ON THE CENTER OF A TRIANGULATED CATEGORY

HENNING KRAUSE AND YU YE

Abstract. We discuss some basic properties of the graded center of a triangulated category and compute examples arising in representation theory of finite dimensional algebras.

1. Introduction

The graded center of a triangulated category T with suspension functor Σ is a ℤgraded ring. The degree n component consists of all natural transformations from the identity functor Id to Σ^{n} which commute modulo the sign (−1)^{n} with Σ. The graded center is the universal graded commutative ring that acts on T . For instance, the Hochschild cohomology HH^{∗}(A) of an algebra A acts on the derived category D(A) via a morphism HH^{∗}(A) → Z^{∗}(D(A)) into the graded center.

It seems that the first systematic use of the graded center appears in work of Buchweitz and Flenner on the Hochschild cohomolgy of singular spaces [BF]. For related work of Lowen and van den Bergh in the setting of differential graded categories we refer to [LV]. Blocks of finite groups and their modular representation theory provide the context for recent work of Linckelmann on the graded center of stable and derived categories [Li]. Closely related is the study of cohomological support varieties which depends on the appropriate choice of a graded commutative ring acting on a triangulated category; see [BIK].

In this article, we prove some structural results and provide complete descriptions of the graded center for some small examples. The article is organized as follows.

In §2, it is shown that for any abelian category A with enough projective objects, there is an isomorphism of graded commutative rings Z^{∗}(D^{b}(A)) ≅ Z^{∗}(D^{b}(Proj(A))). Here Proj(A) denotes the full subcategory of A consisting of all projective objects and the isomorphism is given by restriction.

In §3-4, we deal with derived categories of hereditary categories. Note that for a hereditary category, the derived category and the bounded derived category have the same graded center. In §3, the category mod(R) of finitely generated modules over a Dedekind domain R is considered. We calculate Z^{∗}(D(mod(R))) explicitly. As we show it relates closely to the residue fields of all the maximal ideals of R.

In §4, we consider the module category of a tame hereditary algebra and the category of coherent sheaves on a weighted projective line of non-negative Euler characteristic. We compute the graded centers of their bounded derived categories. Note that our methods do not apply to wild cases. For a weighted projective line of wild type, we only get a subalgebra of the graded center.

The second named author gratefully acknowledges support by the Alexander von Humboldt Stiftung.

He is supported in part by the National Natural Science Foundation of China (Grant No. 10501041).

In §§5-6, we describe the graded centers of D^{b}(mod(k[x]/x^{2})) and \underline{mod}(k[x]/x^{n}) for n ≥ 2 respectively.

2. Morphisms between graded centers

Definition 2.1. Let T be a triangulated category and Σ the suspension functor of T . We define a ℤ-graded abelian group Z^{∗}(T ) = Z^{∗}(T , Σ) as follows. For any n ∈ ℤ, let Z^{n}(T ) denote the collection of all natural transformations η: Id → Σ^{n} which satisfy ηΣ = (−1)^{n}Ση. The composition of natural transformations gives Z^{∗}(T ) the structure of a graded commutative ring, and we call it the graded center of T . Graded commutative here means that ηζ = (−1)^{mn}ζη for all η ∈ Z^{n}(T ) and ζ ∈ Z^{m}(T ).

Remark 2.2. (1) The definition of the graded center Z^{∗}(T ) makes sense for any graded category, that is, an additive category equipped with an autoequivalence. In particular, the choice of the exact triangles of T is not relevant for Z^{∗}(T ).

(2) The degree zero part Z^{0}(T ) is a subring of the usual center Z(T ) of T , which by definition consists of all natural transformations from the identity functor to itself. Note that Z^{0}(T ) = Z(T ) if Σ = Id.

(3) The graded center Z^{∗}(T ) need not be a set in general. However, it will be a set when the category T is small.

(4) For any object M in T we define the graded ring Ext_{T}^{∗} (M, M) by setting

Ext_{T}^{n} (M, M) = Hom_{T} (M, Σ^{n}M)

for any integer n. By definition there is a canonical graded ring morphism

Z^{∗}(T ) ⟶ Ext_{T}^{∗} (M, M)

mapping a natural transformation η: Id → Σ^{n} to the morphism η_{M}: M → Σ^{n}M.

Following Rouquier [Ro] we set ⟨M⟩_{1} to be the full additive subcategory of T which contains M and is closed under finite direct sums, summands and the action of Σ, and for i ≥ 2 we define inductively ⟨M⟩_{i} as the full additive subcategory of T consisting of all objects isomorphic to direct summands of objects Z for which there exists an exact triangle X → Y → Z → ΣX with X ∈ ⟨M⟩_{1} and Y ∈ ⟨M⟩_{n−1}.

Now suppose that M is an object of T with T = ⟨M⟩_{d+1} for some positive integer d. We set N to be the kernel of the canonical morphism Z^{∗}(T ) → Ext_{T}^{∗} (M, M). It can be shown in this case that N is a nilpotent ideal satisfying N^{2^{d}} = 0; see [Li] for a proof. In particular, Z^{∗}(T ) is modulo nilpotent elements a set.

Let F: S → T be an exact functor between triangulated categories. An obvious question to ask is when the functor F induces morphisms between Z^{∗}(S) and Z^{∗}(T ). Recently, Linckelmann gave an affirmative answer for this question in the case that there exists a functor G: T → S which is simultaneously left and right adjoint to F and satisfies some further compatibility conditions [Li]. The answer for general F seems to be not known. The following proposition shows that in some very specific situation, for instance when F is fully faithful, we do obtain some morphisms between the graded centers.

Proposition 2.3. Let T be a triangulated category and S a full triangulated subcategory. (1) The inclusion functor i: S → T induces a morphism of graded rings

i^{∗}: Z^{∗}(T ) ⟶ Z^{∗}(S),

where i^{∗}(η)_{X} = η_{X} for any η ∈ Z^{∗}(T ) and X ∈ S.

(2) The canonical functor π: T → T /S induces a morphism of graded rings

π_{∗}: Z^{∗}(T ) ⟶ Z^{∗}(T /S),

where π_{∗}(η)_{X} = π(η_{X}) for any η ∈ Z^{∗}(T ) and X ∈ T /S.

Proof. The proof is routine. To check that π_{∗} is well defined, one uses the fact that for any commutative diagram in T

[figure: X; s; α; Z; f; β; Y; γ; X′; s′; Z′; f′; Y′]

with Cone(s) and Cone(s′) in S, we have γ ∘ (f/s) = (f′/s′) ∘ α in the quotient category T /S, where we use −−→ to denote the morphisms whose cones are in S. □

Up to now, little seems to be known about the properties of the above morphisms. For example, the question when i^{∗} and π_{∗} are surjective or injective is of special interest to us. Also, one might study the induced morphism of graded rings

(i^{∗}, π_{∗}): Z^{∗}(T ) ⟶ Z^{∗}(S) × Z^{∗}(T /S).

Example 2.4. Let S ⨿ T denote the direct product of two triangulated categories S and T . We view S as a thick subcategory of S ⨿ T and the corresponding quotient is equivalent to T . Then we have Z^{∗}(T ⨿ S) ≅ Z^{∗}(T ) × Z^{∗}(S) via the morphism (i^{∗}, π_{∗}).

For the rest of this section we focus on homotopy categories and derived categories. Firstly we introduce some basic notations and conventions. Let A be any additive category. We denote by C(A) the category of chain complexes in A. Recall that a chain complex in A is a sequence of morphisms in A

X = ( \xymatrix{⋯ \ar[r] & X_{n} \ar[r]^{d_{n}^{X}} & X_{n−1} \ar[r] & ⋯} )

with d_{n}^{X}d_{n+1}^{X} = 0 for all n ∈ ℤ. A morphism of complexes is a chain map f: X → Y consisting of a family of morphisms f_{n}: X_{n} → Y_{n} in A with n ∈ ℤ such that f_{n} ∘ d_{n+1}^{X} = d_{n+1}^{Y} ∘ f_{n+1} for all n, that is, the diagram

\xymatrix{⋯ \ar[r] & X_{n+1} \ar[r]^{d_{n+1}^{X}} \ar[d]_{f_{n+1}} & X_{n} \ar[r]^{d_{n}^{X}} \ar[d]_{f_{n}} & X_{n−1} \ar[r] \ar[d]_{f_{n−1}} & ⋯ \\ ⋯ \ar[r] & Y_{n+1} \ar[r]^{d_{n+1}^{Y}} & Y_{n} \ar[r]^{d_{n}^{Y}} & Y_{n−1} \ar[r] & ⋯}

commutes. We denote by C^{+}(A) the full subcategory of C(A) which consists of all bounded below complexes, that is, the complexes X with X_{n} = 0 for n ≪ 0. Similarly, we denote by C^{−}(A) and C^{b}(A) the full subcategory of bounded above complexes and complexes bounded in both directions, respectively.

If moreover A is abelian, then for any integer n the n-th homology group H_{n}(X) is by defintion Ker(d_{n}^{X})/ Im(d_{n+1}^{X}), and any morphism f of complexes induces morphisms of homology groups H_{n}(f): H_{n}(X) → H_{n}(Y) for all n ∈ ℤ.

The homotopy category K(A) has the same objects as C(A). The morphisms are the equivalence classes of the morphisms in C(A) modulo the null-homotopic morphisms, that is, those with components of the form

d_{n+1}^{Y} ∘ h_{n} + h_{n−1} ∘ d_{n}^{X}

for some family of morphisms h_{n}: X_{n} → Y_{n+1} in A, n ∈ ℤ.

The suspension functor (or shift functor) Σ of C(A) is defined by (ΣX)_{n} = X_{n−1}, d_{n}^{ΣX} = −d_{n−1}^{X} on the objects and by (Σf)_{n} = f_{n−1} on any morphism f. Clearly Σ is not only an autoequivalence but also an automorphism of C(A). Moreover, Σ also induces an automorphism of K(A) and K(A) admits a triangulated structure with suspension functor Σ.

