj).
If X is a graded A-module, we write X i to denote the homogenous component of X in degree i. (If X • is a complex of graded modules, we shall denote the degree i component of the nth term of the complex by (X n ) i .) On any category of graded objects, we define the grading shift functor (n) by X(n) i = X i+n . If x ∈ X is a homogeneous element, we let |x| denote the degree of x.
For a graded module X, we define the support of X to be the set supp(X) = {n ∈ Z|X n = 0}. We also define max(X) = sup(supp(X)) and min(X) = inf (supp(X)). Note that if A is finite-dimensional and X is a finitely generated nonzero A-module, then X is a finite-dimensional k-vector space, therefore supp(X) is a finite, nonempty set and max(X) and min(X) are finite.
Given graded modules X and Y , define Hom • A -grmod (X, Y ) to be the graded vector space whose degree n component is the space Hom A -grmod (X, Y (n)) of degree n morphisms. If X is a graded left B-module for some graded algebra B, then Hom • A -grmod (X, Y ) is a graded right B-module.

Differential Graded Modules
A differential graded algebra is a pair (A, d A ), where A is a graded k-algebra and d A is a degree 1 k-linear differential which satisfies, for all homogenous a, b ∈ A, the equation is a differential graded k-algebra, a differential graded right Amodule (or dg-module, for short) is a pair (X, d X ) consisting of a graded right A-module X and a degree 1 k-linear differential d X : X → X satisfying d X (xa) = d X (x)a + (−1) |x| xd A (a) for all homogeneous elements x ∈ X, a ∈ A. A morphism of differential graded modules is defined to be a homomorphism of graded A modules which commutes with the differential. We denote by A -dgmod the category of finitely-generated right dg-modules. As above, we shall write A -dgMod for the category of arbitrary dg-modules.
Any graded algebra A can be viewed as a differential graded algebra with zero differential. In this case, for any dg-module (X, d X ), d X is a degree 1 graded morphism, and so the kernel and image of d X are dg-submodules of X with zero differential. In this paper, we shall work exclusively with dg-algebras with zero differential.
For dg-modules, we define the grading shift functor (n) : There is a faithful functor : to the dg-module ( X, d X ) whose underlying graded module is X = n∈Z X n (−n) and whose differential d X restricts to d n Identifying graded modules with complexes concentrated in degree zero yields a fully faithful functor A -grmod ֒→ Comp b (A -grmod). The restriction of to A -grmod is fully faithful. Note that X(k) = X(k).
If f, g : X → Y are morphisms of dg-modules, we say f and g are homotopic if there is a degree −1 graded morphism h : We write Ho dg (A) for the category of left dg-modules over A and homotopy classes of morphisms. By formally inverting the quasi-isomorphisms of Ho dg (A), we obtain D dg (A), the derived category of dg-modules. We again use the superscript b (resp., +, −) to denote the full subcategory whose objects are isomorphic to dg-modules with bounded (resp. bounded below, bounded above) support. We write Ho perf dg (A) and D perf dg (A) for the thick subcategories of Ho b dg (A) and D b dg (A), respectively, generated by the dg-module A. We refer to the objects of Ho perf dg (A) and D perf dg (A) as the strictly perfect and perfect dg-modules, respectively.
If P is strictly perfect, then for any dg-module X, we have an isomorphism Hom Ho dg (A) (P, X) ∼ = Hom D dg (A) (P, X). In addition, if A is a finite-dimensional, self-injective graded algebra with zero differential, then Hom Ho dg (A) (X, P ) ∼ = Hom D dg (A) (X, P ). Any perfect dg-module is quasi-isomorphic to a strictly perfect dg-module.
We define the dg-stable module category of A to be the quotient A -dgstab : each of which is the identity on objects. By composing with the inclusion A-grmod ֒→ A-dgmod, we obtain an additive functor A-grmod → A-dgstab whose kernel contains A-grproj. Hence this functor factors through A-grmod ։ A-grstab.
Given a morphism of dg-modules f : X → Y , we define the cone of f to

Functors and Resolutions
Let A be a finite-dimensional, self-injective graded algebra. Let m : A op ⊗ k A ։ A denote the multiplication map, viewed as a morphism of graded (A op ⊗ k A)-modules, and let I = ker(m). We define the functor Ω = − ⊗ A I : A -grmod → A -grmod. Note that Ω has a right adjoint Ω ′ = Hom • A -grmod (I, −). Since I is projective both as a right and left A-module, we have that Ω is exact and Ω(A -grproj) ⊂ A -grproj. Thus Ω lifts to D b (A -grmod) and descends to A -grstab; we also have that Ω(D perf (A -grmod)) ⊂ D perf (A -grmod). The complex P • = 0 → I ֒→ A op ⊗ k A ։ A → 0 is an exact complex of projective right A-modules, hence is homotopy equivalent to zero. Then for any X ∈ A -grmod, we have that X ⊗ A P • is homotopy equivalent to zero, hence exact. But X ⊗ A P • ∼ = 0 → ΩX ֒→ X ⊗ k A ։ X → 0, hence ΩX is the kernel of a surjection from a projective right A-module onto X. Thus Ω is an autoequivalence of A -grstab and is isomorphic to the desuspension functor for the triangulated structure. In A -grstab, ΩX is isomorphic to the kernel of a projective cover of X.
Similarly, for any complex X • ∈ Comp b (A -grmod), we have a short exact sequence of complexes 0 → Ω(X • ) ֒→ X • ⊗ k A ։ X • → 0. From the resulting triangle in D b (A -grmod), we obtain a natural transformation [−1] → Ω. Since By a similar argument, Ω defines a functor A -dgmod → A -dgmod which is exact and preserves direct summands of A. Thus Ω lifts to D b dg (A) and preserves D perf dg (A), and so Ω descends to A -dgstab. We also have a natural transformation Similarly, Ω ′ is exact and preserves projective modules, and so descends to A -grstab and A -dgstab and lifts to D b (A -grmod) and D b dg (A).
Since Ω ′ is right adjoint to Ω, we have that Ω ′ is quasi-inverse to Ω in A -grstab and A -dgstab.
For any X ∈ A -grmod, we can construct a projective resolution (P • X , d • P X ) of X, such that coker(d −n−1 P X ) = Ω n (X) for any n ≥ 0. More specifically, for n ≥ 0, we let P −n X = Ω n X ⊗ k A, and for n ≥ 1 we let d −n P X be the composition ) and define the differential analogously. Joining P • X and I • X via the map P 0 X ։ X ֒→ I 0 X , we can define an acyclic biresolution with B n X = I n X for n ≥ 0 and B n X = P n+1 X for n < 0. We refer to these resolutions as the standard resolutions of X.

