THE GROTHENDIECK GROUP OF THE CATEGORY OF MODULES OF FINITE PROJECTIVE DIMENSION OVER CERTAIN WEAKLY TRIANGULAR ALGEBRAS

THE GROTHENDIECK GROUP OF THE CATEGORY OF MODULES OF FINITE PROJECTIVE DIMENSION OVER CERTAIN

WEAKLY TRIANGULAR ALGEBRAS

E. N. Marcos, H. Merklen and M. I. Platzeck

ABSTRACT. In this paper we study the category of finitely generated modules of finite projective dimension over a class of weakly triangular al- gebras, which includes the algebras whose idempotent ideals have finite pro- jective dimension. In particular, we prove that the relations given by the (relative) almost split sequences generate the group of all relations for the Grothendieck group of P^<∞(Λ) if and only if P^<∞(Λ) is of finite type. A similar statement is known to hold for the category of all finitely generated modules over an artin algebra, and was proven by C.M.Butler and M. Aus- lander ( [?] and [?]).

We assume in this paper that the artin algebra Λ is weakly triangular, that is, that the nonisomorphic indecomposable projective Λ-modulesP₁,· · · ,P_n can be ordered so that HomΛ(Pi,Pj) = 0 fori > j. LetAibe the factor of the projective moduleP_i modulo the trace inP_i of all the other indecomposable projectives. Then A₁,· · · ,A_n play an important role in the study of the categoryP^<∞(Λ) of finitely generated modules of finite projective dimension.

It is known that the finitistic projective dimension of Λ is bounded by the largest of the projective dimensions of the A_i’s ([?, ?]). Moreover, if all the Ai’s have finite projective dimension then P^<∞(Λ) consists of the modules having a filtration with factors amongst the A_i’s. In this case P^<∞(Λ) is a functorially finite subcategory of the category modΛ of finitely generated Λ-modules. Being P^<∞(Λ) closed under extensions it follows that it has relative almost split sequences (see [?]).

These results were proven in [?] for weakly triangular algebras under the assumption that some idempotent ideals of Λ (namely, the traces τQi(Λ), where Q_i = P₁ ⊕ · · · ⊕P_i for all i ≤ n) have finite projective dimension.

This hypothesis implies that all the A_i’s have finite projective dimension, since Ai =Pi/τPˆi(Pi) =Pi/τQi−1(Pi) the trace of a projective module inPi

is an idempotent ideal, for all i. The converse is also true, as it is proven in the first section.

Throughout the remainder of the paper, we assume that Λ satisfies these properties. In the second section we study further properties of P^<∞(Λ) and notions relative to P^<∞(Λ). The modulesA1,· · · ,An are the simple objects

in P^<∞(Λ), and P^<∞(Λ) inherits several properties of modΛ. For example,

every module in P^<∞(Λ) has a well defined relative socle, and the relative injective indecomposable modules inP^<∞(Λ) haveP^<∞(Λ)-simple socle, and coincide with the P^<∞(Λ)-injective envelopes of A₁,· · · ,A_n.

On the other hand, we can consider for every simple module S_i the

P^<∞(Λ)-approximation of the injective envelope of Si. In general this mod-

ule decomposes, but has, up to isomorphism, a unique indecomposable di- rect summand, which is precisely the P^<∞(Λ)-injective envelope of A_i. The multiplicity of this summand is also described, and we show that it can be arbitrarily large.

In the last section we study the Grothendieck group of P^<∞(Λ), finding again an analogy between P^<∞(Λ) and mod Λ.

M. C. Butler proved in [?] that if Γ is an artin algebra of finite represen- tation type then the relations given by the almost split sequences generate the group of all relations for the Grothendieck group of Γ, and M. Auslander proved that the converse also holds ([?]). We prove an analogous result for the relations defining the Grothendieck group of the subcategory P^<∞(Λ). The methods we use are similar to those in [?]. In order to write down the precise statement, given below, we explain some notations that are introduced in this last section.

The statement concerns the defining relations of the Grothendieck group

K₀(P^<∞) as a quotient of the group K₀(P^<∞,0) that we identify with the

free abelian group having ind(Λ) as a basis. The image of a module M in

K₀(P^<∞,0) is denoted by [M]. If C is an indecomposable, non projective

Λ-module, r_C denotes the element [C] + [A]−[B] of K₀(P^<∞,0) associated to the almost split sequence ending up at C. Anda_i will denote the element

[P_i] −[τ_Qj(P_i)]∈K₀(P^<∞,0).