Let D(A) denote the derived category of A, i.e., the localization of K(A) with respect to the quasi-isomorphisms. Note that D(A) is again a triangulated category with suspension functor Σ. One defines K^{∗}(A) and D^{∗}(A) with ∗ ∈ {+, b, −} in a similar way.

Now let A be an abelian category with enough projective objects, and P be the full subcategory consisting of all projective objects. We denote by K^{+,b}(P) the thick subcategory of K^{+}(P) which consists of bounded below complexes X with H_{n}(X) = 0 for almost all n. Clearly, we have K^{b}(P) ⊆ K^{+,b}(P). In some cases, objects in K^{b}(P) are also called perfect complexes.

It is known that the following composition of functors

\xymatrix{K^{+,b}(P) \ar@{^{(}->}[d] \ar@{-->}[rr]^{≅} &  & D^{b}(A) \ar@{^{(}->}[d] \\ K^{+}(P) \ar[r] \ar[rr]^{≅} & K^{+}(A) \ar[r] & D^{+}(A)}

induces equivalences K^{+}(P) →^{∼} D^{+}(A) and K^{+,b}(P) →^{∼} D^{b}(A) of triangulated categories. The quotient category D_{sg}(A) = D^{b}(A)/K^{b}(P) is called the triangulated category of singularities of A, because it is an invariant of the singularities provided that A is the category of sheaves on some variety. We know that D_{sg}(A) = 0 if and only if all objects of A have finite homological dimension. When A = Mod(A) for some self-injective ring A, then D_{sg}(A) is equavalent to the stable module category \underline{Mod}(A) of A.

We are now in a position to state our main result of this section.

Theorem 2.5. Let A be an abelian category with enough projective objects and P the full subcategory consisting of all projective objects. Then the embedding K^{b}(P) → K^{+,b}(P) induces an isomorphism Z^{∗}(K^{b}(P)) →^{∼} Z^{∗}(D^{b}(A)) of graded commutative rings.

To prove the theorem, we need some preparations.

For each n ∈ ℤ the n-th truncation functor ι^{n}: C(A) → C(A) is defined for a complex X by (ι^{n}X)_{i} = X_{i} for i ≤ n and 0 for i > n, and d_{i}^{ι^{n}X} = d_{i}^{X} for i ≤ n and 0 for i > n. Clearly, ι^{n} sends C(A) to C^{−}(A) and C^{+}(A) to C^{b}(A). Note that we have a natural morphism i^{n}: ι^{n}X → X, and sometimes we use i_{X}^{n} to emphasize X. We have (i^{n})_{s} = id for s ≤ n and 0 for s > n. The following lemma is crucial in the proof of the main theorem.

Lemma 2.6. Let X ∈ C(P), Y ∈ C(A) and f: X → Y be a chain map with H_{n}(Y) = 0 for n > 0. Then f is null-homotopic if and only if the composition f ∘i^{n}: ι^{n}X → X → Y is null-homotopic for some n ≥ 0.

Proof. One direction is clear since the null-homotopic morphisms form an ideal. Conversely, suppose that f ∘ i^{n} is null-homotopic for some n ≥ 0. To show that f is also null-homotopic, it suffices to find a family {h_{n}: X_{n} → Y_{n+1} | n ∈ ℤ}, such that f_{n} = d_{n+1}^{Y}h_{n} + h_{n−1}d_{n}^{X} holds for all n. By applying the shift functor, one can assume without loss of generality that f ∘ i^{0} is null-homotopic. Thus there exists a family of morphisms in A, say {h_{n}: X_{n} → Y_{n+1}, n ≤ 0}, such that f_{n} = d_{n+1}^{Y}h_{n} + h_{n−1}d_{n}^{X} for all n ≤ 0.

Since f is a chain map, we have d_{1}^{Y} f_{1} = f_{0}d_{1}^{X} = d_{1}^{Y} h_{0}d_{1}^{X} + h_{−1}d_{0}^{X}d_{1}^{X}, and hence d_{1}^{Y} (f_{1} − h_{0}d_{1}^{X}) = 0, which implies that Im(f_{1} − h_{0}d_{1}^{X}) ⊆ Ker(d_{1}^{Y}) = Im(d_{2}^{Y}), the last equality holds because H_{1}(Y) = 0. Now X_{1} is projective implies that f_{1} − h_{0}d_{1}^{X} factors through d_{2}^{Y}, i.e., there exists h_{1} : X_{1} → Y_{2} such that f_{1} = d_{2}^{Y} h_{1} + h_{0}d_{1}^{X}, thus we get the required h_{1}. Now repeat the argument and the lemma follows. □

Proposition 2.7. Let t ∈ ℤ and η: Id → Σ^{t} be a natural transformation for the category K^{b}(P). Then η extends uniquely to a natural transformation η̃: Id → Σ^{t} for the category K^{+,b}(P).

Proof. First we will construct a morphism η̃_{X}: X → Σ^{t}X for any X ∈ K^{+,b}(P). The idea is to use certain approximations.

Since η is a natural transformation for K^{b}(P), we have for each n a morphism ζ̄^{n} = η_{ι^{n}X}: ι^{n}X → Σ^{t}ι^{n}X. Now we fix a chain map ζ^{0} : ι^{0}X → Σ^{t}ι^{0}X which is a representative of ζ̄^{0}. We can construct inductively the representatives ζ^{n} of ζ̄^{n} for all n ≥ 0, such that ζ_{i}^{n+1} = ζ_{i}^{n} for all n ≥ 0 and i ≤ n.

In fact, suppose that ζ^{n} has been constructed, and let ξ be any representative of ζ̄^{n+1}X. Consider the morphism j: ι^{n}X → ι^{n+1}X which is given by j_{m} = id_{X_{m}} for all m ≤ n and 0 otherwise. Since η is a natural transformation, the diagram

ι^{n}X ⟶^{j} ι^{n+1}X
ζ^{n}↓  ^{ξ}↓
Σ^{t}ι^{n}X ⟶^{Σ^{t}j} Σ^{t}ι^{n+1}X

commutes in the category K^{b}(P), i.e., δ := ξ ∘ j − Σ^{t}j ∘ ζ^{n} is null-homotopic. Explicitly, δ_{i} = ξ_{i} − η_{i} for i ≤ n and δ_{i} = 0 for i ≥ n + 1.

Now there exists a family of morphisms {h_{i}: (ι^{n}X)_{i} → (Σ^{t}ι^{n+1}X)_{i+1} | i ∈ ℤ} with h_{i} = 0 for i > n, such that δ_{i} = d_{i+1}^{Σ^{t}ι^{n+1}X} ∘ h_{i} + h_{i−1} ∘ d_{i}^{ι^{n}X}. The family {h_{i}} can be viewed as a family of morphisms {h_{i}: (ι^{n+1}X)_{i} → (Σ^{t}ι^{n+1}X)_{i+1} | i ∈ ℤ}, thus it gives a null-homotopic morphism δ′: ι^{n+1}X → Σ^{t}ι^{n+1}X, which satisfies δ′_{i} = δ_{i} for all i ≤ n. We are done by setting ζ^{n+1} = ξ − δ′.

Now we define η̃_{X} by (η̃_{X})_{n} = ζ_{n}^{0} for n ≤ 0 and (η̃_{X})_{n} = ζ_{n}^{n} for n > 0. We claim that η̃ is a natural transformation from Id to Σ^{t} for the category K^{+,b}(P).

Note that by construction, η̃_{X} satisfies the following condition: for any n ≥ 0, there exists a representative ζ_{X}^{n} for η_{ι^{n}X}, which is given by (ζ_{X}^{n})_{i} = (η̃_{X})_{i} for i ≤ n, and (ζ_{X}^{n})_{i} = 0 for i ≥ n + 1. In other words, η̃_{X} ∘ i_{X}^{n} = Σ^{t}i_{X}^{n} ∘ η_{ι^{n}X} as chain maps for all n ≥ 0, where i_{X}^{n} denote the natural morphism from ι^{n}X to X as before.

Now let X, Y ∈ K^{+,b}(P) and f: X → Y be any chain map. Assume that η̃_{X} : X → Σ^{t}X and η̃_{Y} : Y → Σ^{t}Y are arbitrary chain maps with the property η̃_{X} ∘i_{X}^{n} = Σ^{t}i_{X}^{n} ∘η_{ι^{n}X} and η̃_{Y} ∘ i_{Y}^{n} = Σ^{t}i_{Y}^{n} ∘ η_{ι^{n}Y} for n ≥ 0. We will show that η̃_{Y} ∘ f = Σ^{t}f ∘ η̃_{X} in K^{+,b}(P). Note that in the cube below, the other five faces are commutative by the construction of η̃_{X}, η̃_{Y}, ι_{X}^{n} and ι_{Y}^{n}.

[figure: X; η̄_{X}; f; Σ^{t}X; Σ^{t}f; ι^{n}X; i_{X}^{n}; η_{ι^{n}X}; Σ^{t}ι^{n}X; Y; η̄_{Y}; Σ^{t}Y; ι^{n}Y; η_{ι^{n}Y}; i_{Y}^{n}; Σ^{t}ι^{n}Y]

By Lemma 2.6 it suffices to show that η̃_{Y} ∘f ∘i_{X}^{n} = Σ^{t}f ∘η̃_{X} ∘i_{X}^{n} for some sufficiently large n. This is equivalent to showing that η̃_{Y} ∘i_{Y}^{n} ∘ι^{n}f = Σ^{t}f ∘Σ^{t}i_{X}^{n} ∘η_{ι^{n}X}. The left hand side is Σ^{t}i_{Y}^{n} ∘ η_{ι^{n}Y} ∘ ι^{n}f, and since η_{ι^{n}Y} ∘ ι^{n}f = Σ^{t}ι^{n}f ∘ η_{ι^{n}X} and Σ^{t}f ∘ Σ^{t}i_{X}^{n} = Σ^{t}i_{Y}^{n} ∘ Σ^{t}ι^{n}f, the equality holds.

Thus by fixing such η̃_{X} for each X, we can extend η to the category K^{+,b}(P). For the uniqueness, we need only to take f = id_{X} in the above argument. This completes the proof. □

Corollary 2.8. Let A and P be as before. Then Z^{t}(K^{b}(P)) = 0 for all t < 0, and therefore Z^{∗}(K^{b}(P)) and Z^{∗}(D^{b}(A)) are positively graded.

