THE GRADED CENTER OF A TRIANGULATED CATEGORY

JON F. CARLSON AND PETER WEBB

To the memory of a wonderful friend, Laci Kovács

Abstract. With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor We show that such natural transformations which have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities.

Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

1. Introduction

The graded center of a triangulated category C is the set of natural transformations Id_{C} → Σ^{n} that commute with the shift functor Σ up to a sign (−1)^{n}. In [12], Linckelmann investigated the graded center of a block algebra of a finite group. His main result showed that the graded center of the derived category of a block is, modulo a nilpotent ideal, noetherian over the cohomology ring of the block. Along the way, Linckelmann showed that there is a large ideal of nilpotent elements in the graded center generated by elements in degree minus one that are supported on only a single Σ-orbit of modules. This result was extended by Linckelmann and Stancu to obtain elements in all degrees each of which is supported on only a single module that is periodic of period one.

Unltimately we would like to be able to characterize the nilpotent elements in the graded center. In that direction, a natural question to ask is whether there can be elements of the graded center that are non-trivially supported on more than one Σ-orbit? By this we mean, do there exist elements in

2000 Mathematics Subject Classification. Primary 16G70; Secondary 18E30, 20C20. Key words and phrases. Auslander-Reiten triangle, stable module category, Serre func-

tor, graded center.

the graded center that vanish on all but a finite number of Σ-orbits, but which have a non-zero composition with some non-isomorphism? A main purpose of this paper is to show that the answer to that question is generally negative.

For the most part, we work in a Hom-finite, Krull-Schmidt, k-linear triangulated category C that is Calabi-Yau. For F : C → C an endofunctor, we define the support of a natural transformation α : Id_{C} → F to be the set of isomorphism classes of objects U with α_{U} : U → F (U) not zero. We prove that the support of such an α is a single object if and only if F is the Serre functor and for any object M, α_{M} is an almost vanishing morphism. Thus such natural transformations have the same form as those contructed by Linckelmann. We show that, under some reasonable assumptions on the Auslander-Reiten quiver, the support of an element of the graded center, which is supported non-trivially on more than one shift orbit in a component of the Auslander-Reiten quiver, has the entire component in its support. Moreover, if C is the stable category of a group algebra of a p-group of wild representation type, then such a element ψ of the graded center has the property that there exists a map γ : U → M of indecomposable modules such that ψ_{M}γ ≠ 0 and γ is not a composition of a finite number of irreducible morphisms. We show, by the example of a finite group with dihedral Sylow 2-subgroup, that this requirement does not hold if the group has tame representation type.

Throughout the paper we assume that k is an algebraically closed field and that C is a Hom-finite, Krull-Schmidt, k-linear triangulated category with shift Σ. For background on Auslander-Reiten theory we refer to standard texts such as [1].

Both authors are grateful for support from the Simons Foundation. The first author also thanks NSA for support during part of the time when this paper was written.

2. Linear functors on a triangulated category

We present some preliminaries on the category of linear functors defined on a triangulated category. Let Fun^{op} C denote the category of contravariant k-linear functors from C to k-vector spaces. The first four results of this section are well known. While they are usually stated for functors on module categories, they hold for k-linear functors on k-linear categories (and even for additive functors on additive categories). In particular they hold for Fun^{op} C, ignoring the triangulated structure of C. There are proofs of these results in [1] stated in terms of functors on the module category of a ring, but the arguments there work for functors on an additive category without change. Our purpose is to point out that these results hold in the generality we consider here.

Recall our assumption that C is a Hom-finite, Krull-Schmidt, k-linear triangulated category with shift Σ, Proposition 2.1. The category Fun^{op} C is an abelian category. Moreover, for each indecomposable object M the representable functor Hom_{C}(−, M) is indecomposable and projective, with endomorphism ring isomorphic to End_{C}(M).

Proof. See IV.6.2(a), IV.6.4(a) and A.2.9 of [1]. It is standard that Fun^{op} C is an abelian category in which kernels, cokernels and exactness are determined by evaluation at the objects of C. The statements about representable functors are a consequence of the linear form of Yoneda’s Lemma, which in our usage says that any morphism from Hom_{C}(−, M) to Hom_{C}(−, N) is induced from a morphism from M to N. □

The simple functors s^{M} ∈ Fun^{op} C are defined in [1, IV.6.7]. For each indecomposable object M of C we have

s^{M} = Hom_{C}(−, M)/ Rad_{C}(−, M)

where Rad_{C}(−, M) is the radical of Hom_{C}(−, M), the subfunctor whose value at an object X is the set of non-isomorphisms from X to M. These functors have the description

s^{M}(N) = \begin{cases}k & if M ≅ N, \\ 0 & otherwise. \end{cases}

The next result is also well known in the context of functors on module categories, and the proof given in the reference carries through verbatim.

Proposition 2.2. (1) The simple objects in Fun^{op} C are all of the form s^{M} for some indecomposable object M in C. The relation M ↔ s^{M} give a one-to-one correspondence between isomorphism types of indecomposable objects in C and isomorphism types of simple objects in Fun^{op} C.

(2) The quotient map Hom_{C}(−, M) → s^{M} is a projective cover, having kernel Rad_{C}(−, M), which is the unique maximal subfunctor of Hom_{C}(−, M).

Proof. See [1, IV.6.8]. □

We say that a functor F is finitely generated if there is an epimorphism Hom_{C}(−, M) → F for some object M. We say that F is finitely presented if there is an exact sequence of functors Hom_{C}(−, M_{1}) → Hom_{C}(−, M_{0}) → F → 0.

The next result is, again, usually only stated for functors on module categories, but it is also true for functors on additive or k-linear categories.

Proposition 2.3. The representable functors Hom_{C}(−, M), where M is indecomposable, are a complete list of the indecomposable finitely generated projective functors in Fun^{op} C.

Proof. See [1, IV.6.5] □

We say that s^{M} is a composition factor of a functor F if there are subfunctors F_{0} ⊂ F_{1} of F so that F_{1}/F_{0} ≅ s^{M}.

Corollary 2.4. Let F be a functor in Fun^{op} C. Then F has s^{M} as a composition factor if and only if F (M) ≠ 0. Furthermore, F has finite composition length if and only if F is non-zero on only finitely many isomorphism classes of indecomposable objects of C, where its value is finite dimensional.

Proof. If F has s^{M} as a composition factor then, since s^{M}(M) ≠ 0, we must have F (M) ≠ 0. Conversely, if F (M) ≠ 0 then by Yoneda’s Lemma there is a non-zero morphism Hom_{C}(−, M) → F showing that the unique simple quotient s^{M} of Hom_{C}(−, M) appears as a composition factor of F. The statement about finite composition length of F follows from the fact that each simple functor is non-zero on a single isomorphism class of indecomposable objects, where its value has dimension 1. □

We turn now to a result for triangulated categories which is not the same as for module categories. It is well known that finitely presented functors on a module category have projective dimension at most 2, because Hom is left exact on such a category [3, Prop. 4.2]. The situation for functors on a triangulated category is quite different.

Proposition 2.5. The only finitely presented functors of finite projective dimension in Fun^{op} C are the representable functors. Equivalently, the only monomorphisms between representable functors are split.