Autoequivalences and Automorphisms
Given any category C and an autoequivalence F : C → C, there is a category C, an automorphism F : C → C, and an equivalence of categories π : C → C such that π • F = F • π.

Left Dg-Modules
In this paper, we work exclusively with right dg-modules. All the results presented are valid for left dg-modules, but minor adjustments must be made to account for numerous unpleasant sign conventions. We describe the necessary adjustments here.
If (A, d A ) is a differential graded k-algebra, we define a left differential graded A-module (or dg-module, for short) to be a pair (X, d X ) consisting of a graded left A-module X and a degree 1 k-linear differential d X : X → X satisfying d X (ax) = d A (a)x + (−1) |a| ad X (x) for all homogeneous a ∈ A, x ∈ X. We let A -dgmod l denote the category of left dg-modules over A.
If A is a graded algebra, we define the algebra (A, •) to be the set A with multiplication given by a • b = (−1) |a||b| (ab). Similarly, if M is a graded right Amodule, we denote by (M , •) the graded right A-module with M as the underlying set and the operation given by m • a = (−1) |m||a| ma. We define M similarly for left graded modules. The functor sending M to M and acting as the identity on morphisms defines an isomorphism between A -grmod and A -grmod.
Let A op denote the opposite algebra. We call A op the graded opposite algebra.
op -dgmod l , and this defines an isomorphism of categories.
As before, there is a faithful functor : . This definition of is equivalent to converting to complexes of right A op -modules, applying the original definition of , and then converting back to left dg-modules over A.
If M is a left dg-module, define the dg-grading shift functor n : The underlying set of (M n , · n ) is M(n), and the operation · n is given by a · n m = (−1) |a|n am. The differential is given by d M n = (−1) n d M . Triangles in the homotopy or derived categories take the form If X and Y are graded modules, we say that a function f : X → Y is a graded skew-morphism of degree n if it is a degree n k-linear map such that f (ax) = (−1) n|a| af (x) for all x ∈ X and all homogeneous a ∈ A. We say two morphisms of left dg-modules f, g : We also note that if A has zero differential, then d A is a graded skew-morphism of degree 1.