Proof. Suppose that η is a natural transformation from Id_{K^{b}(P)} to Σ_{K^{b}(P)}^{t} for some t < 0. We prove that η_{X} = 0 by using induction on the length of the support of X, where the support of X means the interval [i, j], such that i and j are respectively the minimal and maximal integer m with X_{m} ≠ 0. Without loss of generality, we may assume that i = 0 and we use induction on j.

In the case j = 0, clearly Hom_{K^{b}(P)}(X, Σ^{t}X) = 0 for t < 0. Suppose η_{X} = 0 for all j ≤ m and suppose X = (⋯ → 0 → X_{m+1} → ⋯ → X_{0} → 0 → ⋯). By the same argument as in the proof of Proposition 2.7, there is a representative ζ of η_{X}, such that ζ_{i} = 0 for all i ≤ m, and now the assumption t < 0 implies that (Σ^{t}X)_{m+1} = 0, which forces that ζ_{m+1} = 0, thus ζ = 0 and hence η_{X} = 0. □

With the above preparations, we can now prove the main theorem.

Proof of Theorem 2.5. Fix η ∈ Z^{t}(K^{b}(P)). By Proposition 2.7, η extends uniquely to a natural transformation η̃: Id → Σ^{t} for the category K^{+,b}(P), and clearly i^{∗}(η̃) = η, where i^{∗} is induced by the embedding i: K^{b}(P) → K^{+,b}(P). By the same argument as in the last part in the proof of Proposition 2.7, one can show that η̃Σ = (−1)^{n}Ση̃, which implies that η̃ ∈ Z^{n}(K^{+,b}(P)). This proves the surjectivity of i^{∗}. The injectivity of i^{∗} follows from the uniqueness of the extension. □

Remark 2.9. Suppose there are enough injective objects in A and denote by I the full subcategory of injectives. Then we have D^{b}(A) ≅ K^{−,b}(I) and the dual version of the theorem says that there is an isomorphism of graded centers Z^{∗}(K^{b}(I)) ≅ Z^{∗}(D^{b}(A)).

3. Finitely generated modules over Dedekind domains

The following two sections are devoted to studying the graded center of the derived category of some hereditary categories. We look at some basic examples and use explicit calculations. First we discuss the derived category of the category of finitely generated modules mod(R) for any Dedekind domain R. We start with some preparation.

Let R be an arbitrary unitary ring and denote by Z(R) the center of R. Let z ∈ Z(R) and M ∈ mod(R). Then we have a morphism l_{z} ∈ Hom_{R}(M, M), which is given by l_{z}(m) = z · m. This is indeed a morphism of modules since z is in the center of R. Moreover, l_{z} induces a natural transformation from the identity functor to itself for mod(R) as well as for D^{b}(mod(R)).

Now let H be a hereditary abelian category, that is, Ext_{H}^{i}(M, N) = 0 for any M, N ∈ H and i ≥ 2. Consider the derived category of H and observe that any object X ∈ D(H) is isomorphic to ⨁_{i∈ℤ} Σ^{i}(H_{i}(X)). Here, Σ is the shift functor and H_{i}(X) is viewed as a stalk complex concentrated in degree zero. For a simple proof of this, see [Kr, §1]. We have the following easy lemmas.

Lemma 3.1. Left multiplication induces an injective ring homomorphism Z(R) → Z^{0}(D^{b}(mod(R))). Moreover, if R is hereditary, then this is an isomorphism.

Proof. For a proof, we just use the fact that left multiplication gives an isomorphism from Z(R) to the usual center of mod(R), i.e., the ring of natural transformations from the identity functor to itself, and that mod(R) is a full subcategory of D^{b}(mod(R)).

Moreover, if R is hereditary, then all objects of D^{b}(mod(R)) are of the form ⨁_{i∈ℤ} Σ^{i}M_{i} with M_{i} ∈ mod(R) viewed as a stalk complex concentrated in degree zero. Now the lemma follows easily. □

Remark 3.2. Note that the morphism in the lemma need not be an isomorphism; see [Kü] or §5 below.

Lemma 3.3. Let H be an arbitrary hereditary category. Then Z^{∗}(D^{b}(H)) is concentrated in degree 0 and 1. Moreover, the inclusions D^{b}(H) ⊆ D^{−}(H) ⊆ D(H) induce isomorphisms of graded centers Z^{∗}(D^{b}(H)) ≅ Z^{∗}(D^{−}(H)) ≅ Z^{∗}(D(H)).

Proof. We have Hom_{D(H)}(M, Σ^{m}M) = Ext_{H}^{m}(M, M) = 0 for all M ∈ H and m ≥ 2, since H is hereditary. Thus there is no nontrivial natural transformations from Id to Σ^{m} for the category D^{b}(H) for m ≥ 2, and the first part of the lemma follows. The last assertion follows from the fact that any element in the graded center Z^{∗}(D(H)) is uniquely determined by the restriction to the stalk complexes. The minor difference between both derived categories is that any object in D(H) is an infinite direct sum of stalk complexes while objects in D^{b}(H) can always be written as finite direct sums. Similarly, we have Z^{∗}(D^{+}(H)) ≅ Z^{∗}(D(H)). □

Due to the lemma, to study the graded center of the derived categories of hereditary abelian categories, we need only to consider the bounded ones.

Now suppose that H = H_{1} ∨ H_{2}, where H_{1} and H_{2} are full additive subcategory of H, and we use ∨ to indicate that any object of H is a direct sum of an object of H_{1} and an object of H_{2}, and Hom_{H}(M_{2}, M_{1}) = Ext_{H}^{1}(M_{1}, M_{2}) = 0 for all M_{1} ∈ H_{1} and M_{2} ∈ H_{2}. We set Σ^{∗}H_{1} to be the minimal additive subcategory of D^{b}(H) which contains H_{1} and is closed under Σ, in other words the subcategory consisting of all complexes with homologies contained in H_{1}. Note that Σ^{∗}H_{1} is not a triangulated subcategory in general. This will happen if H_{1} is a thick subcategory of H, i.e., H_{1} is closed under extensions, kernels and cokernels, and in this case, H_{1} is also a hereditary abelian category and Σ^{∗}H_{1} ≅ D^{b}(H_{1}). For a proof of this, one uses again the fact that any object in D^{b}(H) is a direct sum of stalk complexes. Since Σ is an autoequivalence of Σ^{∗}H_{1}, we can also define the graded center of Σ^{∗}H_{1} with respect to Σ, and denote it by Z^{∗}(Σ^{∗}H_{1}).

Proposition 3.4. Let H = H_{1} ∨ H_{2} be a hereditary abelian category. Then the restriction map induces an isomorphism of abelian groups

Z^{1}(D^{b}(H)) ≅ Z^{1}(Σ^{∗}H_{1}) × Z^{1}(Σ^{∗}H_{2}).

Proof. We produce an inverse map. First observe that any object in D^{b}(H) can be written uniqueley as X_{1}⊕X_{2} with X_{1} ∈ Σ^{∗}H_{1} and X_{2} ∈ Σ^{∗}H_{2}. Let η_{1} : Id_{Σ^{∗}H_{1}} → Σ_{Σ^{∗}H_{1}} and η_{2} : Id_{Σ^{∗}H_{2}} → Σ_{Σ^{∗}H_{2}} be natural transformations. Then we define η: Id_{D^{b}(H)} → Σ_{D^{b}(H)} by setting η_{X_{1}⊕X_{2}} to be the map (η_{1})_{X_{1}} ⊕(η_{2})_{X_{2}}. We will show that η is indeed a natural transformation. To this end we need to check that for any morphism f: X → Y in D^{b}(H), we have Σf ∘ η_{X} = η_{Y} ∘ f.

Since any object of D^{b}(H) can be uniquely written as ⨁_{i∈ℤ} Σ^{i}(M_{1}^{i} ⊕ M_{2}^{i}) with M_{1}^{i} ∈ H_{1} and M_{2}^{i} ∈ H_{2}, we need only to check the above compatibility for the morphisms of the form f: Σ^{i}M_{1} → Σ^{j}M_{2} and g: Σ^{i}M_{2} → Σ^{j}M_{1} with M_{1} ∈ H_{1} and M_{2} ∈ H_{2}. We claim that Σf ∘ η_{M_{1}} = η_{M_{2}} ∘ f = 0 and Σg ∘ η_{M_{2}} = η_{M_{1}} ∘ g = 0. In fact, since H is hereditary, both sides will vanish unless j = i + 1 or j = i. If j = i + 1, the equalities hold since Ext_{H}^{2}(M, N) = 0 for all M, N ∈ H. Otherwise, if j = i, we have g = 0 and Ext_{H}^{1}(M_{1}, M_{2}) = 0. Now the assertion follows easily, and this completes the proof. □

Now we can begin the study of Z^{∗}(D^{b}(mod(R))) for a Dedekind domain R. A Dedekind domain is an integral domain such that each ideal can be written as a finite product of prime ideals, or equivalently, a noetherian integrally closed domain with Krull dimension at most one. This name was given to such rings in honor of R. Dedekind, who was one of the first to study such rings in the 1870s. The rings of algebraic integers of number fields provide an important class of Dedekind domains, which play a crucial role in algebraic number theory.

The assumption on the Krull dimension implies that each nonzero prime ideal of R is maximal, and that the category H = mod(R) is hereditary and any object M of mod(R) is a direct sum of a torsion-free module and a torsion module. Any finitely generated torsion-free module is projective, and any finitely generated torsion module is a finite direct sum of cyclic modules. Moreover, by the Chinese Remainder Theorem, each cyclic module is a finite direct sum of modules of the form R/𝔭^{l} with 𝔭 a maximal ideal of R and l ≥ 1.