Proof. Given a presentation Hom_{C}(−, M_{1}) → Hom_{C}(−, M_{0}) → F → 0 of a functor F the morphism between the representable functors comes from a morphism α : M_{1} → M_{0} in C, by Yoneda’s Lemma. Complete this morphism to a triangle and rotate it, to get a triangle

M_{2} ⟶^{β} M_{1} ⟶^{α} M_{0} ⟶^{γ} ΣM_{2}.

We get a long exact sequence of representable functors which has F as one of its (co)kernels, giving a long exact sequence

⋯  ⟶ Hom_{C}(−, Σ^{−1}M_{1}) ⟶ Hom_{C}(−, Σ^{−1}M_{0})
⟶ Hom_{C}(−, M_{2}) ⟶ Hom_{C}(−, M_{1}) ⟶ Hom_{C}(−, M_{0})
⟶  F  ⟶  0,

which is a projective resolution of F. By a standard result in homological algebra, F has finite projective dimension if and only if at some stage the kernel in this resolution is projective. Note that every kernel is finitely generated, because it is the image of the next term in the resolution. Thus, if F has finite projective dimension, then one of the morphisms Hom_{C}(−, U) → Hom_{C}(−, V) in the sequence factors as a surjection onto a projective functor followed by an injection from the projective functor as follows:

Hom_{C}(−, U) → Hom_{C}(−, X) → Hom_{C}(−, V)

corresponding to morphisms U ⟶^{ϕ} X ⟶^{θ} V in C. Here we are using the fact that projective functors are representable, by Proposition 2.3. Because the first of these maps of functors is surjective, the identity 1_{X} is an image of a map X → U after composition with ϕ, so that ϕ is a split epimorpism. The injectivity of the second map of functors is exactly the definition that θ is a monomorphism. In a triangulated category all monomorphisms are split, so that θ is a split monomorphism. It follows from this that there are decompositions U ≅ U_{1}⊕X and V ≅ X ⊕V_{1} so that the morphism U → V is the identity on X and 0 on U_{1}. Transferring back to the original triangle we see that it is the sum of two triangles, one of the form Σ^{n}X ⟶^{1} Σ^{n}X → 0 → Σ^{n+1}X for some n, and the other with zero as one of its three morphisms. The first of these produces contractible summands of the resolution of F, and the second produces a resolution which is split everywhere, showing that F is projective because the final map Hom_{C}(−, M_{0}) → F must split.

The equivalence with the statement that monomorphisms between representable functors are split is immediate. If there is a non-split such monomorphism, then its cokernel has projective dimension 1 and is not projective. On the other hand, any non-projective functor of finite projective dimension gives rise to a functor of projective dimension 1 (appearing at the end of the finite projective resolution), and this is presented by a non-split monomorphism of projectives. □

We characterize the existence of Auslander-Reiten triangles in terms of finite presentability of the corresponding simple functors. The result is familiar for functors on module categories, but less so for functors on triangulated categories.

Proposition 2.6. Let C be a Hom-finite, Krull-Schmidt triangulated category, and let M be an indecomposable object in C. The simple functor s^{M} is finitely presented if and only if there is an Auslander-Reiten triangle U → V → M → ΣU. When there exists such an Auslander-Reiten triangle the map of representable functors Hom_{C}(−, M) → Hom_{C}(−, ΣU) has s^{M} as its image.

Proof. If there is such an Auslander-Reiten triangle the long exact sequence

⋯ → Hom_{C}(−, U) → Hom_{C}(−, V) → Hom_{C}(−, M) → Hom_{C}(−, ΣU) → ⋯

provides the start of a resolution

Hom_{C}(−, V) → Hom_{C}(−, M) → s^{M} → 0

because the lifting property of the Auslander-Reiten triangle coupled with the fact that it is not split translates to the statement that the cokernel of Hom_{C}(−, V) → Hom_{C}(−, M) is s^{M}, which is also the image of Hom_{C}(−, M) in Hom_{C}(−, ΣU). This shows that s^{M} is finitely presented.

Conversely, if s^{M} is finitely presented by a 3-term exact sequence of this form, then the morphism Hom_{C}(−, V) → Hom_{C}(−, M) comes from a morphism V → M in C which we may extend to a triangle U → V → M → ΣU.

This triangle satisfies the Auslander-Reiten lifting property at M, and the morphism M → ΣU is not zero since V → M is not a split epimorphism. The triangle gives rise to a long exact sequence of representable functors of the kind at the start of this proof. If the kernel of Hom_{C}(−, V) → Hom_{C}(−, M) has a non-zero projective direct summand, then such a summand is finitely presented and hence representable. By Proposition 2.5, it splits off from Hom_{C}(−, V), as well as from Hom_{C}(−, U). Thus we can remove such a summand and may assume that Hom_{C}(−, V) is a projective cover of Rad_{C}(−, M). In this case the morphism V → M is minimal right almost split, in the terminology of [1]. It is proven by Happel [11, page 36] (in the dual case of a minimal left almost split morphism) that the third term U in the triangle is indecomposable, and hence the triangle is an Auslander-Reiten triangle. □

Given a Hom-finite, Krull-Schmidt triangulated category C over k, a Serre functor on C is a self-equivalence S : C → C for which there are bifunctorial isomorphisms

D Hom_{C}(X, Y) ≅ Hom_{C}(Y, S(X)) for all X, Y ∈ C.

Here D(U) = Hom_{k}(U, k) is the vector space duality. It was shown in [14] that C has a Serre functor S if and only if C has Auslander-Reiten triangles, and that the Auslander-Reiten triangles have the form

Σ^{−1}S(U) ⟶^{α} V ⟶^{β} U ⟶^{γ} S(U)

with Auslander-Reiten translate τ = Σ^{−1}S.

We now point out that the presence of a Serre functor on C makes Fun^{op} C into a self-injective category. We will use, particularly, the fact that representable functors for indecomposable objects have simple socles.

Proposition 2.7. Let C be a Hom-finite, Krull-Schmidt triangulated category with Serre functor S. Then each representable functor Hom_{C}(−, M) is injective (as well as projective), with simple socle s^{S^{−1}(M)}.

Proof. For each object X we have

Hom_{C}(X, M) ≅ D Hom_{C}(M, S(X)) ≅ D Hom_{C}(S^{−1}(M), X).

Because Hom_{C}(S^{−1}(M), −) is a projective covariant functor on C it follows that

D Hom_{C}(S^{−1}(M), −) ≅ Hom_{C}(−, M)

is an injective contravariant functor on C, as well as being projective. Now if

Σ^{−1}M → E → S^{−1}(M) → M

is an Auslander-Reiten triangle, the image of

Hom_{C}(−, S^{−1}(M)) → Hom_{C}(−, M)

is the simple functor s^{S^{−1}(M)} by Proposition 2.6. This is the socle. □

We identify composition factors of functors in Fun^{op} C in the spirit of [4].

Corollary 2.8. Assume that C has a Serre functor S. Let U and M be indecomposable objects of C. The following are equivalent:

(1) The functor s^{U} is a composition factor of Hom_{C}(−, M).

(2) there is a non-zero morphism U → M.

(3) there is a non-zero morphism S^{−1}(M) → U.,

(4) The functor s^{S^{−1}(M)} is a composition factor of Hom_{C}(−, U).