Theorem. The relations given by the almost split sequences generate the defining relations of the Grothendieck group of P^<∞(Λ) , if and only if

P^<∞(Λ) is of finite type. Moreover, if P^<∞(Λ) is of finite type then

a) {a_i :i= 1,· · · , n}S

{r_C :C ∈indP^<∞(Λ), nonprojective }

is a free basis for the kernel of the map fromK0(P^<∞,0)ontoK0(P^<∞).

b) {r_C :C∈indP^<∞(Λ) nonprojective }

is a free basis for K₀(P^<∞).

The first two authors had partial support from CNPq, Brazil, the first named author had also support from FAPESP, and the third, from ANPCyT, Argentina, and Fundaci´on Antorchas, Argentina. During part of the work done, the second author was visiting Universidad del Sur, Bah´ıa Blanca, Argentina, in the frame of project FOMEC, Argentina, and he is thankful for the warm hospitality received during this stay.

1 Preliminaries.

We start this section introducing our assumptions, and some of the notations.

We will denote by R a commutative artinian ring, Λ an artin algebra over R and mod Λ the category of finitely generated left Λ-modules. For the sake of simplicity, we assume that Λ is basic and connected and P₁, . . . ,P_n will be chosen represen- tatives of the isoclasses of indecomposable projective modules.

S_i = topP_i, and I_i the injective envelope of S_i.

In general, for a ring Γ (resp. for an additive category C), indΓ (resp. indC) will denote the full subcategory of mod Γ (resp. C) defined by a family of representatives of the isoclasses of indecomposable modules in mod Γ (resp. in C).

If M and N are Λ-modules, τ_N(M) denotes the trace of N in M, that is, the submodule of M generated by the images of all morphisms from N to M. For each j = 1,2, ..., n, ˆP_j will denote the direct sum of all the P_i for i different from j and Q_j, the direct sum of all the P_i for i = 1, ..., j. Also, A_j will be the quotient P_j/τPˆj(P_j) = P_j/τ_Qj−1(P_j). There is a natural isomorphism A_j ' Λ/τ_Pˆ

j(Λ) , which induces a ring structure on A_j.

Following [?], we will assume also throughout the paper that

Λ is weakly triangular, that is, that the projective modules P1,· · · ,Pn may be ordered in such a way that Λ(Pi,Pj) = 0 provided that i > j. Here we use the simplified notation Λ(M, N) instead of Hom_Λ(M, N), as we will continue to do in the remainder of the paper. In this case, A_i = P_i/τ_Qi−1(P_i).

Whenever we consider a weakly triangular algebra we will as- sume that the indecomposable projective modules are ordered in this way and we will keep these notations.

When Λ is weakly triangular then the local algebra End_Λ(P_i)^op is isomorphic to the factor A_i, where, in general, Γ^op denotes the opposite of the ring Γ. We state this more precisely in the fol- lowing proposition.

Proposition 1. . Let Λ be a weakly triangular artin algebra.

Then the morphism Λ → End_Λ(P_i)^op which associates to λ ∈ Λ the right multiplication x 7→ xλ, induces an algebra isomorphism from A_i = Λ/τ_Pˆ

i(Λ) to End_Λ(P_i)^op, for all i = 1,· · · , n.

The following proposition shows that being weakly triangular is a symmetric property.

Proposition 2. . If Λ is weakly triangular, then so is Λ^op, (for the opposite ordering). Moreover, if i < j then Λ(Ii,Ij) = 0 and so S_j is not a composition factor of I_i .

Proof. The first claim follows from the well known R-module isomorphisms

Λ^op(e_jΛ, e_iΛ) ∼= e_iΛe_j ∼= Λ(Λe_i,Λe_j).

The second statement follows by duality (see [?], 2.2).

We are going to prove (cf. Theorem 1, below) the following useful result. If Λ is weakly triangular then

A₁,· · · ,A_n ∈ P^<∞(Λ) ⇔τ_Qi(Λ) ∈ P^<∞(Λ) ∀i = 1,· · · , n.

In order to prove some lemmas we will need, let us recall the following results of [?]

Let us consider for a projective Λ-module Q the full subcat- egories C^Q_o, C^Q₁ and C^Q_∞ of mod Λ defined in the following way.

The modules inC^Q_o are those having their projective cover in addQ, the full subcategory of mod Λ consisting of direct sum- mands of finite sums of Q. The modules in C^Q₁ are those having a projective presentation in addQ, and, finally C^Q_∞ consists of the modules with a projective resolution in addQ.