We have a decomposition of categories H = H_{+} ∨ H_{0}, where H_{+} denotes the full subcategory of H consisting of all projective modules and H_{0} consists of all torsion modules. Note that H_{0} is a thick subcategory and hence hereditary. Moreover, there exists an Auslander-Reiten translation τ in H_{0}, which is by definition an autoequivalence of H_{0} such that there exists a natural isomorphism D Ext_{H}^{1}(X, Y) ≅ Hom_{H}(Y, τX). This identity is usually called Serre duality and implies the existence of Auslander-Reiten sequences in H_{0}.

Let max(R) denote the set of all maximal ideals of R. Then all the indecomposables in H_{0} are given by {R/𝔭^{l} | 𝔭 ∈ max(R), l ≥ 1}. Denote by H_{𝔭} the subcategory of H_{0} consisting of all 𝔭-torsion modules, i.e., the additive category generated by {R/𝔭^{l} | l ≥ 1} which is also abelian, hereditary and a thick subcategory of H_{0}. Note that H_{𝔭} is τinvariant and H_{0} = ∐_{𝔭∈max(R)} H_{𝔭}.

One can show that H_{𝔭} is equivalent to the subcategory of mod(R_{𝔭}) which consists of torsion R_{𝔭}-modules, where R_{𝔭} is the localization of R at 𝔭; and H_{𝔭} is obviously a full subcategory of mod(R̂_{𝔭}), where R̂_{𝔭} is the completion of R with respect to 𝔭, i.e., the inverse limit \underleftarrow{lim} R/𝔭^{l}. Note that we have an isomorphism R/𝔭 ≅ R_{𝔭}/𝔭R_{𝔭} of residue fields, and we denote it by k_{𝔭}.

Fix an element x ∈ 𝔭 \ 𝔭^{2}. Since R_{𝔭} is a discrete valuation ring, the multiplication with x gives us an isomorphism R/𝔭^{l} ≅ 𝔭/𝔭^{l+1} of R-modules for any l ≥ 1. Thus we have in H_{𝔭} Auslander-Reiten sequences

σ_{𝔭}^{1}: 0 → R/𝔭 → R/𝔭^{2} → R/𝔭 → 0

and

σ_{𝔭}^{l}: 0 → R/𝔭^{l} → R/𝔭^{l−1} ⊕ R/𝔭^{l+1} → R/𝔭^{l} → 0

for all l ≥ 2 , where the morphism from R/𝔭^{l+1} to R/𝔭^{l} is the natural quotient map, and we use the isomorphism R/𝔭^{l} ≅ 𝔭/𝔭^{l+1} induced by the multiplication with x. This says that the Auslander-Reiten quiver of H_{𝔭} is a tube of τ-period 1. For the Auslander-Reiten sequences for Dedekind domains, see also [AL], Example3.1. Note that we have a natural equivalence τ ≅ Id_{H_{𝔭}}.

It is easy to show that Ω(R/𝔭^{s}) ≅ R_{𝔭} for any s ≥ 1, where Ω: mod(R_{𝔭}) → mod(R_{𝔭}) is the syzygy functor. We have a presentation of R/𝔭^{s}

0 ⟶ R_{𝔭} ⟶^{l_{x^{s}}} R_{𝔭} ⟶ R/𝔭^{s} ⟶ 0,

where l_{x^{s}} denotes the multiplication by x^{s}. Note that we have an isomorphism R/𝔭^{s} ≅ R_{𝔭}/𝔭^{s}R_{𝔭}. Now the above exact sequence induces an epimorphism

Hom_{R}(R_{𝔭}, R/𝔭^{i}) → Ext_{R}^{1}(R/𝔭^{s}, R/𝔭^{i}),

and when i ≤ s this induces an isomorphism. Now it is easy to show that the Auslander-Reiten sequence σ_{𝔭}^{s} corresponds to the composition R_{𝔭} → R/𝔭 ≅ 𝔭^{s−1}/𝔭^{s} ↪ R/𝔭^{s}. Moreover, all the Auslander-Reiten sequences with starting term R/𝔭^{s} are given by the morphisms of the form R_{𝔭} → Soc(R/𝔭^{s}) ↪ R/𝔭^{s}, which equals λσ_{𝔭}^{s} for some 0 ≠ λ ∈ k_{𝔭}.

On the other hand, we identify Hom_{D^{b}(H)}(M, ΣN) = Ext_{H}^{1}(M, N) for any abelian category H and any objects M, N ∈ H. Now we can write down the graded center of D^{b}(H_{𝔭}) explicitly by using the notion of a trivial extension.

Let A be an arbitrary ring and M an A-A-bimodule. The trivial extension ring of A by the bimodule M, denoted by T (A, M), is defined to be the ring whose additive ring is A ⊕ M with multiplication given by

(a, m) · (a′, m′) = (aa′, am′ + ma′)

for all a, a′ ∈ A and m, m′ ∈ M. Note that T (A, M) can be identified with the subring of the upper triangular ring

(\begin{matrix}A & M \\ 0 & A \end{matrix})

which consists of all the matrices with equal diagonal entries.

The trivial extension ring is a positively graded ring which is concentrated in degree 0 and 1 with T (A, M)^{0} = A and T (A, M)^{1} = M. Conversely, let A = A^{0} ⊕ A^{1} be an arbitrary positively graded ring which is concentrated in degree 0 and 1. Then A ≅ T (A^{0}, A^{1}) as graded rings, where the A^{0}-bimodule structure on A^{1} is induced by the multiplication of A. If A is commutative and M an A-module, one can also define the trivial extension T (A, M), where M is viewed as an A-A-bimodule. Note that in this case T (A, M) is always graded commutative.

Proposition 3.5. Let R be a Dedekind domain, 𝔭 a maximal ideal of R and k_{𝔭} the residue field. Then as a graded ring,

Z^{∗}(D^{b}(H_{𝔭})) ≅ T (R̂_{𝔭}, ∏_{l∈ℤ,l≥1}k_{𝔭}),

where k_{𝔭} is viewed as a simple R̂-module.

Proof. Note that elements in R̂_{𝔭} are by definition sequences q = (q_{i})_{i∈ℤ,i≥0} with q_{i} ∈ R/𝔭^{i} and satisfying π_{i,j}(q_{i}) = q_{j} for all i > j, where π_{i,j}: R/𝔭^{i} → R/𝔭^{j} is the natural quotient map. Now the collection of morphisms {l_{q_{i}}: R/𝔭^{i} → R/𝔭^{i}, i ∈ ℕ} determines uniquely an element in Z^{0}(D^{b}(H_{𝔭})), where l_{q_{i}} is given by multiplication with q_{i}, and it is easy to show this correspondence gives a bijection between R̂_{𝔭} and Z^{0}(D^{b}(H_{𝔭})), which means that Z^{0}(D^{b}(H_{𝔭})) ≅ R̂_{𝔭}.

Now we consider the degree 1 component of the graded center. For any l ∈ ℤ, l ≥ 1, we define η_{𝔭}^{l} ∈ Z^{1}(D^{b}(H_{𝔭})) by setting (η_{𝔭}^{l})_{R/𝔭^{s}} = 0 for all s ≠ l, (η_{𝔭}^{l})_{R/𝔭^{l}} = σ_{𝔭}^{l} and (η_{𝔭}^{l})_{Σ^{i}(R/𝔭^{s})} = (−1)^{i}Σ^{i}(η_{𝔭}^{l})_{R/𝔭^{s}} for all i, s. To show η_{𝔭}^{l} ∈ Z^{1}(D^{b}(H_{𝔭})), it suffices to show that η_{𝔭}^{l} is a natural transformation. For this, one needs to check that for all i, j, m, n and f : Σ^{i}R/𝔭^{m} → Σ^{j}R/𝔭^{n}, the equality Σf ∘ (η_{𝔭}^{l})_{Σ^{i}R/𝔭^{m}} = (η_{𝔭}^{l})_{Σ^{j}R/𝔭^{n}} ∘ f holds. This is clear since both sides of the above equality vanish, unless f is an isomorphism, where we use the fact that any σ_{𝔭}^{l} is given by an almost split sequence.

The argument above shows that if for each l, we fix an Auslander-Reiten sequence, say λ_{l}η_{𝔭}^{l} for some λ_{l} ∈ k_{𝔭}, with starting term R/𝔭^{l}, then we obtain an element Σ_{l}λ_{l}η_{𝔭}^{l} in Z^{1}(D^{b}(H_{𝔭})). The infinite product makes sense since when applied to any object in D^{b}(H_{𝔭}) it becomes a finite sum.

Conversely, let η ∈ Z^{1}(D^{b}(H_{𝔭})). We claim that η_{R/𝔭^{l}} corresponds to either an Auslander-Reiten sequence or is zero for all l. This can be done by induction on l. Clearly it holds for l = 1. Assume that it holds for all l ≤ n − 1, we will show that it is also true for l = n.

From the exact sequence 0 → 𝔭^{n−1}/𝔭^{n} → R/𝔭^{n} →^{π} R/𝔭^{n−1} → 0 we obtain an exact sequence

Ext_{R}^{1}(R/𝔭^{n}, R/𝔭) → Ext_{R}^{1}(R/𝔭^{n}, R/𝔭^{n}) →^{f} Ext_{R}^{1}(R/𝔭^{n}, R/𝔭^{n−1}) → 0.

It is easy to show that Ker(f) is one dimensional and spanned by the Auslander-Reiten sequences with starting term R/𝔭^{n}. Since η is a natural transformation, the map π: R/𝔭^{n} → R/𝔭^{n−1} yields Σπ ∘ η_{R/𝔭^{n}} = η_{R/𝔭^{n−1}} ∘ π = 0, where the last equality holds since η_{R/𝔭^{n−1}} is given by some Auslander-Reiten sequence or zero. Thus η_{R/𝔭^{n}} corresponds to an Auslander-Reiten sequence by the above argument. The proposition now follows. □

Next we combine Lemma 3.1, Proposition 3.4 and that mod(R) = H_{+} ∨ H_{0} with H_{0} = ∐_{𝔭∈max(R)} H_{𝔭}. This gives the following proposition. Note that H_{+} consists of free modules and hence Z^{1}(Σ^{∗}H_{+}) = 0.