The Dg-Stable Category
Let A be a finite-dimensional, non-positively graded, self-injective k-algebra, viewed as a dg-algebra with zero differential. In this section, we shall provide a description of dg-stable category of A in terms of the graded stable category.
We accomplish this by constructing a category C(A) from the data of A -grstab in Definition 3.5. In Definition 3.7 we define a functor F A : C(A) → A -dgstab and in Theorem 3.9 we show that F A is fully faithful with essential image generating A -dgstab as a triangulated category.
We begin with some simple facts about graded A-modules.
Proof. The first part of the statement follows immediately from the definition of morphisms of graded modules. Since Hom A -grstab (X, Y ) is defined as a quotient of Hom A -grmod (X, Y ), the second part of the statement follows from the first.
Proof. The radical of A is a graded submodule of A (see Kelarev, [4]), and so To establish the reverse inequality, take a nonzero element x ∈ X max(X) . If x / ∈ rad(X), then the image of x in hd(X) is a nonzero element in degree max(X), and we are done. Suppose x ∈ rad(X). Note that since X is finitely generated and A is finite-dimensional, X is also finite dimensional. Thus rad k (X) becomes zero for sufficiently large k. Since x is nonzero, there is a maximum n > 0 such that x ∈ rad n (X) = Xrad n (A). Write x = m i=1 x i a i for some homogeneous a i ∈ rad n (A), x i ∈ X. Without loss of generality, we may assume that all terms are nonzero and that deg(x i a i ) = deg(x) for all i. Since deg(x) = max(X) and A is non-positively graded, we must have that deg(x i ) = max(X) and deg(a i ) = 0 for all i. Since each a i ∈ rad n (A) and x / ∈ rad n+1 (X), there must be some j such that x j / ∈ rad(X). Thus we have obtained a nonzero x j ∈ X max(X) − rad(X), and so max(hd(X)) = max(X).
For the reverse inequality, it suffices to show that soc(X)∩X min(X) = 0. Since A is non-positively graded, X min(X) A ⊂ X min(X) and so X min(X) is a submodule of X. Since X is finite-dimensional, X min(X) has a simple submodule and thus has nonzero intersection with soc(X). Therefore min(soc(X)) = min(X).
Similarly, min(Ω ′ X) = min(Hom • A (I, X)) ≥ min(X)−max(I) ≥ min(X). The last two equations follow from the first two and the definitions of the standard projective and injective resolutions.
Recall from Section 2.5 that the functor Ω(1) is an autoequivalence of A -grstab and A -dgstab. By replacing A -grstab and A -dgstab with equivalent categories A -grstab and A -dgstab (see Section 2.6), we may assume without loss of generality that Ω(1) is an automorphism of both categories. We let Ω −1 denote the inverse of Ω, and we shall identify it with the isomorphic functor Ω ′ .
Going forward, we shall write Ω −n to mean (Ω ′ ) n for n ≥ 0, even on A -grmod and A -dgmod. This is a dangerous abuse of notation as Ω is not invertible in either category. However, adopting this convention allows us to greatly simplify certain expressions and is safe as long as we avoid expressions of the form ΩΩ −1 X outside the stable category.
We obtain the following corollary of Proposition 3.3.
We are now ready to state the main definitions.
Definition 3.5. Let C(A) be the category given by: Remark. If we do not wish to assume that Ω(1) is an automorphism of A -grstab, natural isomorphisms ε n,m : Ω n Ω m → Ω n+m satisfying the appropriate coherence conditions must be inserted into the composition formula.
We note that the sum in the composition formula is finite by Proposition 3.4. It is clear that C(A) is an additive category. In fact, C(A) is precisely the orbit category A -grstab /Ω(1) as defined by Keller, [5]. Keller shows that while such a category need not be triangulated, it can always be included in a "triangulated hull". We shall see that A -dgstab is the triangulated hull of C(A).
We now define the inclusion functor F A : C(A) → A-dgstab. The obvious choice would be for F A to act as the identity on objects and send the morphism (f n ) n : X → Y to the sum of its components n∈Z ψ n,Y • f n , where the ψ n,Y : Ω n Y (n) → Y are isomorphisms chosen so that all the summands share a common domain. However, in order for this process to be functorial, the morphisms ψ n,Y must be satisfy appropriate compatibility conditions. Lemma 3.6. There exists a family of natural isomorphisms {ψ n : For n ≥ 2, recursively define ψ n = ψ 1 • (ψ n−1 • Ω(1)) and analogously for n ≤ −2. It is clear that {ψ n } satisfies i) and ii).
Remark. If we do not assume that Ω(1) is an autormorphism of A -dgstab, we must again insert appropriately chosen natural isomorphisms ε n,m : Ω n Ω m → Ω n+m into condition ii).
Definition 3.7. Let ψ n : Ω n (n) → id A -dgstab be the natural isomorphisms defined in Lemma 3.6. Let F A : C(A) → A-dgstab be the functor given by: 1) F A acts as the identity on objects.
We now state the main theorem. We prove the theorem with a sequence of lemmas below. Definition 3.10. Let X, Y ∈ A -grmod, viewed as dg-modules with zero differential. If X and Y are nonzero, let N = N X,Y := max{n ≤ 0 | max(Ω −n Y (−n)) < min(X)}. Define the bridge complex from X to Y to be the complex R • X,Y = B ≥N Y (see Sections 2.1 and 2.5 for notation) if X and Y are both nonzero, and R • X,Y = 0 otherwise. By Proposition 3.3, N X,Y is well-defined. We will omit the subscript when it is clear from context. By unwinding the definitions, we obtain a quasi-isomorphism of complexes where s has perfect cone. The primary challenge in understanding morphisms in A -dgstab is that perfect dg-modules need not arise from complexes of graded projective modules. However, by restricting our attention to dg-modules with zero differential, we can bypass this difficulty by using the bridge complexes defined above.
. By changing P up to quasiisomorphism, we may assume without loss of generality that P is strictly perfect.
Let p n denote the natural map of complexes P ≥n Y ֒→ P • Y ։ Y and let i n : Y ֒→ C(p n ) denote the natural inclusion of complexes. If n ≥ 1, note that p n is the map from the zero complex to Y and i n is the identity map on Y . Note also that C(p N +1 ) = τ ≤0 R • X,Y and i N +1 = i. We first show that every morphism can be expressed as a roof of the form Then we may choose k << 0 such that k ≤ N + 1 and max(P k−1 Y (−k + 1)) < min(P ). Then the short exact sequence of dg-modules we have that Hom Ho dg (A) (P, P <k Y ) = 0. Since P is strictly perfect, morphisms in the derived and homotopy categories coincide, and so Hom D dg (A) (P, P <k Y ) = 0. We obtain a morphism of triangles in D dg (A): We obtain a morphism of triangles in D b dg (A):