Let Q be a projective Λ-module and Γ = End_Λ(Q)^op. It is known that the functor Λ(Q, ) : modΛ → modΓ induces an equivalence of categories between C^Q₁ and mod Γ. Moreover, it is proven in [?] that this equivalence carries projective resolu- tions of modules in C^Q_∞ into projective resolutions of Γ -modules, proving in particular the following result.

Proposition 3. . [?, Cor. 3.3 a)]. Let Λbe an artin algebra, Qa projective Λ-module and M ∈ C^Q_∞. Then pd_ΛM = pd_ΓΛ(Q, M).

When the artin algebra Λ is weakly triangular we obtain the following corollary.

Corollary 1. . Let Λ be weakly triangular, Q_i = ⊕_j≤iP_j, Γ_i = End_Λ(Q_i)^op. Let M ∈ mod Λ be such that τ_Qi(M) = M. Then proj dim_ΛM = proj dim_ΓiΛ(Q_i, M).

Proof. Since Λ is weakly triangular the subcategories C^Q_oⁱ, C^Q₁ⁱ and C^Q_∞ⁱ coincide. The Corollary follows by observing that τ_Qi(M) =M if and only if M ∈ C^Q_oⁱ.

Lemma 1. . Let us assume that Λ is weakly triangular and that A1,A2,· · · ,An have finite projective dimension. Then Γi = End_Λ(Q_i)^op has the same properties for each i ≤ n.

Proof. We get that Γ_i is weakly triangular from the fact that Λ(Q_i,−) defines an equivalence of categories between addQ_i and the category of projective Γ_i-modules.

We observe next that, if Q is a projective Λ-module and if P, X ∈ C^Q₁, then

τ_Λ(Q,P)Λ(Q, X) = Λ(Q, τ_PX).

It follows then that

A_Γi,t =: Λ(Q_i,P_t)/τ_{⊕j<tΛ(Qi,Pj)}Λ(Q_i,P_t) ∼= Λ(Q_i,A_t).

On the other hand, our hypothesis and Corollary 1 imply that the Γ_i-projective dimension of Λ(Q_i,A_t) is finite.

We state for convenience the following result of [?].

Lemma 2. . ([?], Lemma 2.2. Let us assume that Λ is weakly triangular and that all τQ_i(Λ) (i = 1,2,· · · , n) are in P^<∞(Λ).

Then, M ∈ P^<∞(Λ) implies τQ_i(M) ∈ P^<∞(Λ) for all i ≤ n.

Theorem 1. . Let Λ be a weakly triangular artin algebra. Then, the following conditions are equivalent.

(i) A₁,A₂,· · · ,A_n ∈ P^<∞(Λ) (ii) τQ_i(Λ) ∈ P^<∞(Λ) ∀i = 1,2,· · · , n

Proof. Obviously, (ii) implies (i). To prove the converse we assume that (i) holds, and observe that proving (ii) amounts to proving that τ_Qi(P_j) is in P^<∞(Λ) for all i, j = 1,2,· · · , n. We prove this by induction on j. We observe that the statement holds for j = 1 and assume that it is true for j ≤ k−1. We will prove that τ_Qi(P_k) ∈ P^<∞(Λ)∀i = 1,2,· · · , n by induction on n.

Since we are assuming that the projective dimension ofA_k is finite it follows that τ_Qk−1(P_k) has finite projective dimension.

Thus, by Corollary 1, we get that

pd_Γk−1Λ(Q_k−1, τQ_k−1(Pk)) < ∞.

It follows from Lemma 1 that the induction hypothesis applies to the algebra Γ_k−1. Then we can apply Lemma 2 to the Γ_k−1- module Λ(Q_k−1, τQ_k−1(Pk)) and conclude that

Λ(Q_k−1, τ_Qi(τ_Qk−1(P_k))) = τ_{Λ(Qk−1,Qi)}Λ(Q_k−1, τ_Qk−1(P_k)) has finite projective dimension over Γ_k−1, for all i ≤ k−1. But this finishes the proof, because τQ_i(τQ_k−1(Pk)) = τQ_i(Pk) and, applying Corollary 1 once more, we obtain τQ_i(Pk) ∈ P^<∞(Λ), as desired.

The results proven in the following theorem were proven in [?] for weakly triangular algebras such that pd_Λτ_Qi(Λ) is finite for all i = 1,2,· · · , n. In view of Theorem 1, we replace the last condition by the assumption that A1,· · · ,An have finite projective dimension.