Proposition 3.6. Let R be a Dedekind domain and max(R) the set of all maximal ideals. Then as a graded ring

Z^{∗}(D^{b}(mod(R))) ≅ T (R, ∏_{𝔭∈max(R)} ∏_{l∈ℤ,l≥1}k_{𝔭}),

where each k_{𝔭} ≅ R/𝔭 is viewed as a simple R-module.

4. Tame hereditary algebras and weighted projective lines

This section deals with the derived category for some further classes of hereditary categories. We consider either the category of modules mod(A) of a tame hereditary algebra A or the category Coh(𝕏) for a weighted projective line 𝕏 of non-negative Euler characteristic. Unfortunately, our methods do not work for the wild cases. What we want to emphasize is that tubes are of special importance in our calculations.

Throughout this section, k denotes an algebraically closed field and all categories considered are assumed to be k-linear; therefore the graded centers are k-algebras. Note that most results hold for an arbitrary base field k; however the proofs would require modifications.

We begin by studying tubes. The tubes occurring in this section are different from the ones for Dedekind domains and we will use a different method to deal with them. Note that one can use completed path algebras to unify the proofs.

Let C be a uniserial hom-finite hereditary length k-category. Recall that a length category is an abelian category such that any object has a composition series of finite length. Note that a length category is always a Krull-Remak-Schmidt category, i.e., any object can be written as a finite direct sum of indecomposables and the endomorphism ring of any indecomposable object is local. Following [AR], a locally finite abelian category is called uniserial if any indecomposable object of finite length has a unique composition series.

It follows from Theorem 2.13 in [Si], that any hom-finite length category is equivalent to the category of finite length comodules of some basic coalgebra. And since k is assumed to be algebraic closed, any basic coalgebra is pointed, that is, it can be realized as a subcoalgebra of certain path coalgebra of some quiver.

A quiver Q = (Q_{0}, Q_{1}, s, t) is by definition an oriented graph, where Q_{0} is the set of vertices, Q_{1} the set of edges which are usually called arrows, s and t are two maps from Q_{1} to Q_{0} such that for each arrow α, s(α) and t(α) denote respectively the starting vertex and the terminating vertex of α. A path in Q is a sequence of arrows α_{1}α_{2} ⋯ α_{n} with t(α_{i}) = s(α_{i+1}) for 1 ≤ i ≤ n − 1, s(α_{1}) and t(α_{n}) are called the starting vertex and terminating vertex respectively and n is the length. Each vertex v can be viewed as a path of length 0 which starts and terminates at v.

It is well known that there is a path algebra and a path coalgebra structure on the vector space kQ with basis consisting of all paths in Q and the multiplication and comultiplication are given by composing and splitting the paths. Denote by kQ^{a} and (kQ^{c}, Δ, ϵ) the path algebra and path coalgebra of Q respectively.

We denote the category of k-representations by Rep(Q) and the subcategory of locally nilpotent representations by NRep(Q). As usual we denote the subcategories consisting of finite length objects by rep(Q) and nrep(Q). It is well known that Rep(Q) is equivalent to the module category of the path algebra kQ^{a} and NRep(Q) is equivalent to the comodule category of the path coalgebra kQ^{c}.

Let n, m ∈ ℤ ∪ {−∞, +∞} with n ≤ m. We use A_{[n,m]} to denote the following quiver. The vertices are indexed by {i ∈ ℤ | n ≤ i ≤ m} and for each n ≤ i ≤ m − 1 there is exactly one arrow which starts at the vertex i and terminates at i + 1. Now denote the quivers A_{[−∞,0]}, A_{[−∞,+∞]}, A_{[0,+∞]} and A_{[0,n]} for any n ≥ 1 by A^{∞}, A_{∞}^{∞}, A_{∞} and A_{n} respectively. Also we denote by Z_{n} the basic cycle of length n for any n ≥ 1, i.e., the quiver obtained from A_{n} by gluing the vertices 0 and n.

The following classification is a special case of Theorem 2.10(i) in [CG].

Lemma 4.1. Let C be a uniserial hereditary length k-category. Then C is equivalent to nrep(Q) for some quiver Q, where Q is a disjoint union of quivers of type A^{∞}, A_{∞}^{∞}, A_{∞}, A_{n} or Z_{n}.

The idea of the proof is easy. As shown in [Si], C is equivalent to the category of finite length comodules of some pointed coalgebra C, and that C is hereditary means that C must be a path coalgebra and hence C is given by some nrep(Q). The fact that C is uniserial implies that for each vertex v ∈ Q_{0}, there is at most one arrow starting at v and at most one arrow terminating at v, and hence the lemma follows.

Now suppose that Q is one of A^{∞}, A_{∞}^{∞}, A_{∞} and A_{n}. Then the category nrep(Q) is directed. More explicitly, any indecomposable object M ∈ nrep(Q) is a stone, i.e., End_{C}(M) ≅ k and Ext_{C}^{1}(M, M) = 0. This has the following consequence.

Proposition 4.2. Z^{∗}(D^{b}(nrep(Q))) ≅ k for Q = A^{∞}, A_{∞}^{∞}, A_{∞} and A_{n}.

The only case left is Q = Z_{n}, where n ≥ 1 is a positive integer. It is well known that nrep(Z_{n}) is a tube of τ-period n. We need to fix some notation. Denote by S_{i} the simple representation with respect to the vertex i, and M_{i}^{[l]} the indecomposable representation with socle S_{i} and of length l, for any i and l ≥ 1. In the category nrep(Q), there is neither a nonzero projective nor a nonzero injective object, while in the category NRep(Q) there are enough injective objects, and we denote by M_{i}^{[∞]} or simply M_{i} the indecomposable injective module with socle S_{i}. Note that {M_{i}^{[l]} | 0 ≤ i ≤ n − 1, 1 ≤ l ≤ ∞} gives a complete set of isoclasses of indecomposables in NRep(Q).

There is a monomorphism i_{s}^{l}: M_{s}^{[l]} → M_{s}^{[l+1]} and an epimorphism π_{s}^{[l]} : M_{s}^{[l]} → M_{s−1}^{[l−1]} for any s and l, and any morphism between the indecomposables is a linear combination of compositions of such morphisms. For convenience, we write again π_{s}^{∞} as π_{s}, and we set M_{s}^{[l]} = 0 and i_{s}^{l} = π_{s}^{l} = 0 for l ≤ 0. More generally, we can define monomorphisms

i_{s}^{l,t} = i_{s}^{l+t−1} ∘ ⋯ ∘ i_{s}^{l}: M_{s}^{[l]} ⟶ M_{s}^{[l+t]}, ∀ 0 ≤ s ≤ n − 1, l ≥ 1, t ≥ 1,

and epimorphisms

π_{s}^{l,t} = π_{s−t+1}^{l−t+1} ∘ ⋯ ∘ π_{s}^{l}: M_{s}^{[l]} ⟶ M_{s−t}^{[l−t]}, ∀ 0 ≤ s ≤ n − 1, l ≥ 1, 1 ≤ t ≤ l − 1.

We also set i_{s}^{l,∞} to be the inclusion M_{s}^{[l]} → M_{s} and i_{s}^{l,0} = π_{s}^{l,0} = id_{M_{s}^{[l]}}. Note that we have the equality of morphisms

π_{s}^{l+1} ∘ i_{s}^{l} = i_{s−1}^{l−1} ∘ π_{s}^{l}: M_{s}^{[l]} ⟶ M_{s−1}^{[l]}

for all s and l. The syzygy functor Ω^{−1} is given by Ω^{−1}(M_{s}^{[l]}) = M_{s−l}, Ω^{−1}(i_{s}^{l}) = π_{s−l} and Ω^{−1}(π_{s}^{l}) = id_{M_{s−l}}. In the case n = 1, the subscript s is omitted for simplicity.

Lemma 4.3. Let Q = Z_{n}. Then Z^{0}(D^{b}(nrep(Q))) ≅ k[[ξ]], where ξ is the natural transformation from the identity functor to itself, which is given by

ξ_{M_{s}^{[l]}} = i_{s−n}^{l−n,n} ∘ π_{s}^{l,n}: M_{s}^{[l]} ⟶ M_{s}^{[l]}.

It is easy to check that ξ is a natural transformation. Also the infinite sum ∑_{m≥0} λ_{m}ξ^{m} gives a natural transformation, where λ_{m} ∈ k for all m. Observe that this does make sense, because the sum is indeed a finite sum when applied to any object in nrep(Q). To show that this gives all the natural transformations, one just uses the fact that {ξ_{M_{s}^{[l]}}^{m}, m ≥ 0} spans End_{C}(M_{s}^{[l]}) for any M_{s}^{[l]} ∈ nrep(Q).

Clearly, we have an exact sequence 0 ⟶ M_{s}^{[l]} ⟶^{i_{s}^{l,∞}} M_{s} ⟶^{π_{s}^{∞,l}} M_{s−l} ⟶ 0 for any M_{s}^{[l]}. This induces an epimorphism Hom_{C}(M_{r}^{[m]}, M_{s−l}) ↠ Ext_{C}^{1}(M_{r}^{[m]}, M_{s}^{[l]}), which is an isomorphism when m ≤ l. In particular, we can identify Hom_{C}(M_{s}^{[l]}, M_{s−l}) with Ext_{C}^{1}(M_{s}^{[l]}, M_{s}^{[l]}).

The following lemma is needed to describe the degree 1 component of the graded center of D^{b}(nrep(Q)).

Lemma 4.4. Let Q = Z_{n} and n ≥ 1. If n ≥ 2, then Z^{1}(D^{b}(nrep(Q))) = 0; if n = 1, then as a k-vector space,

Z^{1}(D^{b}(nrep(Q))) ≅ ∏_{l∈ℤ,l≥1}k · η^{l},

where η^{l} is given by (η^{l})_{M^{[l]}} = i^{1,∞} ∘ π^{l,l−1} and (η^{l})_{M^{[a]}} = 0 for a ≠ l.

Proof. First we consider the case n ≥ 2. Fix η ∈ Z^{1}(D^{b}(nrep(Q))). We show that η_{M_{s}^{[l]}} = 0 for all s and l by using induction on l. Clearly η_{M_{s}^{[1]}} = 0 for all s, since there is no self extension for the simple objects if n ≥ 2.