morphism in
A -dgstab. It follows immediately from the above diagram that the roofs s −1 • g It remains to show that k can be replaced by N + 1. Since k ≤ N + 1 by definition, we have an exact sequence of dg-modules 0 → C(p N +1 ) ֒→ C(p k ) ։ (P ≥k Y ) ≤N (1) → 0 arising from the underlying exact sequence of complexes. We also have that The last inequality is true by definition of N. Thus Hom Ho b dg (A) (X, (P ≥k Y ) ≤N (1)) = 0 and, since (P ≥k We obtain a morphism of triangles in D b dg (A): Having found a convenient choice of roofs between X and Y , we now investigate maps between X and τ ≤0 R X,Y in the derived category. This investigation shall yield a method for computing morphisms between zero-differential modules.
Lemma 3.12. Let X, Y ∈ A -grmod. Then we have an isomorphism Proof. Let f ∈ Hom Ho + dg (A) (X, R X,Y ). In order for ξ to be well-defined, we must show that φ −1 • f ∈ Mor(D dg (A)) can be represented by a roof in D b dg (A). By Proposition 3.3, the sequence {min( τ >M R X,Y )} M strictly increases with M. Since X is finitely generated, there exists M >> 0 such that the image of f lies in τ ≤M (R X,Y ). It is clear that the inclusion φ also factors through τ ≤M (R X,Y ), and the inclusion of τ ≤M (R X,Y ) into R X,Y is a quasi-isomorphism. Thus R X,Y can be replaced by the bounded dg-module τ ≤M (R X,Y ) in the roof φ −1 • f , and so we may view φ −1 • f as a morphism in Mor(D b dg (A)). Thus ξ is well-defined.
We now prove surjectivity of ξ. Since R X,Y ∈ Ho + (A -grproj), we have that Hom Ho dg (A) (X, R X,Y ) ∼ = Hom D dg (A) (X, R X,Y ). Post-composition with φ −1 yields an isomorphism: It follows immediately from Lemma 3.11 that the map is surjective. The composition of these two maps is precisely ξ, which is therefore surjective. It remains to show injectivity. Suppose that ξ(f ) = 0. Then there exists a morphism s : We obtain a morphism of triangles in D b dg (A): Since C(s)(−1) is strictly perfect, we can choose to represent α by a morphism in Ho b dg (A). From the above diagram and the fact that In the next three lemmas, we relate morphisms in C(A) to those in A -dgstab via the homotopy category.
. Suppose that Hom A -grmod (X, ker(d n P )(−n)) = 0 for almost all n. Let i n : ker(d n P )(−n) ֒→ P denote the inclusion (of dg-modules). Then the map Φ : is an isomorphism of vector spaces.
Proof. By hypothesis, the sum in the definition of Φ is finite, so Φ is a well-defined k-linear map. It remains to construct Φ −1 . Given f ∈ Hom A -dgMod (X, P ), we have d P •f = f •d X = 0, since X has zero differential. Thus im(f ) ⊂ ker(d P ) = n ker(d n P )(−n). Let π n denote the projection onto the nth summand, and define Φ −1 (f ) = (π n • f ) n ; it is easy to verify that Φ −1 is inverse to Φ. Lemma 3.14. Let all notation and assumptions be as in Lemma 3.13. Assume in addition that P • ∈ Comp(A -grproj) and that P • is exact at each n for which Hom A -grmod (X, ker(d n P )(−n)) is nonzero. Then Φ induces an isomorphism: Φ : n∈Z Hom A -grstab (X, ker(d n P )(−n)) → Hom Ho dg (A) (X, P ) Proof. Take (f n ) n ∈ n∈Z Hom A -grmod (X, ker(d n P )(−n)). By Lemma 3.13, it suffices to show that Φ(f n ) is nullhomotopic if and only if f n factors through a projective module for all n. We also note that d P is A-linear, since d A = 0.
Suppose that Φ(f n ) is nullhomotopic and fix k ∈ Z. Let h : X → P (−1) be a homotopy. Since d X = 0, we have that Φ(f n ) = d P (−1) • h (as morphisms of graded modules). As a graded module, P = n P n (−n); let π n be the projection onto the nth summand. From the proof of Lemma 3.13, we have that f k = π k • Φ(f n ), and so f k = π k • d P (−1) • h. Thus f k factors through the graded projective module P (−1). Now suppose that for each n, f n factors as X an − → Q n bn − → ker(d n P )(−n) for some Q n ∈ A -grproj. We shall define a nullhomotopy of Φ(f n ) by constructing maps h n : X → P n−1 (−n). If f n = 0, let h n = 0. If f n is nonzero, then P • is exact at n, and so P n−1 (−n) surjects onto ker(d n P )(−n) via the differential. Since Q n is projective, b n lifts to c n : Q n → P n−1 (−n). Define h n = c n • a n , as summarized by the diagram below. Viewing P n−1 (−n) as a graded submodule of P (−1), define h := n h n : X → P (−1). Since all but finitely many of the h n are zero, h is a well-defined morphism of graded modules. It is easy to check that Then there is an isomorphism is the natural inclusion of dg-modules for n ≥ N and the zero map for n < N.
Proof. We may assume that X and Y are nonzero. We first show that the hypotheses of Lemmas 3.13 and 3.14 are satisfied by X and R X,Y . By Proposition 3.3 we have that Hom A -grmod (X, ker(d n R X,Y )(−n)) = 0 for all but finitely many n. By construction, R X,Y ∈ Comp(A -grproj) is exact at all n = N. By the definition of N, Hom A-dgmod (X, Ω −N Y (−N)) = 0. Thus the hypotheses of Lemmas 3.13 and 3.14 are satisfied.
χ is precisely the composition of the isomorphism with Φ of Lemma 3.14. Thus χ is an isomorphism.
We are now ready to prove Theorem 3.9.
Lemma 3.16. The functor F A is fully faithful.
Proof. Let X, Y ∈ A -grmod. From Lemmas 3.12 and 3.15, we obtain an isomorphism ξ • χ : . It remains to show that this isomorphism is induced by F A .
We have that ξ It follows easily from the definitions that ψ −n,Y can be represented by roofs of the form where all morphisms are inclusions of dg-modules and have either acyclic or perfect cones. We then obtain commutative diagrams of inclusions: Every map in the above diagrams is either a quasi-isomorphism or has perfect cone. (This is immediate for all maps except i n . It then follows that i n is an isomorphism in A -dgstab, hence has perfect cone.) Thus the above diagrams show that the roof defining ψ n,Y is equivalent to i −1 • φ −1 • i n for all n > N. Having proven Theorem 3.9 we make a few brief remarks on when two graded algebras A and B have equivalent dg-stable categories. If D b (A -grmod) is equivalent to D b (B -grmod), then the equivalence can be expressed as tensoring by a tilting complex. (See Rickard, [9]). This functor is still defined on the derived category of dg-modules and remains an equivalence. Furthermore, this equivalence preserves the perfect dg-modules and thus induces an equivalence between the dgstable categories. Thus, graded derived equivalence implies dg-stable equivalence. However, we can say more: Proof. Since G is a triangulated equivalence, it commutes with Ω. Thus G commutes with the functor Ω(1) and induces a functor C(A) → C(B). Given Y ∈ B -grstab, there exists X ∈ A -grstab such that G(X) ∼ = Y in B -grstab, hence in C(B). Thus the induced functor on the orbit category is essentially surjective. Given X, Y ∈ A -grstab, the map is bijective for each n ∈ Z. Thus G : C(A) → C(B) is an equivalence.
The functor F B • G : C(A) → B -dgstab induces an exact functor G : A -dgstab → B -dgstab by the universal property of the triangulated hull. Since F B • G is fully faithful with image generating B -dgstab, it follows that G is an equivalence.