Proof. This is immediate from Corollary 2.4 and the definition of a Serre functor. □

Following Linckelmann [12] (who attributes the terminology to Happel [11]) we say that the third morphism γ in an Auslander-Reiten triangle

X ⟶^{α} Y ⟶^{β} Z ⟶^{γ} ΣX

is almost vanishing. Notice that the domain and codomain of γ are both indecomposable in this definition. An almost vanishing morphism determines the corresponding Auslander-Reiten triangle by completing it to a triangle and rotating to put it in the right position. Equally, an almost vanishing morphism exists with domain Z (or codomain X) if and only if there is an Auslander-Reiten triangle with Z on the right (or X on the left – since these properties are preserved by Σ).

As an example, when C = stmod(A) is the stable module category of a symmetric algebra, γ is almost vanishing if and only if it represents an almost split sequence of A-modules as an element of Ext_{A}^{1}(W, Ω(U)).

Almost vanishing morphisms have been used in several places in the literature. They underlie the construction of natural transformations in the graded center in [12] and [13]. They provide a construction of ghost maps showing that Freyd’s generating hypothesis fails in general for the stable module category stmod(kG), when G is a finite group [9]. They were used by Happel [11] in constructing Auslander-Reiten triangles in bounded derived categories (where they exist). We present several characterizations of these morphisms. Most of these are well known, but conditions (2) and (3) may be less familiar.

Proposition 2.9. Let C be a Hom-finite, Krull-Schmidt triangulated category with Serre functor S, and let f : X → Y be a morphism between indecomposable objects in C. The following are equivalent:

(1) The map f is almost vanishing.

(2) f is non-zero, and for all objects U, f factors through every non-zero morphism U → Y.

(3) f is non-zero, and for all objects V, f factors through every non-zero morphism X → V.

(4) Whenever g : U → X is not a split epimorphism in C then fg = 0.

(5) Whenever h : Y → U is not a split monomorphism then hg = 0.

(6) The map Hom_{C}(−, f) : Hom_{C}(−, X) → Hom_{C}(−, Y) factors through a simple functor.

Thus morphisms f satisfying any (and hence all) of the above conditions are determined up to scalar multiple. For such a morphism, Y ≅ S(X).

The word ‘split’ is redundant in conditions (4) and (5) because all monomorphisms and epimorphisms in a triangulated category are split.

Proof. We start by observing that the implication (1) ⇒ (6) is part of Proposition 2.6. For the converse (6) ⇒ (1), if (6) holds then the image of Hom_{C}(−, f) must be the simple socle s^{S^{−1}Y} of Hom_{C}(−, Y), by Proposition 2.7, and so Y ≅ S(X), and f is determined up to a scalar multiple. We know that there exists an almost vanishing map g : X → SX, and it has the same property as f. Hence f is a scalar multiple of g, and f is almost vanishing.

(1) ⇒ (2). Suppose that f is almost vanishing, and let U → Y be a nonzero morphism. Then X ≅ S^{−1}(Y) and s^{X} ≅ s^{S^{−1}(Y)} is a composition factor of Hom_{C}(−, U) by Proposition 2.8, and we have morphisms between projective functors Hom_{C}(−, X) → Hom_{C}(−, U) → Hom_{C}(−, Y) with composite mapping to the simple socle of Hom_{C}(−, Y). This means the corresponding composite is almost vanishing and provides a factorization of f as in (2).

(2) ⇒ (1) Suppose f satisfies (2), and let ϕ : S^{−1}(Y) → Y be almost

vanishing. There is a factorization of f as X ⟶^{γ} S^{−1}(Y) ⟶^{ϕ} Y. Since the image of Hom_{C}(−, ϕ) is the simple top s^{S^{−1}(Y)} and f ≠ 0, Hom_{C}(−, γ) : Hom_{C}(−, X) → Hom_{C}(−, S^{−1}(Y)) maps onto the simple top and hence is surjective, by Nakayama’s lemma. Therefore γ is an isomorphism, and f is almost vanishing.

The equivalence of (1) and (3) is similar.

That (1) implies (4) follows because g factors through W in the Auslander-

Reiten triangle Σ^{−1}Y ⟶^{α} W ⟶^{β} X ⟶^{f} Y so that fg = fβg′ for some g′, and this composite is zero because fβ = 0.

To get that (4) implies (1) we complete f to a triangle and rotate to get

a triangle Σ^{−1}Y ⟶^{α} W ⟶^{β} X ⟶^{f} Y. This is an Auslander-Reiten triangle because f ≠ 0, X and Σ^{−1}Y are indecomposable, and condition (4) implies that any morphism g : U → X which is not a split epimorphism factors through W.

The equivalence (1) ⇔ (5) is similar. □

In the next section we consider morphisms f : X → Y for which the image of the natural transformation of representable functors

Hom_{C}(−, f) : Hom_{C}(−, X) → Hom_{C}(−, Y)

has finite composition length. To prepare for this we present some results which identify the occurrence of composition factors.

Proposition 2.10. Assume that C has a Serre functor S. Let f : X → Y be a morphism between indecomposable objects, and let V be an indecomposable object. The following are equivalent.

(1) The functor s^{V} is a composition factor of the image of

Hom_{C}(−, X) ⟶^{Hom(−,f)} Hom_{C}(−, Y).

(2) There is a morphism V ⟶^{γ} X so that fγ ≠ 0.

(3) There are morphisms S^{−1}(Y) ⟶^{ξ} V ⟶^{γ} X so that fγξ is almost vanishing.

Proof. Because Hom(−, V) is projective and has unique simple quotient s^{V} we see that s^{V} is a composition factor of the image of Hom(−, f) if and only if there is a morphism Hom(−, γ) : Hom_{C}(−, V) → Hom_{C}(−, X) whose image is not in the kernel of Hom(−, f). By Yoneda’s Lemma, this happens if and only if there is a morphism γ : V → X so that fγ ≠ 0. By Proposition 2.9 conditions (2) and (3) are equivalent. □

If α : F → G is a natural transformation of functors defined on C, we say that the support of α is the set of isomorphism classes of indecomposable objects M for which α_{M} : F (M) → G(M) is non-zero.

Corollary 2.11. Assume that C has a Serre functor S and let f : X → Y be a morphism between indecomposable objects. The following are equivalent:

(1) The image of Hom_{C}(−, f) has finite composition length.

(2) There are only finitely many isomorphism classes of indecomposable modules V with a morphism γ : V → X so that fγ ≠ 0.

(3) There are only finitely many isomorphism classes of indecomposable

modules V such that there are morphisms S^{−1}(Y) ⟶^{ξ} V ⟶^{γ} X for which fγξ is almost vanishing.

(4) The support of Hom_{C}(−, f) is finite.

(5) Whenever ϕ : S^{−1}(Y) → X is such that fϕ is almost vanishing then ϕ can be expressed as a sum of composites of (finitely many) irreducible morphisms.

Furthermore, if the image of Hom_{C}(−, f) has finite composition length then

its composition factors are the s^{V} for which there are morphisms S^{−1}(Y) ⟶^{ξ} V ⟶^{γ} X, both of which are finite composites of irreducible morphisms, and so that fγξ is almost vanishing.

