DEGENERATION-LIKE ORDERS IN TRIANGULATED CATEGORIES

BERNT TORE JENSEN, XIUPING SU, AND ALEXANDER ZIMMERMANN

Abstract. In an earlier paper we defined a relation ≤_{Δ} between objects of the derived category of bounded complexes of modules over a finite dimensional algebra over an algebraically closed field. This relation was shown to be equivalent to the topologically defined degeneration order in a certain space comproj(A, \underline{d}) for derived categories. This space was defined as a natural generalization of varieties for modules. We remark that this relation ≤_{Δ} is defined for any triangulated category and show that under some finiteness assumptions on the triangulated category ≤_{Δ} is always a partial order.

1. Introduction

For a finite dimensional k-algebra A over an algebraically closed field k one defines for any positive integer d a variety mod(A, d) as the algebraic variety of d-dimensional A-modules. The general linear group Gl_{d}(k) acts naturally by conjugation on mod(A, d) and two points in mod(A, d) correspond to isomorphic modules if and only if they belong to the same orbit under this group action. A module M degenerates to a module N in mod(A, d) if N is in the topological closure of the orbit of M. Riedtmann and Zwara characterized in [7, 10] this degeneration by an algebraic relation namely M degenerates into N if and only if there is a module Z so that N can be embedded into M ⊕ Z with quotient being isomorphic to Z. Zwara proved in [9] algebraically that this so-defined relation on isomorphism classes of modules is transitive.

In [4] we defined a topological space comproj(A, \underline{d}) so that points in this space correspond to right bounded complexes of projective A-modules with fixed homogeneous components in each degree. A base change group acts on this space as well and orbits correspond to isomorphism classes in the derived category. We showed that degeneration in mod(A, d) induces a degeneration in comproj(A, \underline{d}) for suitably defined \underline{d}. Moreover, we defined algebraically a partial order ≤_{Δ} by setting X ≤_{Δ} Y if there is a complex Z so that Y ⟶ X ⊕ Z ⟶ Z ⟶ Y [1] is a distinguished triangle in the derived category. Then, we showed that for two complexes X and Y in the bounded derived category, X ≤_{Δ} Y if and only if X degenerates to Y in comproj(A, \underline{d}) for sufficiently big \underline{d}. By consequence, ≤_{Δ} is a partial order on isomorphism classes of complexes in D^{b}(A). We need that A is finite dimensional over an algebraically closed field in order to be able to apply methods from algebraic geometry.

We show in the present paper that for any commutative ring R the obvious generalization of ≤_{Δ} to an arbitrary triangulated category is a partial order for any triangulated R-linear category T satisfying the following three properties:

1) Hom_{T} (X, Y) is an R-module of finite length for any X, Y

2) for any X, Y ∈ T there is a non zero integer n_{X,Y} so that Hom_{T} (X, Y [n_{X,Y}]) = 0, and 3) idempotent morphisms split in T . These hypotheses are satisfied for the bounded derived category of finitely generated modules over a finite dimensional algebra over a field R. Moreover, they imply that T satisfies the Krull-Remak-Schmidt theorem.

The paper is organized as follows. In Section 2 we give the basic definitions and show by an example that without some finiteness hypothesis on the triangulated categories, it will not be reasonable to have a partial order defined by ≤_{Δ}. In Section 3 we imitate Zwara’s proof [9] in order to show transitivity of our relation ≤_{Δ}. In Section 4 we show that if M ≤_{Δ} N then, Hom_{T} (X, M) has smaller length over R than Hom_{T} (X, N) for any object X. Moreover, as a consequence we show that ≤_{Δ} is anti-symmetric. We finish the proof of the main result Theorem 5 in Section 5.

Date: September 14, 2004; revised October 22, 2004.

2. Basic definitions and an auxiliary results

Definition 1. • Let T be a triangulated category with shift functor [1]. Then, we say for two objects X and Y of T that Y ≤_{Δ} X if there is an object Z of T and a distinguished triangle

X ⟶ Y ⊕ Z ⟶ Z ⟶ X[1]

• We say that idempotent morphisms split in T if for any object X and any e^{2} = e ∈ End_{T} (X) there is an isomorphism X ≃ X_{e} ⊕ X_{e′} so that via this isomorphism the

endomorphism e is mapped to the endomorphism ( \begin{matrix}id_{X_{e}} & 0 \\ 0 & 0 \end{matrix} ) of X_{e} ⊕ X_{e′}.

Remark 2.1. For module categories this partial order is studied by Riedtmann [7] and Zwara [10, 9]. In [4] we extend the main result of Zwara [10] to derived categories.

Example 2.2. (1) Let A be an abelian category admitting countable direct sums and M and N be two objects. Then,

0 ⟶ M ⟶^{id} M ⟶ 0 ⟶ 0

is exact and hence also

0 ⟶ M ⟶^{(id,0)} M ⊕ (⨁_{i=1}^{∞}(N ⊕ M)) ⟶ ⨁_{i=1}^{∞}(N ⊕ M) ⟶ 0

is exact. Now, since

M ⊕ (⨁_{i=1}^{∞}(N ⊕ M)) ≃ ⨁_{i=1}^{∞}(M ⊕ N) ≃ ⨁_{i=1}^{∞}(N ⊕ M) ≃ N ⊕ (⨁_{i=1}^{∞}(M ⊕ N))

we get N ≤_{Δ} M for any two modules M and N in the derived category D^{b}(A) of bounded complexes of objects in A. Hence M ≤_{Δ} N ≤_{Δ} M for any two objects M and N in A even though M and N may be not isomorphic.

This example shows that it does not make sense to try to prove that ≤_{Δ} is a partial order without some finiteness assumption on the category.

(2) Let G = Q_{32} be the generalized quaternion group of order 32 and let ℤG be its integral group ring. In [8] Swan gives a projective non-free ideal 𝔞 of ℤG so that 𝔞 ⊕ ℤG ≃ ℤG ⊕ ℤG. Since 𝔞 is not free, we have that ℤG ≄ 𝔞. The split exact sequence

0 ⟶ 𝔞 ⟶ ℤG ⊕ 𝔞 ⟶ ℤG ⟶ 0

induces an exact sequence

0 ⟶ 𝔞 ⟶ ℤG ⊕ ℤG ⟶ ℤG ⟶ 0

which shows ℤG ≤_{Δ} 𝔞. Likewise, 𝔞 ≤_{Δ} ℤG and we get 𝔞 ≤_{Δ} ℤG ≤_{Δ} 𝔞 in the derived category of ℤG-modules.

Hence it is clear that it will be necessary to have conditions to ensure the Krull Schmidt property in T in order to make ≤_{Δ} a partial order.

We recall a property of triangulated categories.

Proposition 1. [5, Lemma 1.4.3, page 55] [6, Lemma 1.1] Let

A ⟶^{a} C ⟶^{c} E ⟶^{e_{a}} A[1]

and

B ⟶^{b} D ⟶^{d} E ⟶^{e_{b}} B[1]

be distinguished triangles, and let f : A ⟶ B be given so that e_{a}f[1] = e_{b}. Then, there is a mapping g : C ⟶ D so that the following two properties are fulfilled:

(1) (f, g, id_{E}) is a mapping of triangles

A ⟶^{a} C ⟶ E ⟶ A[1]
↓ f ↓ g ∥ ↓ f[1]
B ⟶^{b} D ⟶ E ⟶ B[1]

(2) and

A ⟶^{(f,a)} B ⊕ C \underrightarrow{\binom{b}{−g}} D ⟶ A[1]

is a distinguished triangle for some mapping D ⟶ A[1].

3. Transitivity

In [9] Zwara showed transitivity of the relation defined algebraically on isomorphism classes of A-modules. In this section we shall adapt his proof to our situation of triangulated categories.

Proposition 2. Let T be a triangulated category in which idempotent morphisms split. Suppose that for any object X of T the endomorphism ring End_{T} (X) is an artinian ring. Then, the relation ≤_{Δ} is a reflexive and transitive relation on isomorphism classes in T .

Proof. It is clear that ≤_{Δ} is reflexive, since

X ⟶^{id} X ⊕ 0 ⟶ 0 ⟶ X[1]

is a distinguished triangle.

We want to show that ≤_{Δ} is transitive. Let M, W, N so that M ≤_{Δ} W and W ≤_{Δ} N. Then, there are distinguished triangles

W ⟶^{ω} M ⊕ Z ⟶ Z ⟶ W [1] and N ⟶ W ⊕ Z_{1} \underrightarrow{\binom{g}{f}} Z_{1} ⟶ N[1]

for some objects Z and Z_{1} in T .

First, we shall show that we may assume that f is nilpotent.

Let A = End(Z_{1}). By the Wedderburn-Artin theorem we can write

A/rad(A) ≃ ⊕_{i=1}^{L}Mat_{n_{i}}(Δ_{i}),

where Δ_{i} are division rings for all i ∈ {1, …, L}.

We consider the image \overline{f} ∈ A/rad(A) of f. Suppose that \overline{f} ≠ 0. We write \overline{f} = (\overline{f}_{i})_{i}, where \overline{f}_{i} ∈ Mat_{n_{i}}(Δ_{i}).

By Gauss’ algorithm, for each i there exist invertible elements \overline{g}_{i} and \overline{h}_{i} of Mat_{n_{i}}(Δ_{i}) such that \overline{g}_{i}\overline{f}_{i}\overline{h}_{i} is idempotent with units or zeroes on the diagonal and zero elsewhere. Since rad(A) is nilpotent, we can find an idempotent e ∈ A so that e maps to the idempotent \overline{e} = (\overline{g}_{i}\overline{f}_{i}\overline{h}_{i})_{i} of A/rad(A) (cf e.g. [2, Theorem 1.7.3]). Let G and H be elements of A such that the images \overline{G} and \overline{H} in A/rad(A) are \overline{G} = (\overline{g}_{i})_{i} and \overline{H} = (\overline{h}_{i})_{i}, respectively. Then G and H are invertible and GfH − e ∈ rad(A).

Let F := GfH. We have the following morphism of triangles.

N ⟶ W ⊕ Z_{1} \underrightarrow{\binom{g}{f}} Z_{1} ⟶ N[1]
↓  ↓  ↓  ↓
N ⟶ W ⊕ Z_{1} \underrightarrow{\binom{gH}{F}} Z_{1} ⟶ N[1],

where in the diagram the endomorphism of W ⊕ Z_{1} is ( \begin{matrix}1 & 0 \\ 0 & G^{−1} \end{matrix} ) and the endomorphism

of Z_{1} is H. So we get a triangle N⟶W ⊕ Z_{1} \underrightarrow{\binom{gH}{F}} Z_{1} ⟶ N[1]. We will show that eF e has a right inverse.

Since e is idempotent and since idempotents split in T , there is a decomposition Z_{1} = Z_{(1)} ⊕ Z_{(2)} such that e preserves the decomposition, is identity on Z_{(1)} and 0 on Z_{(2)}. Moreover, we have eF e ∈ End(Z_{(1)}) and eF e − e ∈ rad(End(Z_{(1)})). By [2, Lemma 1.2.2] we know that eF e has a right inverse in End(Z_{(1)}), which we denote by F′. Applying the octahedral axiom on the following diagram of triangles,

[figure: W ⊕ Z_{(1)} ⊕ Z_{(2)}; λ; Z_{(1)} ⊕ Z_{(2)}; N[1]; Z_{(1)}; σ; Z_{(2)}[1]; μ; X]

where λ = ( \begin{matrix}gH \\ F \end{matrix} ) = ( \begin{matrix}g′ & g′′ \\ eF e & eF e′ \\ e′F e & e′F e′ \end{matrix} ), σ = ( \begin{matrix}1 \\ 0 \end{matrix} ) and μ = ( \begin{matrix}g′ \\ eF e \\ e′F e \end{matrix} ) , we get a triangle

N ⟶ X[−1] ⟶ Z_{(2)} ⟶ N[1].

We have ( \begin{matrix}1 & g′F′ & 0 \\ 0 & 1 & 0 \\ 0 & e′F eF′ & 1 \end{matrix} ) ∈ Aut_{T} (W ⊕ Z_{(1)} ⊕ Z_{(2)}) and F′ ∈ Aut_{T} (Z_{(1)}) making the

left most square of the following diagram commutative.

W ⊕ Z_{(1)} ⊕ Z_{(2)} ⟶^{μ} Z_{(1)} ⟶  X  ⟶ (W ⊕ Z_{(1)} ⊕ Z_{(2)})[1]
↓  ↓  ↓
W ⊕ Z_{(1)} ⊕ Z_{(2)} ⟶^{(0,1,0)^{t}} Z_{(1)} ⟶ (W ⊕ Z_{(2)})[1] ⟶ (W ⊕ Z_{(1)} ⊕ Z_{(2)})[1],

Since the vertical mappings are isomorphisms, they induce an isomorphism X ≅ (W ⊕ Z_{(2)})[1].

So the above triangle becomes a triangle

N ⟶ W ⊕ Z_{(2)} ⟶ Z_{(2)} ⟶ N[1].

Since Z_{(2)} is a direct summand of Z_{1}, by induction on the length of End(Z_{1}) we can assume that f is nilpotent.

We continue the proof, showing that the relation is transitive. We get a factorisation

N ⟶ W ⊕ Z_{1} ⟶ M ⊕ Z ⊕ Z_{1}.

This gives

[figure: N; W ⊕ Z_{1}; Z_{1}; ^{\binom{g}{f}}; (M ⊕ Z) ⊕ Z_{1}; Z; Z_{2}]

where the triangle

Z_{1} ⟶^{ι_{1}} Z_{2} ⟶ Z ⟶ Z_{1}[1]

is given by the octahedral axiom. It follows that for the mapping

(M ⊕ Z) ⊕ Z_{1} \underrightarrow{\binom{g_{1}}{f_{1}}} Z_{2}

one has fι_{1} = f_{1}.

We abbreviate M′ := M ⊕ Z and have a new factorisation

N ⟶ M′ ⊕ Z_{1} \underrightarrow{( \begin{matrix}id_{M′} & 0 \\ 0 & ι_{1} \end{matrix} )} M′ ⊕ Z_{2} .

This gives again a diagram

[figure: N; M′ ⊕ Z_{1}; Z_{2}; ^{\binom{g_{1}}{f_{1}}}; M′ ⊕ Z_{2}; Z; Z_{3}]

where the triangle

Z_{2} ⟶^{ι_{2}} Z_{3} ⟶ Z ⟶ Z_{2}[1]

is given by the octahedral axiom. More generally we define inductively complexes Z_{ℓ}, distinguished triangles

Z_{ℓ} ⟶^{ι_{ℓ}} Z_{ℓ+1} ⟶ Z ⟶ Z_{ℓ}[1]

and mappings \binom{g_{ℓ}}{f_{ℓ}} : M′ ⊕ Z_{ℓ} ⟶ Z_{ℓ+1} given by the factorisation

N ⟶ M′ ⊕ Z_{ℓ−1} \underrightarrow{( \begin{matrix}id_{M′} & 0 \\ 0 & ι_{ℓ−1} \end{matrix} )} M′ ⊕ Z_{ℓ}

which induce the diagram

[figure: N; M′ ⊕ Z_{ℓ−1}; Z_{ℓ}; ^{\binom{g_{ℓ−1}}{f_{ℓ−1}}}; M′ ⊕ Z_{ℓ}; Z; Z_{ℓ+1}]

Hence, we get a sequence of mappings (f_{ℓ}, f_{ℓ+1}, id_{Z}) of distinguished triangles

Z_{ℓ}  ⟶^{f_{ℓ}} Z_{ℓ+1}  ⟶^{f_{ℓ+1}}  Z_{ℓ+2}
↓ ι_{ℓ}  ↓ ι_{ℓ+1}  ↓ ι_{ℓ+2}
Z_{ℓ+1} ⟶^{f_{ℓ+1}} Z_{ℓ+2}  ⟶^{f_{ℓ+2}}  Z_{ℓ+3}
↓  ↓  ↓
Z  =  Z  =  Z
↓  ↓  ↓
Z_{ℓ}[1] ⟶^{f_{ℓ}[1]} Z_{ℓ+1}[1] ⟶^{f_{ℓ+1}[1]} Z_{ℓ+2}[1]

Since the composition of morphism of triangles is a morphism of triangles, we get a morphism of distinguished triangles

Z_{1}  ⟶^{f_{1}f_{2}...f_{k}}  Z_{k+1}
↓ ι_{1}  ↓ ι_{k+1}
Z_{2}  ⟶^{f_{2}f_{3}...f_{k+1}} Z_{k+2}
↓  ↓
Z = Z
↓  ↓
Z_{1}[1] ⟶^{f_{1}f_{2}...f_{k}[1]} Z_{k+1}[1]

Now, since f_{1} = ι_{1}f and since ι_{ℓ}f_{ℓ+1} = f_{ℓ}ι_{ℓ+1}, we get f_{1}f_{2} … f_{k} = f^{k}ι_{1}ι_{2} … ι_{k}. Since f is nilpotent we get f_{1}f_{2} … f_{k} = 0 for some k. Using Proposition 1 we get a triangle

Z_{1} \underrightarrow{\binom{0}{ι_{1}}} Z_{k+1} ⊕ Z_{2} ⟶ Z_{k+2} ⟶ Z_{1}[1] .

Now,

Z_{k+2} ≃ cone(\binom{0}{ι_{1}}) ≃ Z_{k+1} ⊕ cone(ι_{1}) ≃ Z_{k+1} ⊕ Z

Since

N ⟶ (M ⊕ Z) ⊕ Z_{k+1}⟶Z_{k+2} ⟶ N[1]

is a triangle, we get M ≤_{Δ} N as required.

4. Anti-symmetry

Proposition 3. Let R be a commutative ring and let T be an R-linear triangulated category. Suppose that X is an object in T so that Hom_{T} (X, Y) is an R-module of finite length length_{R}(Hom(X, Y)) for all Y in T . Then,

N ≤_{Δ} M ⇒ length_{R}(Hom(X, N[j])) ≤ length_{R}(Hom(X, M[j]))

for any integer j.

If X is an object in T so that Hom_{T} (Y, X) is an R-module of finite length for all Y in T , then

N ≤_{Δ} M ⇒ length_{R}(Hom(N[j], X)) ≤ length_{R}(Hom(M[j], X))

for any integer j.

Proof. Let N ≤_{Δ} M. Then, there is an object Z of T so that

M ⟶ N ⊕ Z ⟶ Z ⟶ M[1]

is a distinguished triangle. Apply Hom_{T} (X, −) to this triangle. Abbreviating (X, −) := Hom_{T} (X, −) we get a long exact sequence

... ⟶ (X, M[j]) ⟶ (X, N[j]) ⊕ (X, Z[j]) ⟶ (X, Z[j]) ⟶ ...

For any j one gets

length_{R}(Hom_{T} (X, M[j])) + length_{R}(Hom_{T} (X, Z[j]))
≥ length_{R}(Hom_{T} (X, N[j]) ⊕ Hom_{T} (X, Z[j]))

and so,

length_{R}(Hom_{T} (X, M[j])) ≥ length_{R}(Hom_{T} (X, N[j]))

as soon as length_{R}(Hom_{T} (X, Z[j])) and length_{R}(Hom_{T} (X, M[j]) are finite.

The dual statement follows by the dual arguments, applying Hom_{T} (−, X) to the triangle. This shows the proposition.

Suppose that M ≤_{Δ} N and N ≤_{Δ} M. Then, we know by Proposition 3 that for any object X in T so that for all objects Y one has that Hom_{T} (X, Y) and Hom_{T} (Y, X) are of finite length over R, one has

length_{R}(Hom_{T} (X, N[j])) = length_{R}(Hom_{T} (X, M[j]))

and

length_{R}(Hom_{T} (N[j], X)) = length_{R}(Hom_{T} (M[j], X))

for any integer j.

Proposition 4. Let R be a commutative ring and let T be an R-linear triangulated category in which idempotent morphisms split and so that for any two objects X and Y of T the set Hom_{T} (X, Y) is of finite length as an R-module. Suppose that M and N are two objects in T so that there is n ∈ ℤ \ {0} satisfying Hom_{T} (M, N[n]) = 0.

If for any object X of T one has that the length of Hom_{T} (M, X) as R-module equals the length of Hom_{T} (N, X) as R-module, then M ≃ N.

Remark 4.1. • Bongartz showed a similar result for an abelian category [3]. We see

that his proof can be modified so that it applies to our situation as well.

• If we assume that Hom_{T} (M, M[n]) vanishes instead of Hom_{T} (M, N[n]) the conclusion of Proposition 4 still holds. The proof is similar.

Proof. Suppose M and N are objects satisfying the above hypothesis. Since Hom_{T} (M, N) is of finite length over R, take generators f_{1}, f_{,} …, f_{ℓ} of Hom_{T} (M, N) as an R-module. Let f := (f_{1}, f_{2}, …, f_{ℓ})^{tr} : M^{l} ⟶ N. We show that f is split.

So, f induces an epimorphism

f^{∗} := Hom_{T} (M, f) : Hom_{T} (M, M^{ℓ}) ⟶ Hom_{T} (M, N).

Let K be the cone of f so that M^{ℓ} ⟶^{f} N ⟶ K ⟶ M^{ℓ}[1] is a distinguished triangle. Then,

(M, M^{ℓ}[−j]) ⟶ (M, N[−j]) ⟶ … ⟶ (M, K[−1]) ⟶ (M, M^{ℓ}) ⟶ (M, N) ⟶ (M, K) ⟶ … ⟶ (M, M^{ℓ}[j]) ⟶ (M, N[j])

and

(N, M^{ℓ}[−j]) ⟶ (N, N[−j]) ⟶ … ⟶ (N, K[−1]) ⟶ (N, M^{ℓ}) ⟶ (N, N) ⟶ (N, K) ⟶ … ⟶ (N, M^{ℓ}[j]) ⟶ (N, N[j])

are both exact. By construction (M, M^{ℓ}) ⟶ (M, N) is surjective. Hence,

(M, M^{ℓ}[−j]) ⟶ (M, N[−j]) ⟶ … ⟶ (M, K[−1]) ⟶ (M, M^{ℓ}) ⟶ (M, N) ⟶ 0

and

0 ⟶ (M, K) ⟶ … ⟶ (M, M^{ℓ}[j]) ⟶ (M, N[j])

are both exact.

Suppose Hom_{T} (M, N[n]) = 0 and let j = |n|. If n < 0, then one gets that

0 ⟶ (M, K[n + 1]) ⟶ … ⟶ (M, K[−1]) ⟶ (M, M^{ℓ}) ⟶ (M, N) ⟶ 0

is exact. Since the R-length of Hom_{T} (M, N[n]) equals the R-length of Hom_{T} (N, N[n]), also Hom_{T} (N, N[n]) = 0 and

0 ⟶ (N, K[n + 1]) ⟶ … ⟶ (N, K[−1]) ⟶ (N, M^{ℓ}) ⟶ (N, N)

is exact. Since now the length over R of the corresponding terms coincide for (N, −) and for (M, −), counting the alternate sum of R-lengths, we see that

0 ⟶ (N, K[n + 1]) ⟶ … ⟶ (N, K[−1]) ⟶ (N, M^{ℓ}) ⟶ (N, N) ⟶ 0

is exact. If n > 0, then

0 ⟶ (M, K) ⟶ … ⟶ (M, M^{ℓ}[n]) ⟶ 0

is exact, and since the R-lengths of (M, N[i]) equals the R-lengths of (N, N[i]), also

(N, K) ⟶ … ⟶ (N, M^{ℓ}[n]) ⟶ 0

is exact. But then, counting again the alternate sum of the R-lengths, one gets that

0 ⟶ (N, K) ⟶ … ⟶ (N, M^{ℓ}[n]) ⟶ 0

is exact and consequently

(N, K[−n + 1]) ⟶ … ⟶ (N, K[−1]) ⟶ (N, M^{ℓ}) ⟶ (N, N) ⟶ 0

is exact.

Hence, in any case, f is split and N is a direct factor of M^{n}. The hypothesis on T implies that the Krull-Schmidt theorem holds in T [1, Ch 1, Theorem 3.6]. We see that therefore N and M have an indecomposable non zero direct factor U in common. Let N = U ⊕ N′ and M = U ⊕ M′. By induction on the R-length of Hom(M, M) we are finished.

Corollary 5. Let T be a triangulated category satisfying the assumptions in Proposition 4. Then the relation ≤_{Δ} is an anti-symmetric relation on the isomorphism classes of T .

Proof. Let M ≤_{Δ} N and N ≤_{Δ} M. Then, Proposition 3 shows that the hypothesis of Proposition 4 are fulfilled and therefore N ≃ M. □

5. The main result

We are now ready to formulate our main result.

Theorem. Let R be a commutative ring and let T be an R-linear triangulated category satisfying the following three conditions.

• For any two objects X and Y of T the set Hom_{T} (X, Y) is of finite length as an R-module.

• For any two objects X and Y in T there is an integer n_{XY} ∈ ℤ \ {0} so that Hom_{T} (X, Y [n_{X,Y}]) = 0,

• Idempotent morphisms in T split. Then, the relation ≤_{Δ} defines a partial order relation on the set of isomorphism classes

of objects in T .

Proof. The fact that ≤_{Δ} is reflexive and transitive is Proposition 2. The anti-symmetry is Corollary 5.

Remark 5.1. Examples for triangulated categories T satisfying the hypotheses of Theorem 5 are

• the bounded derived category D^{b}(A) of finitely generated A-modules over an artinian R-algebra A,

• the bounded derived category of coherent sheaves over a projective variety.

References

[1] Hyman Bass, Algebraic K-theory. W. A. Benjamin, Inc., New York-Amsterdam 1968.

[2] David J. Benson, Representations and cohomology I, Cambridge University Press 1990.

[3] Klaus Bongartz, A generalization of a result of M. Auslander, Bulletin of the London Mathematical Society 21 (1989) 255-256.

[4] Bernt Tore Jensen, Xiuping Su and Alexander Zimmermann, Degenerations for derived categories, to appear in Journal of Pure and Applied Algebra.

[5] Amnon Neeman, Triangulated categories, Annals of Mathematics Studies 148; Princeton, NJ, 2001 [6] Liangang Peng and Youjun Tan, Derived categories, tilted algebras, and Drinfel’d doubles. J. Algebra

266 (2003), no. 2, 723–748.

[7] Christine Riedtmann, Degenerations for representations of quivers with relations, Annales scientifiques de l’École Normale Supérieure 4^{ème} série 19 (1986) 275-301.

[8] Richard G. Swan, Projective modules over group rings and maximal orders, Annals of Mathematics 76 (1962) 55-61.

[9] Grzegorz Zwara, A degeneration-like order for modules, Archiv der Mathematik 71 (1998) 437-444.

[10] Grzegorz Zwara, Degenerations of finite-dimensional modules are given by extensions, Compositio Mathematica 121 (2000) 205-218.

B.T.J: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, UK

  E-mail address: bjensen@maths.leeds.ac.uk

X.S and A.Z.: Université de Picardie, Faculté de Mathématiques et UMR 6140 du CNRS, 33 rue St Leu, F-80039 Amiens Cedex 1, France

  E-mail address: xiuping.su@u-picardie.fr and alexander.zimmermann@u-picardie.fr