Theorem 2. . [?] Let Λ be a weakly triangular artin ring such that all the Ai (i = 1,2, ..., n) have finite projective dimension and let M ∈ mod(Λ). Then

1. If M is in P^<∞(Λ) then τQ_j(M) is in P^<∞(Λ) , for all j = 1,2, ..., n, and the factor τQ_j(M)/τQ_j−1(M) is a free A_j-module.

2. M is in P^<∞(Λ) if and only if M admits a filtration with factors in {A_i/i= 1,2, ..., n}.

3. The finitistic projective dimension of Λ is the maximum of the projective dimensions of A₁,· · · ,A_n.

4. P^<∞(Λ) is functorially finite, closed under extensions and

hence has Auslander-Reiten sequences.

It follows from the last example in [?] that the hypothesis of the theorem do not imply that all idempotent ideals of Λ have finite projective dimension.

2 The P

(Λ)-injective modules.

Throughout this section we assume that Λ is weakly triangular and the modules Ai = Pi/τPˆi(Pi) have finite projective dimen- sion, for i = 1,· · · , n. These hypothesis are met by weakly tri- angular algebras having all idempotent ideals of finite projective dimension.

We will study the relative injective modules in P^<∞(Λ) . We know that the modules in P^<∞(Λ) are those admitting a filtra- tion in the set {A₁,· · · ,A_n}, andP^<∞(Λ) is a functorially finite subcategory of mod Λ. The results in this section strengthen and generalize similar results proven in [CMMMP] for algebras with all idempotent ideals projective.

We will refer to notions relative toP^<∞(Λ) , such asP^<∞(Λ) - injective,P^<∞(Λ)-projective,P^<∞(Λ)-simple, etc. We will show

that one can define the P^<∞(Λ)-socle of a module in P^<∞(Λ).

It is clear that the P^<∞(Λ)-projective objects are the projective Λ-modules. On the other hand, using the fact that P^<∞(Λ) is closed under cokernels of monomorphisms, it follows that

the P^<∞(Λ)-injective modules coincide with the Ext-injective

modules in P^<∞(Λ), that is, with the modules I such that Ext¹_Λ(X, I) = 0 for all X in P^<∞(Λ). Or, equivalently, with the modules I such that any exact sequence in P^<∞(Λ) starting at I splits.

Since P^<∞(Λ) is functorially finite we can give a different description of the P^<∞(Λ)-injective modules in terms of the

P^<∞(Λ)-approximations of the indecomposable injective Λ-mod-

ules. Although these approximations may decompose, we will prove that they have only one indecomposable summand, up to isomorphism. The multiplicity of such summand will also be determined.

Definition 1. . We say that the P^<∞(Λ)-socle of M ∈ P^<∞(Λ) is defined and is equal to V when M ∈ P^<∞(Λ) has a unique maximal P^<∞(Λ)-semisimple submodule, V.

We start by proving that the P^<∞(Λ)-socle is defined for all

M ∈ P^<∞(Λ).

Proposition 4. Assume Λ is weakly triangular and the modules A₁,· · · ,A_n have finite projective dimension. Then, for M ∈

P^<∞(Λ), the family of P^<∞(Λ)-semisimple submodules of M

has a maximum element.

Proof. We write again Q_j = ⊕_i≤jP_i. Let M ∈ P^<∞(Λ). We know by Theorem 1 that τQ_j(M) ∈ P^<∞(Λ) and

τ_Qj(M)/τ_Qj−1(M) ' Aⁿ_j^j,

for some nj ≥0 and for all j = 1,2, ..., n.

We will prove the proposition by induction on the minimal number k such that τ_Qk(M) = M. If k = 1 then M ' Aⁿ₁¹ and the result is true. Let k be greater than 1. We assume that the proposition holds for modules N such that τ_Qk−1(N) = N and let M ∈ P^<∞(Λ) be such that τ_Qk(M) = M. We may assume that M is indecomposable. Since τ_Qk−1(M) ∈ P^<∞(Λ) and τ_Qk−1(τ_Qk−1(M)) = τ_Qk−1(M) we can apply the induction hypothesis and conclude that the socle of τ_Qk−1(M) is defined.

We will prove that either A_k ' M or soc_P^<∞_(Λ)(M) is defined and coincides with soc_P^<∞_(Λ)(τQ_k−1(M)). To do so we consider a

P^<∞(Λ)-simple submodule X of M and prove that either M '

A_k or X ⊆ τ_Qk−1(M). If X ' A_i with i < k then the second assertion holds. So we assume X ' A_k , and let j : X → M be the inclusion and π : M → τ_Qk(M)/τ_Qk−1(M) the canonical map. Since the only composition factor ofX isS_k, which is not a composition factor of τ_Qk−1(M) it follows thatX∩τ_Qk−1(M) = 0, so πj is a monomorphism.