Proof. The equivalence of the first four statements is immediate from Proposition 2.10.

(1) ⇔ (5) The image has finite composition length if and only if Rad_{C}^{n}(−, X) is contained in the kernel of Hom_{C}(−, f) for some n. Thus, assuming (1), no morphism γ : Y → X with fγ ≠ 0 lies in Rad_{C}^{n}(Y, X), and so such a morphism cannot be expressed as a sum of composites of n or more irreducible morphisms. Thus (5) holds. Conversely, assume (5) holds. We may take an almost vanishing morphism ϕ : S^{−1}(Y) → X and express it as a sum of composites of irreducible morphisms, deducing that ϕ does not lie in Rad_{C}^{n}(S^{−1}(Y), X) for some n. Since for each non-zero morphism γ : Y → X there is a factorization ϕ = γξ for some ξ, we deduce that every non-zero morphism γ lies outside Rad_{C}^{n}(Y, X). This shows that Rad_{C}^{n}(−, X) is contained in the kernel of Hom_{C}(−, f) and so (1) holds.

The composition factors are as claimed because firstly, by Proposition 2.10, the s^{V} for which fγξ is as described are among the composition factors. The complete set of composition factor arises without the requirement that ξ and γ be finite composites of irreducible morphisms, but we see from (5) that they must be sums of composites of irreducible morphisms. The V which can arise from sums of composites of irreducible morphisms are the same as the V which arise from composites of irreducible morphisms. □

3. Elements of the graded center with finite support

In the context of stable module categories stmod(A) for symmetric algebras A, Linckelmann [12] constructed certain elements of the graded center of stmod(A) of degree −1. In that situation the shift is given by Σ = Ω^{−1}, the inverse of the Heller operator and the Serre functor is S = Ω. For each finitely generated indecomposable non-projective module U, he constructed a natural transformation ζ : Id_{C} → Ω such that ζ_{U} : U → Ω(U) is almost vanishing (i.e. represents an almost split sequence ending in U), and such that ζ(V) = 0 for any finitely generated indecomposable non-projective module V which is not isomorphic to Ω^{n}(U), for any integer n. Linckelmann and Stancu [13] then combined this construction with the existence of periodic modules of period one to produce elements of the graded center in degree 0.

We assume throughout that C is a Hom-finite, Krull-Schmidt, k-linear triangulated category with Serre functor S and Auslander-Reiten translate τM = Σ^{−1}S(M). Our first goal is to show that the natural transformations of the kind constructed by Linckelmann are the only ones with small support.

Proposition 3.1. Let F : C → C be a k-linear endofunctor and suppose α : Id_{C} → F is a natural transformation with support consisting of a single indecomposable object M. Then F (M) = S(M) and α_{M} : M → S(M) is an almost vanishing morphism.

Proof. Let α have support only on M. Consider the image of

Hom_{C}(−, M) ⟶^{Hom_{C}(−,α_{M})} Hom_{C}(−, F (M))

as a subfunctor of Hom_{C}(−, F (M)). If the image only has one composition factor (which must be s^{M}, the simple top of Hom_{C}(−, M)) this composition factor must be the socle of Hom_{C}(−, F (M)). Since the socle is s^{S^{−1}(F (M))} we deduce that M = S^{−1}(F (M)), so F (M) = S(M), and that α_{M} is almost vanishing by Proposition 2.9.

If the image has another composition factor s^{X} for some object X, this also appears as a composition factor of Hom_{C}(−, M). Hence, by projectivity of the representable functor Hom_{C}(−, X), there is a non-zero morphism Hom_{C}(−, X) → Hom_{C}(−, M) so that the composite

Hom_{C}(−, X) → Hom_{C}(−, M) ⟶^{Hom_{C}(−,α_{M})} Hom_{C}(−, F (M))

is non-zero. By Yoneda’s Lemma this corresponds to a homomorphism ϕ : X → M so that α_{M} ∘ ϕ ≠ 0. Since α is a natural transformation it follows that F (ϕ) ∘ α_{X} ≠ 0, so that α has X in its support, contradicting our hypothesis. Hence, the image has only one composition factor, a case we have already considered. □

We apply this to the situation considered by Linckelmann and Stancu [13], namely the stable module category stmod(kG) of a finite p-group G over an algebraically closed field of characteristic p. We show that the elements of the graded center that they construct in positive degree are the only ones with support a single module.

Corollary 3.2. Let G be a finite group, k a field and r an integer. Let ϕ : Id_{stmod(kG)} → Σ^{r} be a natural transformation with support a single indecomposable module {M}. Then Σ^{r} ≅ Ω(M) and ϕ_{M} : M → Ω(M) is an almost vanishing morphism. Thus if r ≠ 1, then M ≅ Ω^{r+1}(M) is a periodic module and ϕ is, up to scalar multiple, one of the natural transformations constructed by Linckelmann and Stancu.

Proof. In stmod(kG) we have S = Ω = Σ^{−1}. According to Proposition 3.1, Σ^{r}M = S(M) and M → S(M) is almost vanishing. The remaining assertions are immediate. □

Linckelmann and Stancu were interested in elements of the graded center of stmod(kG). The degree n elements of this ring are the natural transformations Id_{C} → Σ^{n} which commute (in a sense which includes a sign) with Σ. Such natural transformations have support consisting of a single indecomposable object only if that object is periodic under Σ. Allowing such transformations to have support on a single Σ-orbit of indecomposable objects, which might not be periodic, we obtain the following.

Proposition 3.3. Let M be an indecomposable object of C for which there are no irreducible maps Σ^{r}M → M for any r ∈ ℤ, and let F : C → C be a k-linear endofunctor. Suppose that α : Id_{C} → F is a natural transformation whose support is contained in {Σ^{r}M | r ∈ ℤ}. Then F (M) = S(M), and for each r, the map α_{Σ^{r}M} : Σ^{r}M → F (Σ^{r}M) is almost vanishing. Thus α is one of the natural transformations constructed by Linckelmann in [12].

The hypothesis that there are no irreducible maps Σ^{r}M → M for any r ∈ ℤ holds in many cases of interest. For example, it always holds if M belongs to an Auslander-Reiten quiver component of tree class A_{∞}. By [15] it can be seen to hold most of the time for stmod(kG) when G is a finite group.

Proof. As in the proof of Proposition 3.1, consider the image of

Hom_{C}(−, M) ⟶^{Hom_{C}(−,α_{M})} Hom_{C}(−, F (M)).

This has s^{M} as a composition factor, and if it has more composition factors than this it must have one of the form s^{E} where E → M is an irreducible morphism since such simple functors form the second radical layer of the projective cover of s^{M}. This would mean E does not have the form Σ^{r}M and that α has E in its support, which is not possible. We conclude that the image is the simple functor s^{M}, and as before, F M = S(M) and α_{M} is an almost vanishing morphism. □

We now consider elements of the graded center of C with support larger than a single Σ-orbit of objects. One way to construct such elements is to add two elements which have support on different shift orbits: the resulting natural transformation α : Id_{C} → Σ^{r} has the property that for every morphism f : M → N between indecomposable objects in different shift orbits we have α_{N}f = 0. We consider α with α_{N}f ≠ 0 for some non-isomorphism f. This is equivalent to requiring that the support of the natural transformation Hom_{C}(−, α_{N}) has size at least 2 for some N.