Now assume that the assertion holds for l − 1. Applying the naturality of η to the injection i_{s}^{l−1} : M_{s}^{[l−1]} → M_{s}^{[l]}, we get the equality η_{M_{s}^{[l]}} ∘ i_{s}^{l−1} = Σi_{s}^{l−1} ∘ η_{M_{s}^{[l−1]}} = 0.

We claim that this equality holds only if η_{M_{s}^{[l]}} = 0. Otherwise, if η_{M_{s}^{[l]}} : M_{s}^{[l]} → M_{s−l}

is nonzero, then the dimension of Im(η_{M_{s}^{[l]}}) is at least n since M_{s−l}^{[n]} is the minimal

submodule of M_{s−l} with the same top as M_{s}^{[l]}, thus the dimension of Im(η_{M_{s}^{[l]}} ∘ i_{s}^{l−1}) is at least n−1 and hence nonzero (here we see the difference between n = 1 case and n ≥ 2 case), and now we use the isomorphism Hom_{C}(M_{s}^{[l−1]}, M_{s−l}) ≅ Ext_{C}^{1}(M_{s}^{[l−1]}, M_{s}^{[l]}) to get that the left hand side of the above equality is nonzero, this introduces a contradiction.

Now we assume that n = 1. Note that any Auslander-Reiten sequence with starting term M_{s}^{[l]} is given by a nonzero multiple of η_{M^{[l]}}^{l}. Now we can use the same argument as in the proof of Proposition 3.5. □

Combining Lemmas 4.3 and 4.4, we get the following.

Proposition 4.5. Let Q = Z_{n} and n ≥ 1. If n ≥ 2, then Z^{∗}(D^{b}(nrep(Q))) ≅ k[[ξ]] is a graded k-algebra concentrated in degree 0; if n = 1, we have an isomorphism

Z^{∗}(D^{b}(nrep(Q))) ≅ T (k[[ξ]], ∏_{l∈ℤ,l≥1}k)

of graded algebras, where k is viewed as the unique simple k[[ξ]]-module on which ξ acts trivially. Moreover, we have an isomorphism of graded algebras

T (k[[ξ]], ∏_{l∈ℤ,l≥1}k) ≅ k[[ξ]][η]/(η^{2}),

where ξ is of degree 0 and η is of degree 1.

Remark 4.6. In case that n = 1, we know that nrep(Z_{1}) is equivalent to the category of finite dimensional nilpotent k[x]-modules, which is just the category of finitely generated (x)-torsion modules over the Dedekind domain k[x]. Thus Lemma 3.5 applies and we get the same result. More generally, one can consider the completed path algebra kẐ_{n} of the quiver Z_{n}. Then nrep(Z_{n}) is equivalent to the category of finite dimensional nilpotent modules over kẐ_{n}, and the center of kẐ_{n} is isomorphic to k[[x]].

Remark 4.7. Combining Lemma 4.1 and Propositions 4.2 and 4.5, we have now a description of Z^{∗}(D^{b}(C)) for any uniserial hereditary length k-categroy C.

Next we consider the category of finite dimensional modules over finite dimensional hereditary k-algebras. Since k is assumed to be algebraically closed, we need only consider the path algebras. Now let Q be a finite, connected quiver without oriented cycles and A = kQ the path algebra. Note that in this case, the center of the algebra is the field k. First we consider the finite type case.

Proposition 4.8. Let Q be a quiver such that the path algebra kQ is of finite representation type. Then Z^{∗}(D^{b}mod(kQ)) ≅ k.

Proof. The proof is almost the same as the one of Proposition 4.2. If A is of finite representation type, then any indecomposable A-module M is a stone. In particular, Hom_{D^{b}(A)}(M, Σ^{1}M) = Ext_{A}^{1}(M, M) = 0. □

Next we consider the tame case. Let τ be the Auslander-Reiten translation in mod(A). The Auslander-Reiten quiver of mod(A) consists of the preprojective part, the preinjective part and the regular part. Recall that a A-module M is preprojective if and only if τ^{n}M = 0 for sufficiently large n; and M is preinjecitve if and only if τ^{−n}M = 0 for sufficiently large n. Modules without preprojective and preinjective summands are called regular modules. Denote by P, R and I the full subcategory of preprojective modules, regular modules and preinjective modules respectively. We have the decomposition mod(A) = P ∨ R ∨ I. Let η ∈ Z^{1}(D^{b}(mod(A))), since preprojective and preinjective modules have no self-extensions, we get the following easy lemma by applying Proposition 3.4.

Lemma 4.9. Let Q be a quiver such that the path algebra kQ is of tame representation type, and let R denote the full subcategory of mod(kQ) consisting of regular modules. Then Z^{1}D^{b}(mod(kQ)) ≅ Z^{1}(D^{b}(R)).

Recall that for a tame quiver, the regular part of the Auslander-Reiten quiver is a disjoint union of tubes, and there are neither morphisms nor extensions between different tubes, i.e., R = ∐_{𝔱∈𝔗} R_{𝔱}, where 𝔗 is an index set for all the tubes, R_{𝔱} ≅ nrep(Z_{p(𝔱)}) and p(𝔱) denotes the τ-period of R_{𝔱}. Each tube is an abelian subcategory and we have D^{b}(R) ≅ ∐_{𝔱∈𝔗} D^{b}(R_{𝔱}). Applying Propositions 4.5 and 3.4, we get the following.

Proposition 4.10. Let Q be a tame quiver, and 𝔗_{1} the index set of all homogeneous tubes. Then

Z^{∗}(D^{b}(mod(kQ))) ≅ T (k, ∏_{𝔱∈𝔗_{1}} ∏_{m≥0} k).

Next we consider the weighted projective lines over the field k. Recall that a weighted projective line 𝕏 is defined through the attached category Coh(𝕏) of coherent sheaves, which is a small k-category satisfying certain axioms. This concept was introduced by Geigle and Lenzing in [GL] to study the interaction between preprojective modules and regular modules for tame hereditary algebras. For a definition we refer the readers to [Le, §10], where one can also find most references about this subject.

Weighted projective lines play an important role in the classification of hereditary categories. By a theorem of Happel [Ha], any connected, Ext-finite, hereditary abelian k-category which has a tilting complex is derived equivalent either to the category mod(A) for some finite dimensional hereditary algebra A or the category Coh(𝕏) for some weighted projective line 𝕏.

First we recall some basic facts. Let 𝕏 be a weighted projective line and H = Coh(𝕏) the category of coherent sheaves. The category H has Serre duality, i.e., there exists an equivalence τ: H → H and a natural isomorphism D Ext_{H}^{1}(X, Y) ≅ Hom_{H}(Y, τX). We denote by H_{0} the full subcategory consisting of all objects of finite length. Then H_{0} is a hereditary abelian subcategory and H_{0} = ∐_{x∈C} U_{x} for some index set C, where U_{x} is a tube with finite τ-period p(x). Members in C are called the points of H. Note that there are only finitely many points with p(x) > 1.

We denote by H_{+} the subcategory consisting of all objects without a simple subobject. Objects of H_{+} are called vector bundles. Any indecomposable object of H is either of finite length or a vector bundle. There is a linear form rk : K_{0}(H) → ℤ, called rank, which is τ-invariant, vanishes on objects of H_{0} and takes positive values on objects of H_{+}. Objects of H_{+} of rank one are called line bundles, and by definition H contains a line bundle. For any vector bundle E, we have a filtration E_{0} ⊆ E_{1} ⊂ ⋯ ⊆ E_{r} = E with the line bundle factors E_{i}/E_{i−1}, where r = rk(E).

For any line bundle L and any point x ∈ C, ∑_{S∈U_{x}} dim_{k} Hom_{H}(L, S) = 1, where S runs through all simple objects in U_{x}. Clearly we have H = H_{+} ∨ H_{0}, and therefore any nonzero morphism between line bundles is a monomorphism.

Now we consider the graded center of D^{b}(H). Note that one can define the Euler characteristic χ_{H} for H. If χ_{H} > 0, then H is derived equivalent to the category mod(A) for some finite dimensional tame hereditary algebra A, and in this case, the graded center has been computed. Firstly we have the following easy lemma.

Lemma 4.11. Let 𝕏 be a weighted projective line. Then Z^{0}(D^{b}(Coh(𝕏))) = k.

Proof. We denote Coh(𝕏) by H as before. Since H contains a line bundle, we choose one and denote it by L. Let η: Id_{H} → Id_{H} be a natural transformation. To prove the lemma, it suffices to show that if η_{L} = 0, then η = 0.

Now assume that η_{L} = 0. Let x ∈ C be an arbitrary point, and S ∈ U_{x} the simple object with Hom_{H}(L, S) ≠ 0. Note that such S exists and is unique. By Proposition 4.3, η_{U_{x}} = 0 if and only if η_{S^{[mr+1]}} = 0 for all m ≥ 0, where r = p(x) is the τ-period and S^{[mr+1]} is the object in U_{x} with socle S and of length mr + 1. By using induction we have dim_{k} Hom_{H}(L, S^{[mr+1]}) = m + 1 for any m ≥ 0. We claim that there exists an epimorphism from L to S^{[mr+1]}. Otherwise, all morphisms will factor through S^{[(m−1)r+1]}, and hence dim_{k} Hom_{H}(L, S^{[mr+1]}) = dim_{k} Hom_{H}(L, S^{[(m−1)r+1]}) = m, which gives a contradiction.

Let f: L → S^{[mr+1]} be an epimorphism. Since η is a natural transformation, we have η_{S^{[mr+1]}} ∘ f = f ∘ η_{L} = 0, and hence η_{S^{[mr+1]}} = 0. Now we have shown that η_{H_{0}} = 0. Conversely, using a similar argument, one can show that if η_{H_{0}} = 0, then η_{N} = 0 for any line bundle N. Since any vector bundle E has a filtration with line bundle factors, we get, using the five lemma, that η_{E} = 0. This completes the proof. □

Combined with Proposition 3.4, Lemma 4.4 and Proposition 4.5, we obtain the following embedding of algebras.