Since A_k is a local artinian ring then any monomorphism f : A_k → Aⁿ_k^k splits. This follows from the fact that 0 →A_k → Aⁿ_k^k → Coker(f) → 0 is a projective resolution of Coker(f) and is not minimal.

Since X ' A_k and τ_Qk(M)/τ_Qk−1(M) ' Aⁿ_k^k , for some n_k ≥ 0, it follows that the monomorphism πj splits. Thus j : X →M splits. Since M is indecomposable this proves that M ' X ' A_k, ending the proof of the proposition.

We go on now to study the P^<∞(Λ)-injective modules. The following proposition proves in particular that the P^<∞(Λ)-in- jective indecomposable modules have P^<∞(Λ)-simple socle and are the P^<∞(Λ)-injective envelopes (in the sense that the corre-

sponding inclusions are minimal morphisms) of A1,· · · ,An. Proposition 5. . Assume Λ is weakly triangular and the mod- ules A₁,· · · ,A_n have finite projective dimension. Let

eIi φi

→Ii

be the minimal right P^<∞(Λ)-approximation of the indecompos- able injective Λ-module Ii, and leteIi = `

jeIij, witheIij indecom- posable. Then

1. {eI_ij}_i,j is the set of indecomposable P^<∞(Λ)-injective ob- jects, up to isomorphism. Thus the category P^<∞(Λ) has enough injectives.

2. A_i is not a P^<∞(Λ)-composition factor of eI_j, for i < j. 3. eI_nj ∼= A_n, for all j.

4. eI_ij 'eI_i1, for all i, j.

5. soc_P^<∞_(Λ)(eI_i1) = A_i.

6. eI_i = (eI_i1)ⁿⁱ, where n_i is the smallest number of copies of A_i necessary to cover the A_i-injective envelope I_Ai(S_i) of S_i. That is, n_i is the length of I_Ai(S_i)/rad(I_Ai(S_i)).

1) Since P^<∞(Λ) contains the projective Λ-modules, all the ap- proximations eIi

φi

→ Ii are epimorphisms. We start by proving that all eIi are P^<∞(Λ)-injective. Let

0→eI_i →^f X →^g Y →0

be an exact sequence in P^<∞(Λ). According to the remark at the beginning of this section we only need to prove that this

sequence splits. Since f is a monomorphism and Ii is injective there is a morphism t : X → Ii such that φi = tf. Since X is in P^<∞(Λ) then t : X → I_i factors through the P^<∞(Λ)- approximation eI_i →^φi I_i That is, there is h : X → eI_i such that t = φ_ih. So φ_i = φ_ihf. The minimality of φ_i implies that hf is the identity map, so f splits, as required.

To prove the converse, let J be an indecomposable P^<∞(Λ)- injective module inP^<∞(Λ), and letj : J →I(J) be its injective envelope in mod Λ. Since J is in P^<∞(Λ) the map j factors through the P^<∞(Λ)-approximation

eI(J) →^φ I(J)

Thus there is h : J → eI(J) such that j = φh. Since j is a monomorphism his a monomorphism too and therefore it splits, because J is P^<∞(Λ)-injective. This ends the proof of 1).

2) We observe first that Λ(P_i,I_j) = 0 for i < j, because in this case Λ(Pj,Pi) = 0. So the minimal left P^<∞(Λ) -approximation

eI_j →^φj I_j

vanishes on the trace of Q_i ineI_j, for i < j

Hence, φ_j factors through ˜I_j/τ_Qi(eI_j). Since this quotient is

in P^<∞(Λ) (see Theorem 1) we deduce from the minimality of

φj that the trace of Qi ineIj is 0, proving 2).

3) We know by 2) that the only P^<∞(Λ)-composition factor of eI_n is A_n So for each summandeI_nj of I_n there is an epimorphism f :eI_nj → A_n in mod A_n. SinceeI_nj is indecomposable it follows that f is an isomorphism, proving 3).

4) Since the approximationeIi φi

→Ii is minimal it follows thatSi is a composition factor of all summandseI_ij ofeI_i. Thus τ_Pi(eI_ij) 6= 0

for all j. Since we know by 2) that Hom_Λ(P_k,eI_i) = 0 for all k < i, it follows that τP_i(eIij) = τQ_i(eIij)/τQ_i−1(eIij). We know then by Theorem 1 that τ_Pi(eI_ij) is a free A_i-module. That is, τP_i(eIij) = A^k_iⁱ, for some ki > 0.