We recall that a triangulated category C is d-Calabi-Yau if Σ^{d} is a Serre functor. It follows from [14] that such a category has Auslander-Reiten triangles. In the next result we refer to the Auslander-Reiten quiver simply as the ‘quiver’. We recall that the term mesh denotes a region of this quiver bounded by the objects which appear in the three left terms of an Auslander-Reiten triangle [6].

Theorem 3.4. Let C be a k-linear, Hom-finite, Krull-Schmidt triangulated category. Let α : Id_{C} → Σ^{r} be a natural transformation in the graded center of C. Fix an indecomposable object N of C. We suppose that

(1) C is a d-Calabi-Yau category for some integer d,

(2) for all objects U in the quiver component of N, Σ^{r−d}U and U lie in the same τ-orbit,

(3) every mesh in the quiver component of N has at most two middle terms, and

(4) for all objects U in the same quiver component as N, the support of the natural transformation Hom_{C}(−, α_{U}) : Hom_{C}(−, U) → Hom_{C}(−, Σ^{r}U) is finite, and for some U it has size at least 2.

Then the support of α contains the entire quiver component of N.

Proof. Let U be an indecomposable object in the quiver component of N for which the support of Hom_{C}(−, α_{U}) has size at least 2. i Since Hom_{C}(−, α_{U}) has finite composition length, by Proposition 2.9 and Corollary 2.11 there is a morphism ϕ : Σ^{r−d}U → U, which is a sum of finite composites of irreducible morphisms, such that α_{U}ϕ is almost vanishing. Because the support of Hom_{C}(−, α_{U}) has size at least 2, ϕ is not an isomorphism.

We claim that the composite of morphisms in any path in the Auslander-Reiten quiver from Σ^{r−d}U to U also has the same property as ϕ, and in fact equals ±ϕ. This is because whenever we have a pair of consecutive irreducible morphisms in such a path of the form τV → W → V the Auslander-Reiten triangle τV → E → V → ΣτV has middle term E with at most two indecomposable summands, one of which is W. If E = W ⊕ X for some X we can replace the maps into and out of W by irreducible morphisms τV → X → V, because the composite τV → W ⊕ X → V is zero, so that the new irreducible morphisms have composite (−1) times the composite of the old. Repeating this operation allows us to move from any path from Σ^{r−d}U to U to any other path, changing the composite by (−1) each time. Now ϕ must be a linear combination of composites along these paths, but since the composites are all the same up to sign we deduce that ϕ could be taken to be the composite of the irreducibles along any of the paths.

Since Σ^{r−d}U and U lie in the same τ-orbit, there is a path in the quiver from Σ^{r−d}U to U going through each member of the τ-orbit of U between these two objects. We deduce that for every irreducible morphism with codomain U the domain of that morphism lies in the support of α. This and the fact that α commutes (up to sign) with Σ, and hence with τ, implies that all objects in the component of N lie in the support of α. □

In the next section we present an example of a natural transformation satisfying the conditions of Theorem 3.4 in the context of the stable module category of a group with a dihedral Sylow 2-subgroup in characteristic 2. In general it is not always possible to find such examples, as we now see.

Corollary 3.5. With the same hypotheses as in Theorem 3.4, suppose further that the quiver component containing N has type A_{∞}. Then no such natural transformation α can exist.

Proof. Suppose there were such a natural transformation α. Its support would have to contain the quiver component containing N, and for any choice of indecomposable object N_{0} in this quiver component the proof of Theorem 3.4 shows that there is an irreducible morphism f : U → N_{0} with α_{N_{0}}f ≠ 0. We may choose N_{0} so that it lies on the rim of the quiver, as in the following diagram.

⋮  ⋮  ⋮  ⋮
↘  ↗  ↘  ↗  ↘  ↗
⋯  M_{2}  N_{2}  O_{2}  ⋯
↗  ↘  ↗  ↘  ↗  ↘
L_{1}  M_{1}  N_{1}  O_{1}
↘  ↗  ↘  ↗  ↘  ↗
L_{0}  M_{0}  N_{0}

There is no path of irreducible morphisms from Σ^{r−d}N_{0} to N_{0} with nonzero composite unless r = d. This is because Σ^{r−d}N_{0} is also on the rim, and such a path has composite equal to that of a path which has two irreducible morphisms between consecutive objects on the rim, and the composition of these morphisms is zero. Such a path was necessary to the existence of α in the proof of Theorem 3.4, so this situation cannot occur. When r = d the support of Hom_{C}(−, α_{N_{0}}) has size 1, because there is no finite chain of irreducible morphisms from N_{0} to N_{0} other than the empty chain at N_{0}. This shows that α_{N_{0}} is almost vanishing, so that α_{N_{0}}f = 0, a contradiction. Hence i no α can exist as in Theorem 3.4. □

Corollary 3.6. Let C = stmod(B) be the stable module category of a block with wild representation type of a group algebra kG. Let α be an element of the graded center of C.

(1) If α is supported on only finitely many τ-orbits then α is a sum of elements which are supported on single τ-orbits, each of which is of the kind described in Proposition 3.3. Thus α_{Y} f = 0 for every non-isomorphism f : X → Y between indecomposable objects.

(2) If there is any non-isomorphism f : X → Y between indecomposable objects so that α_{Y} f ≠ 0 then such an f can be found which is not a finite composite of irreducible morphisms. In this case α is not supported on only finitely many τ-orbits.

Proof. We exploit the fact, using a theorem of Erdmann [10], that all quiver components of C have type A_{∞} and satisfy conditions (1), (2) and (3) of Theorem 3.4.

To prove (1), if α were supported on only finitely many τ-orbits then the support of Hom_{C}(−, α_{Y}) would be finite for all indecomposable Y and by Corollary 3.5 such α cannot exist unless this support has size 1 for every Y. This is equivalent to requiring that α_{Y} f = 0 for every non-isomorphism f : X → Y between indecomposable objects, and that α is a sum of natural transformations supported on single τ-orbits.

With the hypothesis of (2), we must have that some Hom_{C}(−, α_{Y}) has infinite support. Finding f : X → Y so that α_{Y} f is almost vanishing as in Proposition 2.9, we find by Corollary 2.11 that f is not a composite of irreducible morphisms. □

Remark 3.7. In the case of modules in a block of wild type in a group algebra, it seems likely that any map f : X → Y as above, that is not a composite of a finite number of irreducible maps, should factor through a module that is not in the quiver component of X, implying that α would have support on more than one quiver component. This is easily verified in some specific cases, but seems difficult to prove in general.

4. An example: groups with dihedral Sylow 2-subgroups

In this section we show, under certain circumstances, that there exist natural transformations in the graded center of the stable module category that are supported on only a single component of the Auslander-Reiten quiver, and which are not sums of the natural transformations constructed by Linckelmann in [12]. Furthermore, our natural transformations satisfy the finiteness condition of Theorem 3.4.