Lemma 4.12. Let 𝕏 be a weighted projective line, H = Coh(𝕏) and C_{1} the set of points of τ-period 1. Then the algebra Z = T (k, ∏_{x∈C_{1}} ∏_{m≥0} k) is isomorphic to a subalgebra of Z^{∗}(D^{b}(H)).

In the tubular case, i.e., χ_{H} = 0, we have H = ⋁_{q∈ℚ∪{∞}} H^{⟨q⟩}, where for each q we have H^{⟨q⟩} ≅ H_{0}. In fact one can define the slope for objects of H, and roughly speaking, for any q ∈ ℚ, H^{⟨q⟩} is just given by objects of slope q, and H^{⟨∞⟩} = H_{0}. With this decomposition of categories, we have the following proposition.

Proposition 4.13. Let 𝕏 be a weighted projective line of Euler characteristic 0, H = Coh(𝕏) and C_{1} the set of points of τ-period 1. Then

Z^{∗}(D^{b}(H)) ≅ T (k, ∏_{q∈ℚ∪{∞}} ∏_{x∈C_{1}} ∏_{m≥0} k).

5. The graded center of D^{b}(mod(k[x]/(x^{2})))

In this section, we will study the ring of dual numbers, which is by definition the k-algebra A = k[x]/(x^{2}), where k is an arbitrary base field. Set C = mod(A) and P the full subcategory of C consisting of projective modules. One has a complete description of the indecomposable objects of D^{b}(C) = K^{+,b}(P), and therefore one can write down the elements in Z^{∗}(D^{b}(C)) explicitly. By Theorem 2.5, we need only to consider the category K^{b}(P).

The indecomposable objects in K^{b}(P) are well understood, for example see [Kü]. They are given by {A_{m}^{n} | −∞ < m ≤ n < ∞}, where A_{m}^{n} is the complex

[figure: ⋯; 0; \underbrace{A}_{n}; x; A; x; ⋯; x; \underbrace{A}_{m}; 0; ⋯] ,

that is, (A_{m}^{n})_{i} = A for m ≤ i ≤ n and 0 otherwise, and d_{i}^{A_{m}^{n}} = x for all m < i ≤ n − 1, where we use x to denote the multiplication map l_{x}. If we allow n to take the value ∞, then we get all indecomposable objects in K^{+,b}(P), in fact A_{m}^{∞} ≅ Σ^{m}S in the derived category, where S is the simple A-module, regarded as a stalk complex concentrated in degree zero. Note that ΣA_{m}^{n} ≅ A_{m+1}^{n+1}. The following lemma is basic for our computations.

Lemma 5.1. Let −∞ < m ≤ n < ∞, −∞ < m′ ≤ n′ < ∞. If (m, n) ≠ (m′, n′), then Hom_{K^{b}(P)}(A_{m}^{n}, A_{m′}^{n′}) is at most one dimensional. The morphisms between indecomposable objects in K^{b}(P) are linear combinations of compositions of the following four classes of morphisms:

(a) π_{m}^{n,n′}: A_{m}^{n} → A_{m}^{n′} for m ≤ n′ ≤ n, (π_{m}^{n,n′})_{m} = x and (π_{m}^{n,n′})_{i} = 0 ∀i ≠ m;

(b) π_{m,m′}^{n}: A_{m}^{n} → A_{m′}^{n} for m ≤ m′ ≤ n, (π_{m,m′}^{n})_{i} = 1 ∀m′ ≤ i ≤ n;

(c) i_{m}^{n,n′}: A_{m}^{n} → A_{m}^{n′} for m ≤ n ≤ n′, (i_{m}^{n,n′})_{i} = 1 ∀m ≤ i ≤ n;

(d) i_{m,m′}^{n}: A_{m}^{n} → A_{m′}^{n} for m′ ≤ m ≤ n, (i_{m,m′}^{n})_{m} = x, and (i_{m,m′}^{n})_{i} = 0 ∀i ≠ m.

The morphisms in the lemma look as follows.

(a) [figure: 0; \overbrace{A}^{n}; ⋯; \overbrace{A}^{n′}; 0; ⋯; \overbrace{A}^{m}; x; 0; 0; A; ⋯; A; 0]

(b) [figure: 0; \overbrace{A}^{n}; 1; ⋯; \overbrace{A}^{m′}; 1; ⋯; \overbrace{A}^{m}; 0; 0; A; ⋯; A; 0]

(c) [figure: 0; A; 1; ⋯; A; 1; 0; 0; \underbrace{A}_{n′}; ⋯; \underbrace{A}_{n}; ⋯; \underbrace{A}_{m}; 0]

(d) [figure: 0; A; 0; ⋯; A; x; 0; 0; \underbrace{A}_{n}; ⋯; \underbrace{A}_{m}; ⋯; \underbrace{A}_{m′}; 0]

The proof is straightforward and left to the reader. For any m ≤ n < ∞, the space Hom_{K^{b}(P)}(A_{m}^{n}, A_{m}^{n}) is two dimensional, and we denote the morphism i_{m,m}^{n} = π_{m}^{n,n} by x_{m}^{n}. Now let η: Id_{K^{b}(P)} → Id_{K^{b}(P)} be a natural transformation. Clearly, η is uniquely given by some datum {μ_{m}^{n}, λ_{m}^{n} ∈ k, −∞ < m ≤ n < ∞} with η_{A_{m}^{n}} = μ_{m}^{n} · 1 + λ_{m}^{n}x_{m}^{n}.

Proposition 5.2. Let η: Id_{K^{b}(P)} → Id_{K^{b}(P)} be a natural transformation and {μ_{m}^{n}, λ_{m}^{n}} the corresponding datum.

(1) We have μ_{m}^{n} = μ_{m′}^{n′} for any m, m′, n and n′. Conversely, any datum of the form {μ, λ_{m}^{n} ∈ k, ∞ < m ≤ n < ∞} arises as the datum of some natural transformation η: Id_{K^{b}(P)} → Id_{K^{b}(P)} by setting η_{A_{m}^{n}} = μ+λ_{m}^{n}x_{m}^{n} for any m and n.

(2) If η ∈ Z^{0}(K^{b}(P)), then λ_{m}^{n} = λ_{m+r}^{n+r} for any m, n and r, and any elements in Z^{0}(K^{b}(P)) is obtained in this way.

(3) As an algebra, Z^{0}(K^{b}(P)) ≅ T (k, ∏_{r≥0} k), where T (k, ∏_{r≥0} k) is viewed as a graded algebra concentrated in degree 0.

Proof. (1) Apply the naturality of η to get i_{m}^{n,n′} ∘ η_{m}^{n} = η_{m}^{n′} ∘ i_{m}^{n,n′} and π_{m,m′}^{n} ∘ η_{m}^{n} = η_{m}^{n′} ∘ π_{m,m′}^{n}. From this follows that μ_{m}^{n} = μ_{m′}^{n′} for any m, m′, n and n′.

Conversely, for any datum of the form {μ, λ_{m}^{n} ∈ k, ∞ < m ≤ n < ∞}, we claim that the above constructed η is indeed a natural transformation. In fact, one can easily show that the equalities f ∘ η_{m}^{n} = η_{m′}^{n′} ∘ f hold for those morphisms f: A_{m}^{n} → A_{m′}^{n′} listed in Lemma 5.1. Now using the fact that K^{b}(P) is a Krull-Remak-Schmidt category and any morphism is some linear combination of compositions of morphisms listed in Lemma 5.1, we get that η_{Y} ∘ f = f ∘ η_{X} holds for any morphism f: X → Y in the category K^{b}(P). Thus η is a natural transformation.

(2) Use the fact that by definition η ∈ Z^{0}(K^{b}(P)) if and only if Ση = ηΣ, and this is equivalent to the requirement that λ_{m}^{n} = λ_{m+1}^{n+1} for any m and n.

(3) is an easy consequence of (2). In fact we can explicitly write down the elements in Z^{0}(K^{b}(P)). For any r ≥ 0, let η_{r} ∈ Z^{0}(K^{b}(P)) denote the natural transformation obtained by setting (η_{r})_{A_{m}^{n}} = x_{m}^{n} for m − n = r and 0 otherwise. Thus as vector spaces

Z^{0}(K^{b}(P)) = k · 1 ⊕ ∏_{r≥0} k · η_{r}.

By direct computation, the multiplication satisfies η_{r}η_{r′} = 0 for any r and r′ and the isomorphism in (3) follows. □

Now we consider the natural transformations from the identity functor to Σ^{t} for any positive integer t > 0. Note that Hom_{K^{b}(P)}(A_{m}^{n}, Σ^{t}A_{m}^{n}) = 0 for any m, n with n < m+t, and in the case n ≥ m + t, the morphism space is one dimensional with basis element f_{t;m}^{n} = i_{m+t}^{n,n+t} ∘ π_{m,m+t}^{n}. Let ζ: Id_{D^{b}(P)} → Σ^{t} be a natural transformation; it is uniquely determined by the datum {λ_{t;m}^{n}, n ≥ m + t}, where ζ_{A_{m}^{n}} = λ_{t;m}^{n}f_{t;m}^{n}. Applying the naturality of ζ to the morphisms i_{m}^{n,n′} and π_{m,m′}^{n}, one gets f_{t;m}^{n} = f_{t;m′}^{n′} for any m, m′, n and n′. Thus we get the following lemma.

Lemma 5.3. Let t > 0. All natural transformations from Id_{K^{b}(P)} to Σ^{t} form a one dimensional k-space with a basis element ζ_{t}, where ζ_{t} is given by (η_{t})_{A_{m}^{n}} = f_{t;m}^{n} for all n ≥ m + t and 0 otherwise. Moreover, the multiplication satisfies ζ_{t}ζ_{t′} = ζ_{t+t′} for any t, t′ > 0 and ζ_{t}η_{r} = η_{r}ζ_{t} = 0 for any t > 0 and r ≥ 0, where the η_{r} are given as in the proof of Proposition 5.2.