We can therefore consider monomorphisms A_i → eI_ij, A_i → eI_i1. Since both eI_ij and eI_i1 are P^<∞(Λ)-injective and indecom- posable, we get that they are isomorphic. This proves 4).

5) Using that Λ(A_j,I_i) = 0 and the minimality of the approx- imation φ_i it follows that A_j is not contained in ˜I_i, for j 6= i.

Thus, to prove 5) we only need to prove that the integer ki

above considered is 1. We start by observing the following con- sequence of 2). If M ∈ P^<∞(Λ) has P^<∞(Λ)-composition fac- tors in {A_i,· · · ,A_n}, then all the modules that occur in a mini- mal injectiveP^<∞(Λ)-coresolution ofM have the same property.

This fact follows by induction on the length of the coresolution, and using 2). We know that the length of the coresolution is fi- nite because the projective dimension of the modules in P^<∞(Λ) is bounded.

To prove 5) we proceed by decreasing induction on i, using that the statement is true for i = n, by 3). Let

0 →A_i → E₁ → E₂ → · · · → E_t →0 be a minimal P^<∞(Λ)-injective coresolution of A_i.

Let m_k denote the number of times thateI_i1 occurs as a sum- mand of E_k, and let n_i be the multiplicity of A_i as P^<∞(Λ) -composition factor of eI_i1. We want to prove that n_i = 1. From the above observation we know that the E_i’s have P^<∞(Λ)- composition factors in {A_i,· · · ,An}. So the only possible sum- mands of Ei areeIi1,· · · ,eIn1. From 2) we know that Ai is not a

P^<∞(Λ)-composition factor of eI_j1 for j > i. Thus the multiplic-

ity of Ai in Ek is mkni.Then, from the above coresolution, we obtain

1 +

t

X

k=1

(−1)^km_kn_i = 0

from which it follows that n_i divides 1. So n_i = 1, as required.

6) Since Ai = Λ/τPˆiΛ then the Ai-injective envelope of Si is I_Ai(S_i) ={x ∈ I_i :τ_Pˆ

iΛ.x= 0}.

Let π : A^k_iⁱ → I_Ai(S_i) be the A_i-projective cover of I_Ai(S_i), and let j : I_Ai(S_i) → I_i be the inclusion map. Then jπ factors through the minimal approximation φ_i : ˜I_i → I_i. Since the do- main of π is P^<∞(Λ) -semisimple then it factors also through

the P^<∞(Λ)-socle Aⁿ_iⁱ of ˜Ii. But this says that Aⁿ_iⁱ also covers

IA_i(Si) =j(IA_i(Si)), implying that φi |_Ani

i factors as ππ⁰, where π⁰ is a split epimorphism. Let Aⁿ_iⁱ ∼= A^k_iⁱ ⊕ Aⁿ_i^i−ki be the in- duced factorization. Since ˜Ii1 is the P^<∞(Λ)-injective envelope of Ai we obtain a decomposition ˜Ii '˜I^k_i1ⁱ⊕tildeIⁿ_i1^i−ki so that φi

vanishes on the second summand tildeIⁿ_i1^i−ki. Since the approx- imation φ_i is minimal, this implies that n_i −k_i = 0, as desired.

Proposition 5 shows that though the approximation of an indecomposable injective module has only one indecomposable summand up to isomorphism, it is not indecomposable. In fact the multiplicity of such summand can be arbitrarily large, as shown in the following example.

Example 1. . Let Λ be a basic radical square zero local artin algebra of length n. LeteI be the minimalP^<∞(Λ)-approximation of the indecomposable injective Λ-module I. Then the multiplic- ity of the indecomposable P^<∞(Λ)-injective module as a sum- mand of eI is n−1.

3 On the defining relations for the Grothen- dieck group.

As in the preceding section we assume throughout this one that the artin algebra Λ is weakly triangular and the modules Ai = Pi/τPˆi(Pi) have finite projective dimension, for i = 1,· · · , n.

We know that under these hypotheses P^<∞(Λ) is functorially finite, and has therefore almost split sequences.

In [?] M. Auslander has shown for an artin algebra Λ that the relations given by the almost split sequences generate the group of all relations for the Grothendieck group of Λ if and only if A is of finite representation type. The sufficiency of this latter condition had been previously proved by Butler in [?]. In this section we will prove this statement for the category P^<∞(Λ), using methods similar to those used by Auslander in [?].