We assume throughout that k is an algebraically closed field of characteristic 2 and that G is a finite group with a dihedral Sylow 2-subgroup having order at least 8. The group algebra kG has tame representation type, and a primary fact in the example is that the Auslander-Reiten quiver component which contains the trivial module has tree class A_{∞}^{∞} [15] and consists entirely of endotrivial modules (see [2]). By definition, a kG-module M is endotrivial provided Hom_{k}(M, M) ≅ k ⊕ P where P is a projective kG-module. We note that a kG-module is endotrivial if and only if its restriction to every elementary abelian p-subgroup is endotrivial and that the tensor product of two endotrivial modules is again an endotrivial module (See [8]).

Suppose that S is a Sylow 2-subgroup of G and that E_{1} and E_{2} are representatives of the two conjugacy classes of elementary abelian subgroups of order 4 in S. The Auslander-Reiten quiver containing the trivial kGmodule has the form

\xymatrix{… \ar[dr] &  & U_{4,0} \ar[dr]_{γ′_{(4,0)}} &  & U_{2,−2} \ar[dr]_{γ′_{(2,−2)}} &  & … \\  & U_{4,2} \ar[ur]^{γ_{(4,2)}} \ar[dr]_{γ′_{(4,2)}} &  & U_{2,0} \ar[ur]^{γ_{(2,0)}} \ar[dr]_{γ′_{(2,0)}} &  & U_{0,−2} \ar[dr]_{γ′_{(0,−2)}} \ar[ur] \\ … \ar[ur] \ar[dr] &  & U_{2,2} \ar[ur]^{γ_{(2,2)}} \ar[dr]_{γ′_{(2,2)}} &  & U_{0,0} \ar[ur]^{γ_{(0,0)}} \ar[dr]_{γ′_{(0,0)}} &  & U_{−2,−2} \\  & U_{2,4} \ar[ur]^{γ_{(2,4)}} \ar[dr]_{γ′_{(2,4)}} &  & U_{0,2} \ar[ur]^{γ_{(0,2)}} \ar[dr]_{γ′_{(0,2)}} &  & U_{−2,0} \ar[ur]^{γ_{(−2,0)}} \ar[dr] \\ … \ar[ur] &  & U_{0,4} \ar[ur]^{γ_{(0,4)}} &  & U_{−2,2} \ar[ur]^{γ_{(−2,2)}} &  & …}

where U_{0,0} ≅ k, U_{i,i} ≅ Ω^{i}(k) and U_{i,j} is an endotrivial module with the property that U_{i,j} ↓_{E_{1}} ≅ Ω^{i}(k_{E_{1}}) and U_{i,j} ↓_{E_{2}} ≅ Ω^{j}(k_{E_{2}}) (See [2, 15]). The almost split sequence ending in the trivial module has the form

\xymatrix{0 \ar[r] & Ω^{2}(k) \ar[r] & U_{2,0} ⊕ U_{0,2} \ar[r] & k \ar[r] & 0}

Note that all modules in the component of the trivial module have odd dimension since they are endotrivial. If M is an indecomposable kG-module of odd dimension, then by [2] the almost split sequence ending in M is (modulo projective summands)

\xymatrix{0 \ar[r] & Ω^{2}(k) ⊗ M \ar[r] & U_{2,0} ⊗ M ⊕ U_{0,2} ⊗ M \ar[r] & M \ar[r] & 0}

In particular, we see that U_{i,j} ⊗ U_{s,t} ≅ U_{i+s,j+t} ⊕ P for some projective module P.

Lemma 4.1. For any n there is an exact sequence having the form

[figure: E_{n} :; 0; Ω^{2n}(k); ^{(α, β)}; U_{2n,0} ⊕ U_{0,2n}; ^{(\begin{matrix}γ \\ δ \end{matrix})}; k; 0]

where α = γ_{(2n,2)} … γ_{(2n,2n−2)}γ_{(2n,2n)}, β = γ′_{(2,2n)} … γ′_{(2n−2,2n)}γ′_{(2n,2n)}, etc. That is, each map is the obvious composition of irreducible maps in the Auslander-Reiten quiver.

Proof. The modules in the sequence are positioned in the Auslander-Reiten quiver as the vertices of a diamond. By an argument similar to the one used to prove Theorem 3.4 we see that the two composites of irreducible morphisms from Ω^{2n}(k) to k, obtained by going round the two sides of the diamond, are equal of opposite sign. This shows that the composite of the two middle morphisms in the sequence is zero. We will show that U_{2n,0} ⊕ U_{0,2n} → k is surjective and that the left side of the sequence is the kernel of this surjection. In what follows, we write Rad_{kG}^{n} for the n^{th} radical of Hom_{kG}.

We use functorial methods to establish this. We claim that for any module M in a stable component of the Auslander-Reiten quiver of kG-modules of type A_{∞}^{∞}, the composition factors of Hom_{kG}(−, M)/ Rad_{kG}^{∞}(−, M) are the s^{V} for which there is a path of irreducible morphisms from V to M. More specifically, the composition factors of Rad_{kG}^{n}(−, M)/ Rad_{kG}^{n+1}(−, M) are the s^{V} for which there is a path of n irreducible morphisms V to M, each s^{V} taken with multiplicity 1. This may be proved by considering the projective resolutions of simple functors, such as

0 → Hom_{kG}(−, τM) → Hom_{kG}(−, L_{1}) ⊕ Hom_{kG}(−, L_{2})
→ Hom_{kG}(−, M) → s^{M} → 0

where 0 → τM → L_{1} ⊕ L_{2} → M → 0 is an almost split sequence. For each n ≥ 2 this restricts to an exact sequence

0 → Rad_{kG}^{n−2}(−, τM) → Rad_{kG}^{n−1}(−, L_{1}) ⊕ Rad_{kG}^{n−1}(−, L_{2})
→ Rad_{kG}^{n}(−, M) → 0

since the morphisms are obtained by composition with an irreducible morphism. Hence we obtain for each n ≥ 1 an exact sequence

0 → Rad_{kG}^{n−2}(−, τM)/ Rad_{kG}^{n−1}(−, τM)
→ Rad_{kG}^{n−1}(−, L_{1})/ Rad_{kG}^{n}(−, L_{1}) ⊕ Rad_{kG}^{n−1}(−, L_{2})/ Rad_{kG}^{n}(−, L_{2})
→ Rad_{kG}^{n−1}(−, M)/ Rad_{kG}^{n}(−, M) → 0

where we take Rad^{−1} = Rad^{0}. We also know that the composition factors of

Rad_{kG}^{n−2}(−, τM)/ Rad_{kG}^{n−1}(−, τM)

are the composition factors of

Rad_{kG}^{n−2}(−, M)/ Rad_{kG}^{n−1}(−, M)

with τ applied and that each indecomposable representable functor has a simple top. This provides a system of equations which allows us to compute the composition factors by recurrence: in a Grothendieck group,

Rad_{kG}^{0}(−, M)/ Rad_{kG}^{1}(−, M) =s^{M}
Rad_{kG}^{1}(−, M)/ Rad_{kG}^{2}(−, M) = Rad_{kG}^{0}(−, L_{1})/ Rad_{kG}^{1}(−, L_{1})
+ Rad_{kG}^{0}(−, L_{2})/ Rad_{kG}^{1}(−, L_{2})
=s^{L_{1}} + s^{L_{2}}
Rad_{kG}^{2}(−, M)/ Rad_{kG}^{3}(−, M) = Rad_{kG}^{1}(−, L_{1})/ Rad_{kG}^{2}(−, L_{1})
+ Rad_{kG}^{1}(−, L_{2})/ Rad_{kG}^{2}(−, L_{2})
− Rad_{kG}^{0}(−, τM)/ Rad_{kG}^{1}(−, τM)
=s^{L_{11}} + s^{τM} + s^{L_{22}} + s^{τM} − s^{τM}
=s^{L_{11}} + s^{τM} + s^{L_{22}}

where 0 → τL_{i} → L_{ii} ⊕ τM → L_{i} → 0, i = 1, 2 are almost split sequences; and so on. We conclude that each irreducible morphism, such as L_{1} → M, induces a monomorphism

Hom_{kG}(−, L_{1})/ Rad_{kG}^{∞}(−, L_{1}) → Hom_{kG}(−, M)/ Rad_{kG}^{∞}(−, M).