Morphisms Concentrated in One Degree
Let A be a non-positively graded finite-dimensional self-injective algebra over a field k. Let F A : C(A) → A -dgstab be the functor of Definition 3.7. Having shown in the previous section that F A is fully faithful, we now investigate conditions on A that guarantee essential surjectivity.
Since the image of C(A) generates A -dgstab as a triangulated category, F A is essentially surjective if and only if the essential image Im(F A ) is a triangulated subcategory of A -dgstab, if and only if C(A) admits a triangulated structure compatible with F A . In general, this is not the case.
The primary obstacle to essential surjectivity is that there is no natural candidate for the cone of a morphism (f n ) n : X → Y for which more than one f n is nonzero. The cone of such a morphism will correspond to a dg-module that does not arise from a chain complex and need not be isomorphic to a chain complex modulo projectives.
However, by imposing restrictions on the algebra A, we can prevent this scenario from occurring. In this case, we shall see that F A is essentially surjective, hence an equivalence.
Note that Im(F A ) is closed under cones (and thus triangulated) if and only if all of its objects are nice. In fact, it suffices for all the indecomposable objects of Im(F A ) to be nice: Applying the octahedron axiom to the composition f 1 = (f 1 f 2 ) • i 1 , we obtain the following diagram: (1) h The bottom-most triangle is exact, so C(f 1 f 2 ) is the cone of g : X 2 → C(f 1 ). Since X 1 is nice and Y ∈ Im(F A ), it follows that C(f 1 ) ∈ Im(F A ). Since X 2 is nice, C(f 1 f 2 ) ∈ Im(F A ). Thus X 1 ⊕ X 2 is nice.
The following condition is sufficient to guarantee that all indecomposables are nice.

Lemma 4.3 (One Morphism Rule). Suppose for all indecomposable X, Y ∈
A -grmod, Hom A -grstab (X, Ω n Y (n)) = 0 for at most one n ∈ Z. Then every indecomposable object of Im(F A ) is nice. In particular, Changing each Y i up to isomorphism, we may assume without loss of generality that Hom A -grstab (X, Ω n Y i (n)) = 0 for n = 0. Then any morphism (f n ) n : X → M in C(A) is concentrated in degree 0 and thus can be identified with the morphism f 0 in A -grstab. Since F A is fully faithful, any morphism f : X → M in A -dgstab can be represented by a morphism in A -grmod.
Choosing a monomorphism i : X ֒→ I, where I is injective, we obtain a short exact sequence of graded A-modules 0 → X C is a cone of f and lies in the image of F A (since it is in A -grmod). Thus X is nice.
The second statement follows immediately from Lemma 4.2 and the preceding remarks.
Remark. The hypotheses of Lemma 4.3 are quite restrictive. However, we note that if A is concentrated in degree 0 (that is, ungraded), then the One Morphism Rule is trivially satisfied.
In this case, any indecomposable object X ∈ A -grmod is concentrated in a single degree n, and so Ω n X(n) is concentrated in degree 0. Thus every object of C(A) is isomorphic to an object concentrated in degree zero, and Hom C(A) (X, Y ) ∼ = Hom A -stab (X, Y ) for any two such objects X and Y . Thus C(A) is equivalent to A -stab.
Furthermore, a dg-module over A is the same as a complex of A-modules.
Thus, in the case where A an ungraded finite-dimensional, self-injective algebra, Theorem 3.9 and Lemma 4.3 precisely yield Rickard's Theorem [8] that A -stab ∼ = D b (A -mod)/D perf (A -mod). Thus it is appropriate to view C(A) as the differential graded analogue of the additive definition of the stable module category.

Nakayama Algebras
Definition 4.4. A Nakayama algebra is a finite-dimensional algebra for which all indecomposable projective and injective modules are uniserial.
Since every indecomposable module has an indecomposable projective cover, it follows that every indecomposable module over a Nakayama algebra is indecomposable.
Proposition 4.5. Let A be a finite-dimensional, self-injective Nakayama algebra, graded in non-positive degree. Let X ∈ A -grmod be indecomposable and not projective. Let p X : P X ։ X be a projective cover of X and let i X : X ֒→ I X be an injective hull of X. Let K = ker(p X ) and C = coker(i X ). Then max(K) ≤ min(X), and max(X) ≤ min(C).

Proof. For any
Since X is indecomposable, P X is indecomposable, hence uniserial, and we have that K = ker(p X ) = rad l(X) (P X ) and X ∼ = P X /rad l(X) (P X ). Let M = rad l(X)−1 (P X )/rad l(X)+1 (P X ). Then hd(M) = L l(X)−1 (P X ) = soc(X) and soc(M) = L l(X) (P X ) = hd(K) are simple. Thus, max(K) = max(hd(K)) = max(soc(M)) = min(soc(M)) = min(M) ≤ max(M) = max(hd(M)) = max(soc(X)) = min(soc(X)) = min(X) The proof of the second inequality is precisely dual, using the socle layers of I X . Proof. Let X, Y ∈ A -grmod be indecomposable, and suppose that there is a nonzero morphism f : X → Ω m (Y )(m) in A -grstab for some m ∈ Z. Changing Y up to isomorphism in Im(F A ) ⊂ A -dgstab, we may assume that m = 0. Then there is a nonzero morphism from X to Y in A -grmod, and so max(hd(X)) ≥ min(soc(Y )).