We will denote by K₀(P^<∞) the Grothendieck group of the categoryP^<∞(Λ) and byK₀(P^<∞,0) the free abelian group with basis a complete set of indecomposable objects in P^<∞(Λ).

We observe thatK₀(P^<∞,0) is the quotient of the free abelian group generated by the isomorphism classes [M] of elements M

in P^<∞(Λ) by the subgroup generated by all relations of the

form [A] + [C]−[B], where 0 → A → B → C → 0 is a short split exact sequence in P^<∞(Λ) . We denote also by [M] the element of K0(P^<∞,0), and by (M) the element of K0(P^<∞), determined by the module M in P^<∞(Λ) .

Then K₀(P^<∞) is the quotient of K₀(P^<∞,0) by the sub-

group generated by the relations of the form [A] + [C] − [B]

such that there is an exact sequence 0 → A → B → C → 0 in P^<∞(Λ).

In all that follows ES will denote this latter subgroup and AR the subgroup of K0(P^<∞,0) generated by the elements of the form [A]+[C]−[B] such that there is a (relative) almost split sequence 0 → A → B → C → 0 in P^<∞(Λ). We will say that

P^<∞(Λ) is of finite type if P^<∞(Λ) contains only a finite number

of nonisomorphic indecomposable objects. We will prove that AR = ES if and only if P^<∞(Λ) is of finite type.

We fix here some further notations which will be used in all that follows.

Following [?] we consider the bilinear form < , > on

K₀(P^<∞,0) which satisfies that < [M],[N] > is the length of

the R-module Λ(M, N), for all M, N in P^<∞(Λ).

For each indecomposable and not projective module C in

P^<∞(Λ), r_C denotes the element [A] + [C]−[B] of K₀(P^<∞,0),

where 0 → A → B → C → 0 is the almost split sequence in

P^<∞(Λ) ending at C. Dually, if A ∈ P^<∞(Λ) is indecomposable

and not injective we write l_A = [A] + [C]−[B] , where 0 → A → B → C → 0 is the almost split sequence in P^<∞(Λ) beginning at A.

For each i ≤ n we consider in K₀(P^<∞,0) the element a_i = [P_i]− [τPˆi(P_i)]. Then the element in K₀(P^<∞) corresponding to ai is (Ai).

The following proposition follows directly from the fact that the modules in P^<∞(Λ) are those having a filtration with factors amongst the A_i’s.

Proposition 6. . K₀(P^<∞) is a free abelian group with basis the classes of the modules A₁,· · · ,A_n. Hence K₀(P^<∞,0) is generated by ES and the elements a₁,· · · , a_n.

For nonprojective indecomposable modules inP^<∞(Λ) the el-

ements rC above defined have the following important property:

If X ∈ P^<∞(Λ) then < [X], rC >= 0 if and only if X is not

isomorphic to C. A dual result holds for the elements l_A.

Many of the results and arguments here are a direct gen- eralization of those in [?]. However, there are some essential differences, so that the proof in [?] cannot be directly carried over. One important difference is the following. For each pro- jective indecomposable module P_i Auslander considers the ele- ment r_Pi = [P_i]−[rP_i] in the Grothendieck group of Λ. Then if X is an indecomposable Λ-module, < [X], r_Pi >6= 0 implies thatX is isomorphic to Pi. However, the element we consider in K0(P^<∞,0) instead of rP_i is ai = [Pi]−[τPˆi(Pi)], and it doesn’t have the corresponding property. To see this, we consider the algebra Λ and the module M of the example in [?] p. 2529.

Example 2. . Let Λ be the k-algebra given by the following quiver Q with the relation β² = 0.

q

r r

α β

2 1

Let M be given by the following representation

)

r r

(0 1) 0

0 1 0

k k²

Then a₁ = [P₁]−[τ_P2(P₁)] and

Λ(M,P1) 6= 0, Λ(M, τP₂(P1)) = 0.

Thus < [M], a1 >6= 0, with M not isomorphic to P1.

We also observe that in this example the P^<∞(Λ)-approxima- tion of S1 is not A1 but M, as observed in [?],p 2529.

The following result will be useful to decide if two elements

of K₀(P^<∞,0) are equal.

Proposition 7. . Let x = P

M a_M[M], y = P

M b_M[M]

in K₀(P^<∞,0), where M runs over the indecomposable modules

in P^<∞(Λ). Then the following conditions are equivalent.

• (i) x = y

• (ii) < [N], x >=< [N], y > for all indecomposable module

N in P^<∞(Λ).