Hence, so does every composite of irreducible morphisms induce such a monomorphism. By counting composition factors we see that

0 → Hom_{kG}(−, Ω^{2n}(k))/ Rad_{kG}^{∞}(−, Ω^{2n}(k))
→ Hom_{kG}(−, U_{2n,0} ⊕ U_{0,2n})/ Rad_{kG}^{∞}(−, U_{2n,0} ⊕ U_{0,2n})
→ Hom_{kG}(−, k)/ Rad_{kG}^{∞}(−, k)

is exact (and the last cokernel has composition factors inside the diamond we are considering in the quiver).

We may now deduce that the morphism U_{2n,0} ⊕ U_{0,2n} → k is surjective, because it induces a non-zero map of representable functors and hence must be non-zero, to a module of dimension 1. Let K be the kernel of this morphism. Thus 0 → K → U_{2n,0} ⊕ U_{0,2n} → k → 0 is exact and our task is to show that K is Ω^{2n}(k). Then

0 → Hom_{kG}(−, K) → Hom_{kG}(−, U_{2n,0} ⊕ U_{0,2n}) → Hom_{kG}(−, k)

is exact (by left exactness of Hom), hence so is the similar sequence we get after factoring out Rad^{∞} from each term. Since the composite

Ω^{2n}(k) → U_{2n,0} ⊕ U_{0,2n} → k

is zero we get a morphism Ω^{2n}(k) → K (by the universal property of the kernel). This passes to a map

Hom_{kG}(−, Ω^{2n}(k))/ Rad_{kG}^{∞}(−, Ω^{2n}(k)) → Hom_{kG}(−, K)/ Rad_{kG}^{∞}(−, K)

which is an isomorphism since both terms act as the kernel in the sequences of Rad^{∞} quotients. It follows from this that the irreducible morphisms to Ω^{2n}(k) and to (the summands of) K are the same, so that the summands of Ω^{2n}(k) and of K occupy the same positions in the Auslander-Reiten quiver. Thus K is indecomposable, and the map Ω^{2n}(k) → K is an isomorphism. We deduce that the sequence

0 → Ω^{2n}(k) → U_{2n,0} ⊕ U_{0,2n} → k → 0

is exact. □

We notice that the sequence E_{n} represents an element

μ_{n} ∈ Ext_{kG}(k, Ω^{2n}(k)) ≅ \underline{Hom}_{kG}(k, Ω^{2n−1}(k)) ≅ Ĥ^{1−2n}(G, k).

It is not necessary for our development, but perhaps interesting to note that, considered as an element in Tate cohomology Ĥ^{1−2n}(G, k), μ_{n} is perpendicular (under Tate duality) to the subspace of Ĥ^{2n−2}(G, k) spanned by the transfers from the proper 2-subgroups of the Sylow subgroup of G.

The important thing is that multiplication by μ_{n} induces a natural transformation from the identity functor to Ω^{2n−1} in the stable category stmod(kG). That is, we first chose a cocycle μ_{n} : k → Ω^{2n−1}(k) representing μ_{n}. The class of the cocycle as a map in the stable category is unique. Then for any M we have a composition map μ_{n,M} given by

[figure: M; ≅; k ⊗ M; μ_{n}⊗1; Ω^{2n−1}(k) ⊗ M; Ω^{2n−1}(M)]

where the first map sends m to 1 ⊗ m, and the last is the isomorphism in the stable category. This is well defined in the stable category and does not depend on the choice of a cocycle representing μ_{n} or the choice of a splitting Ω^{2n−1}(k) ⊗ M ≅ Ω^{2n−1}(M) ⊕ P for some projective module P. Thus, we see that μ_{n,−} is an element of the graded center of the stable category of kG-modules.

Next we note the following relevant fact.

Proposition 4.2. Suppose that ϕ : M → N is a homomorphism of indecomposable kG-modules such that M and N do not lie in the same component of the Auslander-Reiten quiver. Then μ_{n,N}ϕ = 0, and Ω^{2n}(ϕ)μ_{n,M} = 0 in the stable category stmod(kG).

Proof. There is an isomorphism \underline{Hom}_{kG}(M, N) ≅ \underline{Hom}_{kG}(M ⊗ N^{∗}, k) that is natural in both M and N. Hence, letting X = M ⊗ N^{∗}, and θ : X → K be the homomorphism corresponding to ϕ, it is only necessary to show that μ_{n}θ = 0 in \underline{Hom}_{kG}(X, Ω^{2n−1}(k)) ≅ Ext_{kG}(X, Ω^{2n}(k)). That is, we need to prove that the map θ factors through the middle term of the sequence in the diagram:

\xymatrix{ &  &  &  & X \ar[d]^{θ} \ar[dl]^{σ} \\ E_{n} : & 0 \ar[r] & Ω^{2}n(k) \ar[r] & U_{2n,0} ⊕ U_{0,2n} \ar[r] & k \ar[r] & 0.}

In other words, it must be shown that there exist two maps σ_{1} : X → U_{2n,0} and σ_{2} : X → U_{0,2n} such that

θ = σ_{1}γ′_{(2,0)} … γ′_{(2n−2,0)}γ′_{(2n,0)} + σ_{2}γ_{(0,2)} … γ_{(0,2n−2)}γ_{(0,2n)}

Observe that by our hypotheses, no indecomposable direct summand Y of X is in the Auslander-Reiten component of the trivial module k. If it were otherwise, then Y would have odd dimension, implying that M and N would also have odd dimension [5]. Moreover, Y would be an endotrivial module, requiring that Y ⊗ N have only a single non-projective summand. However, k is a direct summand of N^{∗} ⊗ N, and hence M must be the unique non-projective direct summand of Y ⊗ N. Recall from [2] that the Auslander-Reiten component of N consist of the non-projective summands of Y ⊗N for Y in the Auslander-Reiten component of k. Thus we would have that M is in the same Auslander-Reiten component as N, contradicting our hypotheses.