An Example of the Failure of Essential Surjectivity
Let A = k[x, y]/(x 2 , y 2 ), where k = C. We grade A by putting x in degree 0 and y in degree −1. It is easy to check that A is symmetric, hence self-injective. Up to grading shift, A has a single simple graded module, S, which has dimension one and upon which both x and y act by zero. Therefore, up to grading shift, the only indecomposable projective module is A itself.
The representation theory of A is closely related to that of the Kronecker quiver, We let B denote the path algebra of this quiver, with a in degree 0 and b in degree −1. B has two simple modules S 1 and S 2 , one corresponding to each vertex. There is a one-to-one correspondence between the indecomposable graded A-modules, excluding the projective module, and the graded B-modules, excluding the simple module S 2 . (See Chapter 4.3 of [1] for the ungraded case. Note that the graded case follows from the same argument.) The classification of graded indecomposable B-modules is known. (For instance, see Seidel [11], Section 4.) Transferring these results to A-modules, we obtain the following classification of the indecomposable graded A-modules. Up to shift, these are: 1) The indecomposable projective module, A.
2) For n ≥ 0, the module K n , which is of dimension 2n + 1. As a graded vector space, K n = V ⊕ W , where V = n i=0 k(i), W = n i=1 k(i), and x and y act by mapping V into W via the matrices respectively. Note that in A -grstab we have that K n ∼ = Ω n S for all n ≥ 0. We shall use the notation Ω n S going forward.
3) For n < 0, the module K n , which is of dimension 2n + 1. As a graded vector , and x and y act by mapping V into W via the matrices respectively. Once again, we note that K n ∼ = Ω n S in A -grstab for all n < 0. We shall use the notation Ω n S going forward. 4) For n > 0, the module M 0,n , which is of dimension 2n.
respectively. Note that for any of the modules described in 2-5 above, hd(X) ∼ = V and soc(X) ∼ = W as graded modules, each with x and y acting by 0.
The following computations are straightforward; we leave them to the reader. Below, n ≥ 0 and m ≥ 1. dim In C(A), functors Ω and (−1) are isomorphic, so our list of indecomposable objects shrinks. In A -grstab, note that ΩM 0,n = M 0,n and ΩM ∞,n = M ∞,n (1); thus M 0,n ∼ = M 0,n (1) and M ∞,n ∼ = M ∞,n (2) in C(A). Thus, a complete list of indecomposable objects in C(A) up to isomorphism is: 1) S(n), for n ∈ Z.
The sizes of the following Hom sets in A -dgstab are an immediate consequence of the above computations for A -grstab and some simple counting arguments. dim dim Hom A -dgstab (M ∞,m , M ∞,1 (1)) = 1 m ≡ 0 mod 2 0 m ≡ 1 mod 2 We are now ready to construct an object K of A -dgstab lying outside of C(A). From the above computations, we have that dim Hom A -dgstab (S, S(2)) = 2; for a basis we can take the unique (up to a nonzero scalar) morphisms f −1 : S → Ω −1 (S)(1) ∼ = S(2) and f −2 : S → Ω −2 S ∼ = S(2). Let g = f −1 + f −2 , and let K be the cone of g in A -dgstab. We shall show that K does not lie in the image of F A . Proposition 4.7. dim Hom A -dgstab (K, S(n)) = 1 for all n ≥ 3 Proof. Consider the triangle S g − → S(2) → K → S(1) which defines K. Choosing some n ≥ 2, we apply Hom A -dgstab (−, S(n)) and observe the resulting long exact sequence. We will show that g(−k) * : Hom A -dgstab (S(2 − k), S(n)) → Hom A -dgstab (S(−k), S(n)) is injective for all k ≥ 0. From this, it will follow from the long exact sequence that for all k ≥ 0, and we will have dim Hom A -dgstab (K, S(n)) = 1 for all n ≥ 3.
Applying the functor (k), it suffices to show that g * : Hom A -dgstab (S(2), S(r)) → Hom A -dgstab (S, S(r)) is injective for all r ≥ 2, where r = n + k. Interpreting f −1 and f −2 as morphisms in A -dgstab, we have that g * = f * −1 + f * −2 . If we are given a nonzero morphism h s : S(2) → Ω s S(r+s) in A -grstab, a straightforward computation shows that both and It follows immediately that f * −1 and f * −2 are injective. We now show that g * is injective. Let (h s ) s : S(2) → S(r) in C(A). Note that h s can be nonzero only when −r Now suppose that g * (h s ) = 0. If (h s ) s = 0, let N be the maximum s such that h s is nonzero. By injectivity of f * −1 , we have that N < −⌈ r 2 ⌉ + 1, and by injectivity of f * −2 , we have that N > −r + 2. But then As this contradicts the definition of N, we must have that h s = 0 for all s, and so g * is injective for all r ≥ 2. Thus dim Hom A -dgstab (K, S(n)) = 1 for all n ≥ 3.  Proof. Again consider the triangle S g − → S(2) → K → S(1) defining K and write g = f −1 + f −2 . Applying Hom A -dgstab (−, M ∞,1 ), we again show that g * (k) : is an isomorphism for all k. As in Proposition 4.7, we shall apply (−k) and work instead with g * : (2), it suffices to consider the cases k = 0 and k = 1.

Brauer Tree Algebras
In this section we shall prove that the functor F A of Theorem 3.9 is an equivalence whenever the algebra A is any non-positively graded Brauer tree algebra. We shall work over an algebraically closed field k.
A Brauer tree consists of the data Γ = (T, e, v, m), where T is a tree, e is the number of edges of T , v is a vertex of T , called the exceptional vertex, and m is a positive integer, called the multiplicity of v. To any Brauer tree Γ, we can associate a basic finite-dimensional symmetric algebra A Γ . For the details of this process, we refer to [10].
An important special case is S = (S, n, v, m), the star with n edges and exceptional vertex at the center. In this case, the algebra A S is a Nakayama algebra whose indecomposable projective modules have length nm + 1.
The following theorems are due to Bogdanic:  From these facts, we obtain the following result: Proof. If Γ is the star, then A Γ is a Nakayama algebra and the result follows immediately from Lemma 4.6. If Γ is not the star, let S denote the star with the same number of edges and multiplicity as Γ. By Theorem 5.2 there is a nonpositive grading on A S such that soc(A S ) is in degree −d. Then by Theorem 5.1, D b (A Γ -grmod) and D b (A S -grmod) are equivalent as triangulated categories. By a theorem of Rickard [8], this induces a triangulated equivalence G : A Γ -grstab → A S -grstab which commutes with grading shifts. By Proposition 3.18, G induces an equivalence between C(A Γ ) and C(A S ). Since A S is a Nakayama algebra, it satisfies the hypotheses of Lemma 4.3, hence A Γ does as well. Thus F A Γ is an equivalence. Proof. This follows from the use of Proposition 3.18 in the previous theorem.
6 The Dg-Stable Category of the Star with n Vertices For n ≥ 2, d ≥ 0, let A = A n,d denote the graded Brauer tree algebra, with socle in degree −d, corresponding to the star S with n edges and exceptional vertex of multiplicity one. This specifies A up to graded Morita equivalence; we will choose a specific grading once we have adopted some more notation in the section below. By the results of Section 5, A -dgstab is equivalent to C(A). We shall identify the two categories throughout this section.

Notation, Indexing, and Grading
We index the edges of S by the set Z/nZ = {1, · · · , n}, according to their cyclic order around the center vertex. We define a total order ≤ on Z/nZ by 1 < 2 < · · · < n. This order is of course not compatible with the group operation on Z/nZ. If P is a statement with a truth value, we define δ P to be 1 if P is true and 0 if P is false. For x, y ∈ Z, define x, y to be the closed arc of the unit circle starting at e 2π √ −1 n x and proceeding counterclockwise to e 2π √ −1 n y . Thus x, x denotes a point, rather than the full circle.
With these definitions, the Ext-quiver of A is a directed cycle, C, of length n. C has vertices e i and edges e i e i+1 a i for all i ∈ Z/nZ. A is isomorphic to, and will be identified with, the quotient of the path algebra of C by the ideal generated by paths of length n + 1. Changing A up to graded Morita equivalence, we determine the grading on A by defining deg(a i ) = −dδ i=n . We denote by S i the simple A-module corresponding to e i , in degree 0. We denote by P i the indecomposable projective module with head S i and socle S i (d).
The indecomposable A-modules are uniserial and determined, up to isomorphism, by their head and socle. For i, j ∈ Z/nZ, let M i j denote the indecomposable module with head S i and socle S j (dδ j<i ). More specifically, for 1 ≤ i, j ≤ n, we define M i j to be the module e i A/e i J l , where J is the Jacobson radical of A and l = δ i>j n + 1 + j − i is the length of M i j . The non-projective indecomposable objects of A -grmod, up to grading shifts and isomorphism, are precisely M i j for i, j ∈ Z/nZ.
Even when working in A -grstab, it will be helpful to define the "length" of M i j , for 1 ≤ i, j ≤ n, to be l(M i j ) = δ i>j n + 1 + j − i. Finally, we note that for 1 ≤ r, j ≤ n, the module M j+1−r j (−dδ j =r ) has length r and socle S j in degree zero; we shall make extensive use of this module later on.

Structure of A -grstab
One of the desirable features of Brauer tree algebras is that the A-module homomorphisms X → Y can be determined combinatorially from the composition towers of X and Y , allowing quick and easy computation of morphisms. For a more general and explicit description of this procedure, we refer to Crawley-Boevey [3]. These techniques generalize easily to graded modules.
The following results about A -grstab follow from straightforward computation and are well-known. We state them without proof.
Proposition 6.1. The (distinct) indecomposable objects of A -grstab are precisely M i j (k), for any i, j ∈ Z/nZ, k ∈ Z.
We shall refer to the statement a, j ⊂ i, b as the arc containment condition.
For describing composition, it will be helpful to choose a collection of generators for the above Hom spaces. Fortunately, there are natural choices. For i = j + 1, define the canonical injection Note, in particular, that α a,a b,b is the identity map.
Proposition 6.4. The indecomposable maps in A -grstab are precisely the canonical surjections and injections. Composition in A -grstab is given by the formula: if a, f ⊂ e, b 0 otherwise Proposition 6.5. In A -grstab, the following formulas hold: Analogous formulas hold for the α a,i b,j .
can be completed into the exact triangle: , and

Structure of A -dgstab
Since Ω ∼ = (−1) in A -dgstab, and Ω is periodic in A -grstab, it follows that (1) is periodic in A -dgstab. The period depends both on n and d. This period is the same for all indecomposable modules except when n is odd, in which case the indecomposable modules of length n+1 2 have their period halved. Proposition 6.7. In A -dgstab, M i j ∼ = M i j ((n + 1)d + 2n) for all i, j ∈ Z/nZ. If n is odd, then we also have M i Proof. By Proposition 6.5 we have that M i j (−2n) ∼ = Ω 2n M i j = M i j (d(n + 1)), from which the first formula follows. Similarly, if n is odd, then M i , from which the second formula follows. Definition 6.8. Define the period of r ∈ {1, · · · , n} to be We also define the period of M i j to be P (l(M i j )). We define the period P (X) of an arbitrary object X to be the maximum period of its indecomposable components.
For any X ∈ A -dgstab, let ψ : X → X((n + 1)d + 2n) denote the map induced by the natural isomorphism id → ((n + 1)d + 2n), whose unique nonzero component is the identity map in degree −2n. For any X ∈ A -dgstab that can be expressed as a direct sum of modules of length n+1 2 , let ψ 1/2 : X → X( (n+1)d 2 +n) denote the isomorphism whose unique nonzero component is the identity map in degree −n.
Thus Proposition 6.7 states that for any 1 ≤ i, j ≤ n, M i j ∼ = M i j (P (r)) in A -dgstab, where r = j + 1 − i is the length of M i j . One consequence of periodicity is that we can express any M i j as a suitable shift of some M 1 l . Furthermore, l can always be chosen to lie in the range 1 ≤ l ≤ n+1 2 , since l(ΩM i j ) = n + 1 − l(M i j ). Proposition 6.9. Let 1 ≤ i, r ≤ n and 1 ≤ l ≤ n+1 2 . The following identities hold in A -dgstab: 0 otherwise f l,r,j is an isomorphism if and only if l = r = j, in which case it is the identity map. g l,r,j is an isomorphism if and only if l = r = j = n+1 2 , in which case g l,l,l = ψ 1/2 . In particular, the indecomposable modules listed in Proposition 6.10 are pairwise non-isomorphic.
For r = n+1 2 and any value of l, the morphisms f l,r,j and g l,r,j are defined for the same values of j and represented by the same morphism in A -grstab. More specifically, for each such j, g l,r,j = ψ 1/2 • f l,r,j For l = n+1 2 and any value of r, the morphisms f l,r,j+r− n+1 2 and g l,r,j are defined for the same values of j, and their unique nonzero components differ only by an application of Ω n and a grading shift. More precisely, for each such j, ψ • f l,r,j+r− n+1 2 = g l,r,j ( (n+1)d 2 + n) • ψ 1/2 Apart from the above identities, all the f l,r,j and g l,r,j are distinct, in the sense that their unique nonzero components cannot be transformed into one another by applying powers of Ω and grading shifts.
To verify the composition formulas, we translate them into statements about A -grstab and use Proposition 6.4.
We start with Equation (10). A tedious but straightforward computation using Proposition 6.5 shows that the only possible nonzero component of . Then by Proposition 6.4, this composition is nonzero if and only if 1, q + j − r ⊂ (q + j − r) + 1 − c, l , in which case it is equal to α 1,(q+j−r)+1−c l,q+j−r . Since the codomain of this morphism has length c, it follows that the resulting morphism, if nonzero, is equal to f l,c,q+j−r . It remains to verify that the arc containment condition is equivalent to the desired inequality. If q +j −r < 1, then the restrictions on q, j, r, and l imply that l −n < q +j −r < 1, hence both the desired inequality and the arc containment condition are false. If q + j − r ≥ 1, the restrictions on q, j, l, r, and r imply that 1 ≤ q + j − r ≤ n and c + l − n ≤ 1. We can then apply Proposition 6.11 and conclude the arc containment condition is equivalent to the inequality 1 ≤ q + j − r ≤ min(c, l). Thus Equation (10) holds.
Proceeding to Equation (11) . This composition is nonzero if and only if 1, q + j − r ⊂ (q + j − r) + c, l , in which case it is equal to α 1,(q+j−r)+c l,q+j−r . The codomain of this component has length n + 1 − c, hence the composition, if nonzero, is equal to g l,c,q+j−r . The same argument as above shows that the arc containment condition is equivalent to the inequality max(1, 1 + l − c) ≤ q + j − r ≤ l. Thus Equation (11)  . This composition is nonzero if and only if 1, q+ j − c ⊂ q + j, l , in which case it is equal to α 1,q+j l,q+j−c . The codomain of this component has length n + 1 − c, hence the composition, if nonzero, is equal to g l,c,q+j−c . The same argument as above shows that the arc containment condition is equivalent to the inequality max(1, 1 + l − c) ≤ q + j − c ≤ l. Thus Equation (12) holds.
For Equation (13), the only possible nonzero component of g r,c,q ((d + 2)(n + 1 − j) − 1) • g l,r,j : M 1 l → M 1 c ((d + 2)(2(n + 1) − q − j) − 2) is α j+r,q+j j,q+j+c−1 (−dδ j =n+1−r ) • α 1,j+r l,j . This composition is nonzero if and only if 1, q + j + c − 1 ⊂ q + j, l , in which case it is equal to α 1,q+j l,q+j+c−1 . It is clear that the desired inequality implies the arc containment condition; we now show the converse. Due to the restrictions on q, j, and c, we have that 2 ≤ q + j ≤ q + j + c − 1 ≤ l + n and q + j ≤ n + 1. Thus if q + j ≤ l, we have that 1 < q + j ≤ l, and the arc containment condition fails. Thus we must have that l < q + j ≤ n + 1. If l < q + j + c − 1 ≤ n + 1, then the arc containment condition fails, hence we must also have n + 1 ≤ q + j + c − 1. The desired inequality follows immediately. Thus the arc containment condition and the desired equality are equivalent. If both hold, the codomain of the nonzero component has length c. We also have that 1 ≤ q + j + c − (n + 1) ≤ min(c, l). Thus the composition is equal to ψ • f l,c,q+j+c−(n+1) . To explain the presence of ψ in this formula, note that the grading shift of the codomain of the composition is (d + 2)(2(n+ 1) −q −j) −2 = [(d + 2)(c −(q + j + c −(n+ 1)))] + [(n+ 1)d + 2n] The factor of ψ accounts for the second bracketed term.
Computing the cones of the morphisms in Theorem 6.12 is straightforward, since the computations can be done in A -grstab.