Note that Σζ_{t} = (−1)^{t}ζ_{t}Σ if and only if either char(k) = 2 or char(k) ≠ 2 and t is even. Combined with the last lemma and Proposition 5.2, we get the graded center of D^{b}(mod(A)).

Proposition 5.4. Let k be an arbitrary base field. Then as a graded algebra,

Z^{∗}(D^{b}(mod(k[x]/(x^{2})))) ≅ T (k[ζ], ∏_{r≥0} k),

where k is identified with k[ζ]/(ζ) as a k[ζ]-module,

Z^{0}(D^{b}(mod(k[x]/(x^{2})))) ≅ T (k, ∏_{r≥0} k),

and ζ is of degree 2 if char(k) ≠ 2, and of degree 1 if char(k) = 2.

6. The graded center of the stable category \underline{mod}(k[x]/(x^{n}))

Another important class of triangulated categories are the stable categories of selfinjective algebras. We calculate the graded centers in some special cases, namely for the algebras of the form k[x]/(x^{n}) with n ≥ 2. These calculations are based on the fact that the indecomposable objects and their morphisms are well understood. Note that some algebras of the form k[x]/(x^{n}) are Brauer tree algebras, and we refer to [KL] for the calculation of the graded centers of their stable module categories.

Let A = k[x]/(x^{n}) with k an arbitrary base field. It is well known that A is uniserial and that all the indecomposable objects in mod(A) are of the form A_{l} = A/x^{l}A = x^{n−l}A with 1 ≤ l ≤ n. There are epimorphisms π_{r}^{l} = l_{x^{l−r}}: A_{l} ↠ A_{r} for l ≥ r and monomorphisms i_{l}^{r}: A_{l} ↪ A_{r} for l ≤ r. For any l and r, Hom_{A}(A_{l}, A_{r}) has a basis {f_{s}^{l,r} = i_{s}^{r} ∘ π_{s}^{l} | 1 ≤ s ≤ min(l, r)}. Moreover, the syzygy functor Ω is given by Ω(A_{l}) = A_{n−l} and Ω(f_{s}^{l,r}) = π_{n−r}^{n−s} ∘ i_{n−l}^{n−s} = f_{n−r−l+s}^{n−l,n−r} for all 1 ≤ l ≤ n − 1, r, s ≤ l.

Now let C = \underline{mod}(A) be the stable category. One knows that C is a triangulated category with suspension functor Σ = Ω^{−1} = Ω. In particular, we have Ω^{2} = Id_{C} in C. The indecomposable objects in C are given by A_{l} = A/x^{l}A = x^{n−l}A with 1 ≤ l < n, and f̄_{s}^{l,r} = 0 if and only if l + r − n ≥ s. Consequently, Hom_{C}(A_{l}, A_{n−l}) = Hom_{A}(A_{l}, A_{n−l}).

For any self-injective ring R, let Z(R) denote the graded center of R. There is a canonical morphism from Z(R) to Z^{0}(\underline{mod}(R)). As we will show below, this map is not injective in general. The more interesting question is that whether it is surjective. In the case A = k[x]/(x^{n}), the answer is yes. In fact, for an arbitrary uniserial self-injective algebra, all natural transformations from the identity functor to itself for the stable category come from the center of the algebra.

Proposition 6.1. Let A = k[x]/(x^{n}) with n ≥ 2 and C = \underline{mod}(A). Then Z^{0}(C) ≅ k[x]/(x^{[ \frac{n}{2} ]}), where [\frac{n}{2}] denotes the maximal integer which is no larger than \frac{n}{2}.

Proof. Note that A_{[ \frac{n}{2} ]} is of special importance since End_{C}(A_{[ \frac{n}{2} ]}) is of maximal dimension among the indecomposable objects. Let η be a natural transformation from Id_{C} to Id_{C}. We will show that η is uniquely determined by η_{A_{[ \frac{n}{2} ]}}.

We fix some a ∈ A such that η_{A_{[ \frac{n}{2} ]}} = l̄_{a}, where l_{a} is given by the multiplication

with a as before. Since η is a natural transformation, we have π_{[ \frac{n}{2} ]}^{l} ∘ η_{A_{l}} = l̄_{a} ∘ π_{[ \frac{n}{2} ]}^{l} for

l > [\frac{n}{2}] and i_{l}^{[ \frac{n}{2} ]} ∘ η_{A_{l}} = l̄_{a} ∘ i_{l}^{[ \frac{n}{2} ]} for any l < [\frac{n}{2}]. Now it is easy to show that η_{A_{l}} = l̄_{a} for any l, since the solutions of the equations above are unique. Therefore we have an epimorphism from A to Z^{0}(C), and easy computations show that l_{x^{[ \frac{n}{2} ]}} = 0 in C. □

Next we will compute the natural transformations from the identity functor to Ω. The following lemma is easy.

Lemma 6.2. Let ζ: Id_{C} → Ω be a natural transformation. Then for any 1 ≤ l < n, we have ζ_{A_{l}} = λ_{l}·f̄_{1}^{l,n−l} for some λ_{l} ∈ k. And conversely, any family {λ_{l}, 1 ≤ l < n} induces a natural transformation ζ by setting ζ_{A_{l}} = λ_{l} · f̄_{1}^{l,n−l} for any l. Moreover, ζ ∈ Z^{1}(C) if and only if λ_{l} = −λ_{n−l} for any l.

Proof. We use induction on l. Clearly we have ζ_{A_{1}} = λ_{1} · f̄_{1}^{1,n−1}. Now assume that ζ_{A_{s}} = λ_{s} · f̄_{1}^{s,n−s} for some λ_{s} ∈ k, and consider the inclusion i_{s}^{s+1}. One gets ζ_{A_{s+1}} ∘ ī_{s}^{s+1} = π̄_{n−s−1}^{n−s} ∘ ζ_{A_{s}} = 0, and hence ζ_{A_{s+1}} = λ_{s+1} · f̄_{1}^{s+1,n−s−1}. The remaining part is straightforward. □

Now let ζ_{s} denote the natural transformation given by (ζ_{s})_{A_{l}} = δ_{s}^{l}f̄_{1}^{s,n−s} for any 1 ≤ l < n. We also denote by t the identity map from Id_{C} to Ω^{2} = Id_{C} but viewed as an element in Z^{2}(C).

Let Z̃^{∗}(C) be the ℤ-graded space with Z̃^{n}(C) consisting of all natural transformations from Id_{C} to Ω^{n}. Note that Z̃^{∗}(C) forms a graded algebra and Z^{∗}(C) is a subalgebra of Z̃^{∗}(C).

Observe that the case n = 2 is slightly different. In fact in this case, not only Ω^{2} but also the shift functor Ω itself is equivalent to the identity functor. We deal with this case separately. With the above notations, we get the following results.

Proposition 6.3. Let C = \underline{mod}(k[x]/(x^{2})). Then Z̃^{∗}(C) = k[ζ_{1}, ζ_{1}^{−1}] with ζ_{1} of degree 1. We have Z^{∗}(C) = Z̃^{∗}(C) if char(k) = 2, and Z^{∗}(C) = k[ζ_{1}^{2}] if char(k) ≠ 2.

Note that ζ_{1}^{2} equals t as defined above, and clearly ζ_{1}^{−1} is of degree −1. The proof follows directly from Proposition 6.1 and Lemma 6.2.

Proposition 6.4. Let C = \underline{mod}(k[x]/(x^{n})) and n ≥ 3. Then we have

Z̃^{∗}(C) = k[x, ζ_{1}, ⋯, ζ_{n−1}, t, t^{−1}]/⟨x^{[ \frac{n}{2} ]}, xζ_{s}, ζ_{s}x, ζ_{s}ζ_{s′}⟩,

where x, each ζ_{s} and t are in degree 0, 1 and 2 respectively. Moreover, Z^{∗}(C) is the subalgebra generated by x, t, t^{−1}, and ζ_{s} − ζ_{n−s} with 1 ≤ s ≤ [\frac{n}{2}] if either n is odd or char(k) ≠ 2; if char(k) = 2 and n is even, then Z^{∗}(C) is the subalgebra generated by x, t, t^{−1}, ζ_{[ \frac{n}{2} ]} and ζ_{s} − ζ_{n−s} with 1 ≤ s ≤ [\frac{n}{2}].

Corollary 6.5. Let C = \underline{mod}(k[x]/(x^{n})) and n ≥ 3. Then as a graded algebra,

Z^{∗}(C) = k[x, ζ_{1}, ⋯, ζ_{l}, t, t^{−1}]/⟨x^{[ \frac{n}{2} ]}, xζ_{s}, ζ_{s}x, ζ_{s}ζ_{s′}⟩

with x, each η_{s} and t in degree 0, 1 and 2 respectively, where l = [\frac{n−1}{2}] if either n is odd or char(k) ≠ 2, and l = [\frac{n}{2}] if char(k) = 2 and n is even.

Remark 6.6. For a self-injective algebra A, one has D^{b}(mod(A))/K^{b}(proj A) ≅ \underline{mod}(A). We have already seen that Z^{∗}(D^{b}(mod(A))) ≅ Z^{∗}(K^{b}(proj A)), but what can we say about the ring homorphism π_{∗}: Z^{∗}(D^{b}(mod(A))) → Z^{∗}(\underline{mod}(A))?

For the algebra A = k[x]/(x^{2}) we can describe π_{∗} explicitly, since both graded centers are known. Recall that

Z^{∗}(D^{b}(mod(k[x]/(x^{2})))) = (k ⊕ ∏_{r≥0} k · η_{r})[ζ]/⟨η_{r}η′_{r}, η_{r}ζ⟩

and Z^{∗}(\underline{mod}(A)) = k[t, t^{−1}]. We know that in this case, π_{∗} is neither injective nor surjective. Explicitly, Im(π_{∗}) = k[t] and Ker(π_{∗}) = ∏_{r≥0} k · η_{r}.

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Henning Krause, Institut für Mathematik, Universität Paderborn, 33095 Paderborn, Germany.

   E-mail address: hkrause@math.upb.de

Yu Ye, Department of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, PR China.

   E-mail address: yeyu@ustc.edu.cn