Because the row in the diagram

\xymatrix{ &  &  & X \ar[d]^{θ} \ar[dl] \\ 0 \ar[r] & Ω^{2}(k) \ar[r] & U_{2,0} ⊕ U_{0,2} \ar[r] & k \ar[r] & 0.}

is an almost split sequence, there are maps μ_{1} : X → U_{2,0} and μ_{2} : X → U_{0,2} such that θ = γ_{(2,0)}μ_{1} + γ_{(0,2)}μ_{2}. We can iterate this process. That is, in the next iteration, we write μ_{1} = γ_{(4,0)}ν_{1} + γ_{(2,2)}ν_{2} for ν_{1} : X → U_{4,0} and ν_{2} : X → U_{2,2} using the fact that 0 → U_{4,2} → U_{4,0} ⊕ U_{2,2} → U_{2,0} → 0 is an almost split sequence.

In this way, for some m > n, we write θ as a sum of maps of the form ζσ where σ : X → U_{2m−2j,2j} and ζ : U_{2m−2j,2j} → k is a composition of irreducible maps and 0 ≤ j ≤ m. Next we note that γ′_{(2i,2j−2)}γ_{(2i,2j)} = γ_{(2i−2,2j)}γ′_{(2i,2j)}. Thus, since m > n, the map ζ factors either through U_{2n,0} or through U_{0,2n}. It follows that θ factors through U_{2n,0} ⊕ U_{0,2n} → k as asserted. This proves half of the proposition. The proof of the other half is dual to this one. □

Armed with this proposition, we can prove the main theorem for this example.

Theorem 4.3. Suppose that G is a finite group with a dihedral Sylow 2subgroup of order at least 8, and that k is a field of characteristic 2. Suppose that D is a component of the Auslander-Reiten quiver of kG that contains a module of odd dimension. Then for any n > 0, there exists a natural transformation ψ : Id → Ω^{2n−1} in the stable category stmod(kG) with the property that ψ is supported only on the set of modules in D.

Proof. Let M be a module in D having odd dimension. Recall that the collection of indecomposable modules in D coincides with the collection of non-projective direct summands of modules of the form M ⊗ U_{2i,2j} for i and j in ℤ [2]. Hence, every indecomposable module in D has odd dimension.

Now define the natural transformation ψ by the following rule. For M an indecomposable kG-module, let

ψ_{M} = \begin{cases}μ_{n,M} & if M is in D, \\ 0 & otherwise. \end{cases}

To prove the theorem we must show that, given a homomorphism φ : M → N, for M and N indecomposable modules, the diagram

[figure: M; φ; ψ_{M}; N; ψ_{N}; Ω^{2n−1}(M); Ω^{2n−1}(φ); Ω^{2n−1}(N)]

commutes. This is clear from the definitions if either both M and N are in D or both are not in D. If one of M and N is in D and the other is not, then we need only appeal to Proposition 4.2. □

We now show that the natural transformation just constructed satisfies the conditions of Theorem 3.4, thereby showing that the circumstances of this theorem can actually arise in a non-trivial way.

Proposition 4.4. The natural transformation ψ : Id → Ω^{2n−1} just constructed satisfies the conditions of Theorem 3.4. Moreover, if f : V → M is a map of indecomposable modules such that ψ_{M}f ≠ 0, then f factors as a sum of composites of finitely many irreducible maps.

Proof. We know when C = stmod(kG) that τ = Ω^{2}, Σ = Ω^{−1} and S = Ω. Thus stmod(kG) is a (−1)-Calabi-Yau category. The fact that for each indecomposable U, the only objects of the form Σ^{t}U in the component of U lie in the same τ-orbit as U, as well as the fact that each mesh has at most 2 middle terms, follow from [7] and [15].

We show that for all indecomposable modules M, the natural transformation Hom_{C}(−, ψ_{M}) has finite support. When M is not in D this is clear, so we suppose M lies in D. The construction of ψ_{M} = μ_{n,M} : M → Ω^{2n−1}(M) is that it is the third homomorphism in a triangle in stmod(kG) of the form

Ω^{2n}(M) → (U_{2n,0} ⊕ U_{0,2n}) ⊗ M → M → Ω^{2n−1}(M)

corresponding to a short exact sequence of kG-modules

0 → Ω^{2n}(M) → (U_{2n,0} ⊕ U_{0,2n}) ⊗ M → M → 0.

The argument of Lemma 4.1 showed that the sequence of functors

0 → Hom_{kG}(−, Ω^{2n}(M))
→ Hom_{kG}(−, (U_{2n,0} ⊕ U_{0,2n}) ⊗ M) → Hom_{kG}(−, M)

is exact, and the final cokernel has composition factors lying in the diamond of the Auslander-Reiten quiver determined by the modules M, Ω^{2n}(M), U_{n,0}⊗ M and U_{0,n} ⊗ M, including the right-hand edge of this diamond, but not the left-hand edge or the modules U_{n,0} ⊗ M and U_{0,n} ⊗ M. This cokernel is the image of Hom_{stmod(kG)}(−, ψ_{M}), so that condition (1) of Corollary 2.11 is satisfied. This shows that Hom_{C}(−, ψ_{M}) has finite support.

The final statement follows from part (5) of Corollary 2.11 and condition (2) of Proposition 2.9. If ψ_{M}f ≠ 0 then ψ_{M}fξ is almost vanishing for some ξ. Thus fξ is a sum of composites of irreducible morphisms and hence so is f. □

References

[1] I. Assem, D. Simson and A. Skowronski, Elements of the representation theory of associative algebras Vol. 1: Techniques of representation theory, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006.

[2] M. Auslander and J. F. Carlson, Almost split sequences and group algebras, J. Algebra, 103(1986), 122-140.

[3] M. Auslander and I. Reiten, Stable equivalence of dualizing R-Varieties, Adv. Math 12 (1974), 306-366.

[4] M. Auslander and I. Reiten, Uniserial functors, Representa tion theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 197 9), pp. 1-47, Lecture Notes in Math. 832, Springer, Berlin, 1980.

[5] D. J. Benson and J. F. Carlson, Nilpotent elements in the Green ring, J. Algebra, 104(1986), 329-350.

[6] K. Bongartz and P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), 331378.

[7] M.C.R. Butler and M. Shahzamanian, The construction of almost split sequences, III: modules over two classes of tame local algebras, Math. Annalen 247 (1980), 111- 122.

[8] J. F. Carlson and J. Thévenaz, Torsion endotrivial modules, Algebras and Representation Theory, 3 (2000), 303-335.

[9] J.F. Carlson, S.K. Chebolu and J. Mináč, Freyd’s generating hypothesis with almost split sequences, Proc. Amer. Math. Soc. 137 (2009), 2575-2580.

[10] K. Erdmann, On Auslander-Reiten components for group algebras, J. Pure Appl. Algebra, 104(1995), 149-160.

[11] D. Happel, Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Math. Soc. Lecture Note Ser. 119, Cambridge University Press, Cambridge, 1988.

[12] M. Linckelmann, On graded centres and block cohomology, Proc. Edinb. Math. Soc. 52 (2009), 489-514.

[13] M. Linckelmann and R. Stancu, On the graded center of the stable category of a finite p-group, J. Pure Appl. Algebra 214 (2010), 950-959.

[14] I. Reiten and M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295-366.

[15] P.J. Webb, The Auslander-Reiten quiver of a finite group, Math. Z. 179 (1982), 97-121.

E-mail address: jfc@math.uga.edu

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

E-mail address: webb@math.umn.edu

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA