HOMOLOGICAL CONJECTURES IN REPRESENTATION THEORY OF FINITE-DIMENSIONAL ALGEBRAS

DIETER HAPPEL

Let A be a finite-dimensional k-algebra over an algebraically closed field k. We denote by mod A the category of finitely generated left A-modules. For an A-module AX we denote by pd_{A} X (resp. id_{A} X) the projective (resp. injective) dimension of X. With D = Hom_{k}(−, k) we denote the standard duality with respect to the ground field. Then _{A}D(A_{A}) is an injective cogenerator for mod A. To formulate some of the homological conjectures we need some more notation. Let _{A}I ⊂ mod A be the full subcategory containing the finitely generated injective A-modules. Let K^{b}(_{A}I) be the homotopy category of bounded complexes over _{A}I. Let D^{b}(A) be the derived category of bounded complexes over mod A. We consider K^{b}(_{A}I) as a full subcategory of D^{b}(A).

We define K^{b}(_{A}I)^{⊥} = {X ∈ D^{b}(A) | Hom(I, X) = 0 for all I ∈ K^{b}(_{A}I)}.

The following is the well-known hierarchy of some of the homological conjectures:

(1) Auslander Conjecture: Let _{A}X be an A-module. There exists an integer n such that if Ext_{A}^{i}(X, Y) = 0 for i sufficiently large, then Ext_{A}^{i}(X, Y) = 0 for i ≥ n.

(2) Finitistic Dimension Conjecture: fd(A) = sup{pd_{A} X | pd_{A} X < ∞} is finite.

(3) Vanishing Conjecture: K^{b}(_{A}I)^{⊥} = 0

(4) Nunke Condition: For an A-module _{A}X there is i ≥ 0 such that Ext_{A}^{i}(_{A}D(A_{A}),_{A} X) ≠ 0.

(5) Generalized Nakayama Conjecture: For a simple module _{A}S there is i ≥ 0 such that Ext_{A}^{i}(_{A}D(A_{A}),_{A}S) ≠ 0.

(6) Nakayama Conjecture: If in a minimal injective resolution of _{A}A

0 → _{A}A → I_{0} → I_{1} → ⋯

all I_{j} are projective, then A is a selfinjective algebra. Note that there is a similar hierarchy of dual conjectures. But it is not clear

that if A satisfies one of the conjectures (1) up to (5), then A satisfies the dual conjecture.

In the first section we briefly recall the well-known relationship between these conjectures. In the second section we report about some of the recent investigations on these conjectures. In the third section we will announce some reduction techniques using triangulated categories. For the proofs we refer to the references given.

1991 Mathematics Subject Classification. 16E10,18E30.

Key words and phrases. finitistic dimension, Nakayama conjecture, derived categories.

This article is an expanded version of a lecture given in October 1990 at the Université de Sherbrooke. I would like to thank I. Assem for the invitation and kind hospitality during this visit.

1. Elementary remarks

Let us also introduce

fd′(A) = sup{id_{A} X | id_{A} X < ∞}

It is easy to construct examples such that fd(A) ≠ fd′(A). In fact consider the family of algebras A_{n} for n ∈ ℕ given as the quiver algebra of

[figure: 1; 2; α_{n}; 3; α_{n−1}; α_{n−2}; ⋯; n − 1; α_{3}; n; α_{2}; α_{1}]

bound by α_{1}^{2} = α_{1}α_{2} = ⋯ = α_{n−1}α_{n} = 0. Then it is easy to see that fd(A) = n−1 and fd′(A) = 0.

Recall the following fact.

Lemma. Let A be a finite-dimensional algebra. Then fd(A) = 0 if and only if Hom_{A}(D(A_{A}), S) ≠ 0 for all simple A-modules S.

Proof. In fact, let S be a simple A-module with Hom_{A}(D(A_{A}), S) = 0, then pd_{A} τ^{−}S = 1, where τ^{−} denotes the Auslander-Reiten translation [AR2]. Conversely assume that fd(A) > 0. Then there is a module Y with pd_{A} Y = 1, hence Hom_{A}(D(A_{A}), τY) = 0. Thus there exists a simple S with Hom_{A}(D(A_{A}), S) = 0.

Let us recall the relationship of the conjectures mentioned above. It is shown in [H4] that (2) implies (3) and that (3) implies (4). The implication (4) to (5) is clear, and that (5) implies (6) was observed in [AR1]. The following relationship between (1) and (2) is due to Auslander [A2]. For the convenience of the reader we provide a proof. By A^{op} we denote the opposite algebra.

Proposition. Let A be a finite-dimensional algebra. If the Auslander conjecture holds for the enveloping algebra A^{e} = A ⊗_{k} A^{op}, then the finitistic dimension conjecture holds for A.

Proof. This follows from the following identity for Hochschild-cohomolgy (see for example [CE], IX, 4.4). Let X, Y be A-modules. Then

H^{i}(A, Hom_{k}(X, Y)) ≃ Ext_{A}^{i}(X, Y)

where H^{i}(A,_{A}M_{A}) = Ext_{A^{e}}^{i}(A, M) for a bimodule M. Let X be an A-module with pd_{A} X < ∞ and Y an arbitrary A-module. The identities above show that then Ext_{A^{e}}^{i}(A, Hom_{k}(X, Y)) = 0 for i sufficiently large. By assumption there is n such that if Ext_{A^{e}}^{i}(A, Hom_{k}(X, Y)) = 0 for i sufficiently large then we know that Ext_{A^{e}}^{i}(A, Hom_{k}(X, Y)) = 0 for i ≥ n. We infer that pd_{A} X ≤ n. Thus fd(A) ≤ n.

The following is a most probably incomplete list of references for the early treatment of these and related questions [A1], [AB1], [AB2], [AR1], [B1], [B2], [J1], [J2], [Na], [Nu], [Mu], [Se], [Sm], [T1], [T2].

There exist also conjectures dealing with infinite-dimensional modules, see [B1], [ZH4]. We refer to [ZH3] and [ZH4] for some recent interesting developments.

2. Special classes of algebras

In recent years some of the conjectures above have been verified for particular classes of finite-dimensional algebras.

2.1 Monomial algebras. The results given here are obtained in [GKK] and [IZ].

Let \overrightarrow{Δ} be a finite quiver and let k\overrightarrow{Δ} be the path algebra of \overrightarrow{Δ} over k. Let I be an ideal in k\overrightarrow{Δ} generated by paths of length at least two and containing all paths of length s for some integer s. Then A = k\overrightarrow{Δ}/I is called a monomial algebra. It has been shown in [GHZ] that projective resolutions of monomial algebras are rather well-behaved. This can also be seen by the following remarkable property of monomial algebras (compare [ZH2] or [ZH4]). If w is a path in \overrightarrow{Δ} we denote by w̄ the residue class in A. Clearly there are only finitely many w such that w̄ ≠ 0. We denote by M(w) = Aw̄ the A-module generated by w̄.

Proposition. Let A be a monomial algebra and f: P → Q a map between projective A-modules. Then Ker f = ⨁_{w} M(w).

The following was shown in [GKK] and [IZ].

Theorem. Let A be a monomial algebra. Then fd(A) < ∞.

In 2.3 we will give some remarks about the proof in [IZ]. For further investigations on homological properties of monomial algebras we refer to [ZH2].

2.2 Algebras with vanishing conditions. We follow the articles of [GZH] (see also [FS] and [ZH1]) and [DH].

For a finite-dimensional algebra A we denote by J the Jacobson radical of A.

Theorem. Let A be a finite-dimensional algebra with J^{3} = 0 then fd(A) < ∞.

The main idea of the proof in [GZH] is that the projective resolutions of modules of Loewy-length two can be controlled by a linear map which can be shown to be nilpotent.

Methods of linear algebra are also used in the next result from [DH].

Theorem. Let A be a finite-dimensional algebra such that there is an integer s with J^{2s+1} = 0 and A/J^{s} representation-finite. Then the generalized Nakayama conjecture holds for A.

We point out that variations of the proof can be used to verify the generalized Nakayama conjecture for other classes of algebras satisfying suitable conditions. For details we refer to [DH].

2.3 Finiteness conditions. It is trivial that all these conjectures hold for a representation-finite (i.e. there are only finitely many indecomposable modules up to isomorphism) algebra. There are some concepts which generalize this.

Let _{A}X be an A-module and let

0 → X → I_{0} ⟶^{μ_{o}} I_{1} ⟶^{μ_{1}} I_{2} ⋯

be a minimal injective resolution. Set Ω^{−i}X = Ker μ_{i} for 0 ≤ i < ∞. A module _{A}X is called cosyzygy-finite or ultimately closed (compare [J1]) if {Ω^{−i}X, i ≥ 0} ⊆ add Y for an A-module Y, where add Y is the additive category of direct sums of direct summands of Y. We refer to [CF] for some related notions. The following result is due to [IZ].

Proposition. Let A be a finite-dimensional algebra such that A/J is cosyzygyfinite, then fd(A) < ∞.

Proof. Let S be a simple A-module. Since S is cosyzygy-finite there is an integer r_{S} such that Ω^{−r_{S}}S ∈ add(⨁_{i<r_{S}} Ω^{−i}S). Choose r_{S} minimal with this property and let r = max {r_{S} | S simple}. Note that for a simple module S which satisfies id_{A} S = s < ∞ we have that r > s. If fd(A) = ∞ there is an A-module X with ∞ > pd_{A} X = t + 1 > r. So there is a simple S with Ext_{A}^{t+1}(X, S) ≠ 0. But then id_{A} S = ∞ by the remark above.

Now Ext_{A}^{t+1}(X, S) ≃ Ext_{A}^{1}(X, Ω^{−t}S) shows that there is an indecomposable direct summand Y_{t} of Ω^{−t}S such that Ext_{A}^{1}(X, Y_{t}) ≠ 0. By the choice r there is an integer m < r such that Y_{t} is a direct summand of Ω^{−m}S. But then Y_{t} is a direct summand of Ω^{−s}S for infinitely many s. Thus 0 ≠ Ext_{A}^{1}(X, Ω^{−s}S) ≃ Ext_{A}^{s}(X, S) for infinitely many s, in contrast to pd_{A} X < ∞.

The proof of the result in 2.1 is then obtained by showing that for a monomial algebra A the simple modules are cosyzygy-finite (see 2.1).

We point out that a similar proof shows that fd(A) < ∞ if _{A}A is cosyzygy-finite. It is easy to construct examples where the simples do not have this property. In

fact, let A = k[x, y]/(x^{2}, y^{2}). Then the unique simple A-module S is not cosyzygyfinite.

A quite different approach was taken in [AR3] (see also [AR4]) while using the concept of contravariantly finite subcategories as introduced in [AS1] and [AS2]. We recall the relevant notions. Let D be a full subcategory of mod A. It is always assumed to be closed under direct sums, direct summands and isomorphisms. The subcategory D is called contravariantly finite in mod A if every X ∈ mod A has a right D-approximation, i.e. there is a morphism F_{X} → X with F_{X} ∈ D such that the induced morphism Hom_{A}(D, F_{X}) → Hom_{A}(D, X) is surjective for all D ∈ D.

If the A-module X admits a right D-approximation, then X clearly admits a minimal right D-approximation (i.e. a right approximation of minimal length).

The subcategory D is called resolving if D is closed under extensions, kernels of surjective maps and contains _{A}A. Note that for a contravariantly finite subcategory which is resolving every right approximation is surjective [AR3].

Theorem. For a contravariantly finite resolving subcategory D the objects in D consist of the summands of modules which have a filtration with composition factors the minimal right-approximations of the simple modules.

An easy but important observation is that the subcategory

P(A) = {_{A}X | pd_{A} X < ∞}

is a resolving subcategory.

Corollary. If P(A) is contravariantly finite, then fd(A) < ∞.

There are examples of algebras A such that P(A) is not contravariantly finite [ITS].

A different sort of finiteness condition can be obtained as follows. Consider the following subcategory S of mod A.

S = {_{A}X | pd_{A} X < ∞, each proper submodule U satisfies pd_{A} U = ∞ }

Proposition. If the length of the indecomposable modules in S is bounded, then fd(A) < ∞.

Proof. We sketch the proof using some ideas from [Sc]. Clearly it is enough to show that the projective dimension of modules in S is bounded. For a given integer r we consider the algebraic variety M_{r}(A) of A-modules of length r. Consider the subset P_{t} ⊆ M_{r}(A) formed by those modules X which satisfy pd_{A} X ≤ t. As in [Sc] it follows that P_{t} is an open subset in the Zariski-topology. So we obtain an ascending chain of open subsets P_{0} ⊆ P_{1} ⊆ P_{2} ⊆ ⋯ which has to become stationary, since M_{r}(A) is finite-dimensional. So there exists t_{r} with pd_{A} X ≤ t_{r} for all _{A}X ∈ M_{r}(A).

By assumption there is an integer m such that the length of indecomposable modules in S is bounded by m. Hence fd(A) ≤ max {r_{t} | 1 ≤ t ≤ m}, which shows the assertion.

2.4 Bounds for the finitistic dimension. An interesting question is to obtain good bounds for the finitistic dimension. In the cases discussed above this problem was solved. Let us list the corresponding results.

Let A be a monomial algebra. Consider the following set W of paths in \overrightarrow{Δ}.

W = {w | w̄ ≠ 0, pd_{A} M(w) < ∞}

Let r = max {pd_{A} M(w) | w ∈ W} if W is non-empty and r = −1 if W is empty. The result in 2.1 clearly has the following application.

Corollary. Let A be a monomial algebra. Then fd(A) ≤ r + 2.

We refer to [ZH4] for a slightly different definition of W to obtain a sharper bound.

In [GZH] the following bound for fd(A) was found for an algebra A with J^{3} = 0. For this let S_{1}, …, S_{n} be a complete set of simple A-modules. We may assume that there is m such that pd_{A} S_{i} = ∞ for 1 ≤ i ≤ m and that pd_{A} S_{i} < ∞ for m < i ≤ n. Let d = max {pd_{A} S_{i} | m < i ≤ n}.

Proposition. Let A be a finite-dimensional algebra with vanishing radical cube. Then fd(A) ≤ 2m + d + 1.

In the situation of 2.3 the theorem gives the following bound for fd(A) if P(A) is contravariantly finite. Again let S_{1}, …, S_{n} be a complete set of simple A-modules. Let F_{i} ∈ P(A) be the minimal right P(A)-approximation of S_{i}.

Corollary. If P(A) is a contravariantly finite subcategory. Then fd(A) ≤ max{pd_{A} F_{i} | 1 ≤ i ≤ n}.

3. Reduction techniques

The following is a summary of results in [H3], [H4] and [H6].

For the convenience of the reader we recall some of the terminology for complexes which we have to use.

Let 𝔞 be an arbitrary additive subcategory of mod A.

A complex X^{•} = (X^{i}, d_{X}^{i})_{i∈ℤ} over 𝔞 is a collection of objects X^{i} from 𝔞 and morphisms d^{i} = d_{X}^{i}: X^{i} → X^{i+1} such that d^{i}d^{i+1} = 0. A complex X^{•} = (X^{i}, d_{X}^{i}) is bounded below if X^{i} = 0 for all but finitely many i < 0. It is called bounded above if X^{i} = 0 for all but finitely many i > 0. It is bounded if it is bounded below and bounded above. It is said to have bounded cohomology if H^{i}(X^{•}) = 0 for all but finitely many i ∈ ℤ , where by definition H^{i}(X^{•}) = Ker d_{X}^{i}/ Im d_{X}^{i−1}. Denote by C(𝔞) the category of complexes over 𝔞, by C^{−,b}(𝔞) (resp.C^{+,b}(𝔞), resp. C^{b}(𝔞)) the full subcategories of complexes bounded above with bounded cohomology (resp. bounded below with bounded cohomology, resp. bounded above and below).

If X^{•} = (X^{i}, d_{X}^{i})_{i∈ℤ} and Y^{•} = (Y^{i}, d_{Y}^{i})_{i∈ℤ} are two complexes, a morphism f^{•}: X^{•} → Y^{•} is a sequence of morphisms f^{i}: X^{i} → Y^{i} of 𝔞 such that

d_{X}^{i}f^{i+1} = f^{i}d_{Y}^{i}

for all i ∈ ℤ. The translation functor is defined by

(X^{•}[1])^{i} = X^{i+1} , (d_{X[1]})^{i} = −(d_{X})^{i+1} .

The mapping cone C_{f^{•}} of a morphism f^{•}: X^{•} → Y^{•} is the complex

C_{f^{•}} = ((X^{•}[1])^{i} ⊕ Y^{i}, d_{C_{f}}^{i})

with ’differential’

d_{C_{f}}^{i} = (\begin{matrix}−d_{X}^{i+1} & f^{i+1} \\ 0 & d_{Y}^{i} \end{matrix}) .

We denote by K^{−,b}(𝔞), K^{+,b}(𝔞) and K^{b}(𝔞) the homotopy categories of the categories of complexes introduced above. Note that all these categories are triangulated categories in the sense of [V].

Recall that two morphisms f^{•}, g^{•}: X^{•} → Y^{•} are called homotopic, if there exist morphisms h^{i}: X^{i} → Y^{i−1} such that f^{i} − g^{i} = d_{X}^{i}h^{i+1} + h^{i}d_{Y}^{i−1} for all i ∈ ℤ.

We have denoted by _{A}P (resp. _{A}I) the full subcategory of mod A formed by the projective (resp. injective) A-modules. Then we identify the derived category D^{b}(A) of bounded complexes over mod A with K^{−,b}(_{A}P) or with K^{+,b}(_{A}I). In case A has finite global dimension this yields the identification of D^{b}(A) with K^{b}(_{A}P) or with K^{b}(_{A}I), since the natural embedding of K^{b}(_{A}P) into K^{−,b}(_{A}P) is an equivalence in this case. We identify the derived category D^{−}(A) of complexes bounded above over mod A with K^{−}(_{A}P) and we identify the derived category D^{+}(A) of complexes bounded below over mod A with K^{+}(_{A}I). For a more detailed analysis of the derived category we refer to [G],[Gr] and [V].

3.1 Auslander-Reiten triangles. In [H1] we introduced the notion of an Auslander-Reiten triangle in a triangulated category. We first recall the relevant definitions.

Let C be a triangulated category such that Hom_{C}(X, Y) is a finite-dimensional k-vector space for all X, Y ∈ C and assume that the endomorphism ring of an indecomposable object is local. This assumption ensures that C is a Krull-Schmidt category. We denote by X[1] the value of the translation functor on the object X of C.

A triangle X ⟶^{u} Y ⟶^{v} Z ⟶^{w} X[1] in C is called an Auslander-Reiten triangle if the following conditions are satisfied: (AR1) X, Z are indecomposable, (AR2) w ≠ 0, (AR3) If f: W → Z is not a retraction, then there exists f′: W → Y such that

f′v = f. We will say that C has Auslander-Reiten triangles if for all indecomposable objects Z ∈ C there exists a triangle satisfying the conditions above.

Recall that a translation quiver \overrightarrow{Γ} = (Γ_{0}, Γ_{1}, τ) is given by a (locally finite) quiver (Γ_{0}, Γ_{1}) (Γ_{0} denotes the vertex set, Γ_{1} denotes the set of arrows) together with an injective map τ: Γ′_{0} → Γ_{0} defined on a subset Γ′_{0} ⊆ Γ_{0} such that for any z ∈ Γ′_{0}, and any y ∈ Γ_{0} the number of arrows from y to z is equal to the number of arrows from τz to y. The map τ is called the translation. If Γ′_{0} = Γ_{0} and τ is a bijection we say that \overrightarrow{Γ} is a stable translation quiver.

If the triangulated category C has Auslander-Reiten triangles then \overrightarrow{Γ}(C) has the structure of a translation quiver (see [H2]).

We refer to [H1] and [H2] for some properties of Auslander-Reiten triangles.

Theorem. Let A be a finite-dimensional k-algebra. Then the following are equivalent.

(i) pd_{A} D(A_{A}) < ∞ (ii) K^{b}(_{A}P) has Auslander-Reiten triangles.

We recall that the finitistic dimension conjecture implies the following for a finite-dimensional algebra A (see [AR3] or [H4]).

(∗) id_{A} A < ∞ if pd_{A} D(A_{A}) < ∞.

The theorem and its dual now imply.

Corollary. Let A be a finite dimensional algebra such that pd_{A} D(A_{A}) < ∞. If the translation τ on \overrightarrow{Γ}(K^{b}(_{A}P)) is surjective, then id_{A} A < ∞.

We refer to 3.2 for problems related to (∗).

3.2 Tilting invariance. A module T ∈ mod A is called a (generalized) tilting module if the following conditions are satisfied:

(i) pd_{A} T < ∞ (ii) Ext_{A}^{i}(T, T) = 0 for all i > 0 (iii) There is a long exact sequence 0 → _{A}A → T_{0} → ⋯ → T_{m} → 0 with

T_{j} ∈ add T.

A module _{A}T satisfying the properties (i) and (ii) is called a partial tilting module.

We refer to [H1],[H2] and [Mi] for an outline of tilting theory in this case and to [Ri1], [Ri2] and [Ri3] for the general notion of derived equivalence.

If T is a tilting module and B = End_{A} T. Then it is known that gl.dim A < ∞ if and only if gl.dim B < ∞. The next result is a generalization of this.

Theorem. Let A be a finite-dimensional algebra and T a tilting module. Let B = End_{A} T. Then fd(A) < ∞ if and only if fd(B) < ∞.

We now come back to the property (∗) in 3.1.

If _{A}M is an A-module we may decompose _{A}M = ⊕_{i=1}^{s}M_{i}^{n_{i}} with M_{i} indecomposable, M_{i} ≄ M_{j} for i ≠ j and n_{i} > 0. In this case we denote the number s of non-isomorphic indecomposable direct summands of M by δ(M).

It is easy to see that if _{A}T is a tilting module, then δ(T) = rk K_{0}(A) = n, where K_{0}(A) is the Grothendieck group of A.

It is not known that a partial tilting module which satisfies δ(T) = rk K_{0}(A) = n is a tilting module, unless pd_{A} T ≤ 1 by a result in [Bo]. It is shown in [RS] that this holds if A is representation-finite.

Note that (∗) is a special case of this problem.

3.3 Recollement. Let C, C′ and C′′ be triangulated categories. Following [BBD] (see also [GV]) a recollement of C relative to C′ and C′′ is given by

[figure: C′; i_{∗}; i_{!}; C; i^{∗}; i^{!}; j^{!}; j^{∗}; C′′; j_{!}; j_{∗}]

such that (RI) (i^{∗}, i_{∗}), (i_{!}, i^{!}), (j_{!}, j^{!}) and (j^{∗}, j_{∗}) are adjoint pairs of exact functors and that

i_{∗} = i_{!}, j^{!} = j^{∗} (RII) j^{∗}i_{∗} = 0 (RIII) i^{∗}i_{∗} ≃ id, id ≃ i^{!}i_{!}, j^{∗}j_{∗} ≃ id and id ≃ j^{!}j_{!} (RIV) For X ∈ C there are triangles

j_{!}j^{!}X → X → i_{∗}i^{∗}X → j_{!}j^{!}X[1]
i_{!}i^{!}X → X → j_{∗}j^{∗}X → i_{!}i^{!}X[1].

(The morphisms in (RIII) and (RIV) are the adjunction morphisms.)

We refer to [BBD] for properties of recollements and to [Kö] for necessary and sufficient conditions that D^{−}(A) has a recollement relative to D^{−}(A′) and D^{−}(A′′) for some finite-dimensional algebras A, A′, A′′.

In particular we mention the following result from [Kö]. If D^{−}(A) has a recollement relative to D^{−}(A′) and D^{−}(A′′) for some finite-dimensional algebras A, A′, A′′ and one of the algebras A, A′, A′′ has finite global dimension then D^{b}(A) has a recollement relative to D^{b}(A′) and D^{b}(A′′).

Theorem. Let A be a finite-dimensional algebra and assume that D^{b}(A) has a recollement relative to D^{b}(A′) and D^{b}(A′′) for some finite-dimensional algebras A′, A′′. Then fd(A) < ∞ if and only if fd(A′) < ∞ and fd(A′′) < ∞.

We point out that the theorem above generalizes a theorem in [W].

We give two examples in which this theorem may be applied. We stress that there exist proofs of these results avoiding the use of triangulated categories.

Let A′, A′′ be finite-dimensional algebras and let_{A′}M_{A′′} be a bimodule. Consider the triangular matrix algebra A of the form

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

with multiplication

(\begin{matrix}a′ & m \\ 0 & a′′ \end{matrix}) (\begin{matrix}b′ & m′ \\ 0 & b′′ \end{matrix}) = (\begin{matrix}a′b′ & a′m′ + mb′′ \\ 0 & a′′b′′ \end{matrix})

where a′, b′ ∈ A′, m, m′ ∈ M and a′′, b′′ ∈ A′′.

Then D^{−}(A) has a recollement relative to D^{−}(A′) and D^{−}(A′′).

Assume that A′ or A′′ has finite global dimension. Then D^{b}(A) has a recollement relative to D^{b}(A′) and D^{b}(A′′). In particular, fd(A) < ∞ if fd(A′) < ∞ and fd(A′′) < ∞.

In the next example we will use the concept of perpendicular categories as introduced in [GL], see also [H5].

Let X ∈ mod A with pd_{A} X ≤ 1. We define the right perpendicular category X^{⊥} to be the full subcategory of modA whose objects Z satisfy

Hom_{A}(X, Z) = 0 = Ext_{A}^{1}(X, Z).

It is straightforward to see that X^{⊥} is an abelian category, which is closed under extensions and that the inclusion functor X^{⊥} ↪ mod A is exact. The next result states some useful properties of X^{⊥} under additional assumptions. For the proof we refer to [GL] or [H5].

Theorem. Let X ∈ mod A such that pd_{A} X ≤ 1 and Ext_{A}^{1}(X, X) = 0, then there exists _{A}Q ∈ X^{⊥} such that X^{⊥} ≃ mod A_{0}, with A_{0} = End_{A} Q. If X is indecomposable, then rk K_{0}(A_{0}) = rk K_{0}(A) − 1.

Now assume that A admits a simple A-module S with pd_{A} S = 1. Then D^{−}(A) has a recollement relative to D^{−}(A′) and D^{−}(A′′), where A′ = End_{A} Q for a projective generator _{A}Q of S^{⊥} and A′′ = k. Since gl.dim k = 0 we infer that D^{b}(A) has a recollement relative to D^{b}(A′) and D^{b}(A′′). In particular, fd(A) < ∞ if fd(A′) < ∞.

3.4 Grothendieck groups. The generalized Nakayama conjecture is related to a problem about Grothendieck groups of triangulated categories. First we need a reformulation of the generalized Nakayama conjecture which is due to [AR1].

(5’): Let _{A}M be a generator for mod A with Ext_{A}^{i}(M, M) = 0 for all i > 0, then AM is projective.

The following is shown in [AR1]. The generalized Nakayama conjecture holds for all finite-dimensional algebras if and only if the conjecture (5’) holds for all finite-dimensional algebras.

Let us indicate one direction. We assume that (5’) holds for all finite-dimensional algebras with rk K_{0}(A) = n − 1 and we claim that the generalized Nakayama conjecture holds for all finite-dimensional algebras with rk K_{0}(A) = n. In fact, let A be an algebra with rk K_{0}(A) = n and let

⋯ → P_{2} → P_{1} → P_{0} → D(A_{A}) → 0

be a minimal projective resolution of D(A_{A}). Assume that there is a simple Amodule S with Ext_{A}^{i}(D(A_{A}), S) = 0 for all i. Let P (S) be the projective cover of S. Then P (S) is not a direct summand of P_{i} for all i. Let _{A}A = P ⊕ P (S)^{r} such that P (S) is not a summand of P. Let B = End_{A} P. Then it follows from [Ri1] that we have a full exact embedding of triangulated categories D^{−}(B) → D^{−}(A). By the choice of P we infer that K^{b}(_{A}I) is contained in D^{−}(B). Using the obvious identifications we may consider D(A_{A}) as B-module. But δ(D(A_{A})) = n and Ext_{B}^{i}(D(A_{A}), D(A_{A})) = 0 for all i > 0 yields a contradiction to (5’).

We recall now the definition of the Grothendieck group of a triangulated category [Gr]. For this let C be a triangulated category. Let F be the free abelian group on the isomorphism classes of objects in C. The isomorphism class of an object X ∈ C is denoted by [X]. Let F′ be the subgroup of F generated by [X] + [Z] − [Y] for all triangles X → Y → Z → X[1] in C. Then by definition the Grothendieck group of C is K_{0}(C) = F/F′.

Let F: C′ → C be an exact functor of triangulated categories. Then there is an induced map K_{0}(F): K_{0}(C′) → K_{0}(C).

For example consider the embedding of K^{b}(_{A}P) into D^{b}(A). Then the induced map on the level of Grothendieck groups turns out to be the Cartan map (see [B3] for a definition). In particular we see that K_{0}(F) need not to be injective, if F is an embedding. We also state the following.

Proposition. Let μ: D^{b}(A) → D^{−}(A) be the canonical embedding. Then K_{0}(μ) = 0.

It was shown in [Gr] that K_{0}(D^{b}(A)) ≃ K_{0}(A), which is isomorphic to ℤ^{n}, with n = δ(_{A}A).

Following [V] we call a full triangulated subcategory C′ of a triangulated subcategory C an épaisse subcategory, if C′ is closed under direct summands. We consider the following condition:

(5”): Let C be an épaisse subcategory of D^{b}(A) such that K_{0}(A) is finitely generated. Then rk K_{0}(C) ≤ n.

Remark. If (5”) holds then the generalized Nakayama conjecture holds.

Proof. By the mentioned result of [AR1] it is enough to verify condition (5’). Let AM be a generator which satisfies Ext_{A}^{i}(M, M) = 0 for all i > 0. Then it is easy to see that we obtain a full embedding K^{b}(add M) → D^{b}(A) (compare [H1]). So we may consider K^{b}(add M) as an épaisse subcategory of D^{b}(A). A straightforward calculation shows that K_{0}(K^{b}(add M)) ≃ ℤ^{δ(M)}. So δ(M) ≤ n by (5”). Since M is a generator we know that _{A}A is a direct summand of M, hence M is projective.

Note that the proof actually shows that for an arbitrary A-module M which satisfies Ext_{A}^{i}(M, M) = 0 for all i > 0 the number δ(M) ≤ n in case (5”) holds.

It is easy to construct counterexamples to the condition (5”) if we leave out the assumptions that C is épaisse or that K_{0}(C) is finitely generated. For instance let A be a finite-dimensional tame hereditary algebra and let C = D^{b}(R) be the derived category of the abelian subcategory R ⊂ mod A of regular A-modules. Clearly C is an épaisse subcategory of D^{b}(A), but K_{0}(C) is not finitely generated. We thank H. Lenzing for this example, which led to a reformulation of a more optimistic version of (5”).

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Fakultät für Mathematik, Universität Bielefeld, Postfach 8640, 4800 Bielefeld 1, FRG