• (iii) < x,[N] >=< y,[N] > for all indecomposable module

N in P^<∞(Λ).

Proof. It is clear that (i) implies (ii) and (iii). If (iii) is true, we have that < x, r_C >=< y, r_C > for every non-projective indecomposable C in P^<∞(Λ). Since < −, r_C > annihilates all summands of x and y except the one with index C we obtain thataM = bM if M is not projective. So in order to deduce (i) we may assume that x = Pn

i=1αi[Pi] and y = Pn

i=1βi[Pi]. The fact that Λ is weakly triangular implies that < x,[P_n] > = α_n. <

[P_n],[P_n] > and < y,[P_n] >= β_n < [P_n],[P_n] >. Since we are assuming that (iii) holds we get α_n = β_n. We can then apply the same argument to x₁ = x − α_n[P_n], y₁ = y − β_n[P_n] and use the projective module P_n−1 to conclude that α_n−1 = β_n−1. Iterating the procedure we prove that (i) holds.

If (ii) is true, using thel_A’s instead of the r_C’s, the question of proving (i) is reduced to the case when the modules which occur

in both x and y are P^<∞(Λ)-injective. But then (i) follows from Prop. 2 by successive application of < P1,− >,· · · , <Pn,− >.

The next proposition describes a linearly independent family

in K₀(P^<∞,0). We will prove later that this family is a basis of

K₀(P^<∞,0) if and only if P^<∞(Λ) is of finite type.

Proposition 8. . The subset {a_i : i = 1,· · · , n}S

{r_C : C is indecomposable nonprojective in P^<∞(Λ)} of K0(P^<∞,0) is linearly independent. Moreover, any linear combination x of the elements r_C is of the form

x = X

C

< C, x > / < C, r_C > r_C

Proof. The proof uses arguments similar to those above. Con- sider a linear combination of elements in the given set. That is, let

x = X

i

kiai +X

C

bCrC

with k_i, b_C integers. We observe that, since < D, r_C >= 0 for any indecomposable D in P^<∞(Λ) not isomorphic to C,

< [P_i],− > annihilates all summands except the one of index i in the first sum. Thus, since < [P_i], a_i >6= 0 we obtain that

k_i =< [P_i], x > / < [P_i], a_i > .

Hence, the first sum above is determined by x, and we will denote it with x. That is, we write x = P

ikiai

We obtain next in a similar way that

b_C =< [C], x−x > / < [C], r_C >,

and the proof is complete.

For the proof of the main theorem in this section we introduce the following notation. Let us observe that if M is a module

in P^<∞(Λ), then the number of times m_i that the P^<∞(Λ)-

composition factor A_i appears in M, is exactly

< [P_i],[M] > / < [P_i], a_i >=< [P_i],[M] > / < a_i, a_i > . This can be proven by induction, applying the functor Λ(P_i,−) to an exact sequence of the form 0 →N → M → Ai →0.

In what follows, given M in P^<∞(Λ), we denote by ˆM the module ⊕_iA^m_i ⁱ, where m_i is the multiplicity of A_i in M.

Then M and Mˆ define the same element P

im_i[A_i] in

K₀(P^<∞). Moreover, we can prove the following result.

Lemma 3. . The kernel ES of the natural map fromK₀(P^<∞,0)

onto K0(P^<∞) is generated by the elements of the form [M]−

[ ˆM], with M in P^<∞(Λ).

Proof. Let M in P^<∞(Λ). We know that [M]−[ ˆM] is in ES.

Consider now an exact sequence 0 → A → B → C → 0 in

P^<∞(Λ). Since the P^<∞(Λ)-multiplicity of A_i in B is the sum

of P^<∞(Λ)-multiplicities of Ai inA and C, then [ ˆB] = [ ˆA] + [ ˆC].

Thus [A] + [C] − [B] = [A] + [C]− [B] −([ ˆA] + [ ˆC]− [ ˆB]) = ([A]−[ ˆA]) + ([C]−[ ˆC])−([B]−[ ˆB]), proving the lemma.

We prove next the main theorem in this section.

Theorem 3. . The relations given by the almost split se- quences generate the defining relations of the Grothendieck group of P^<∞(Λ), that is AR = ES, if and only if P^<∞(Λ) is of finite type. Moreover, if P^<∞(Λ) is of finite type then

a) {a_i : i = 1,· · · , n}S

{r_C : C is indecomposable nonprojec- tive in P^<∞(Λ)} is a free basis for K0(P^<∞,0)

b) {r_C : C is indecomposable nonprojective in P^<∞(Λ)} is a free basis for the kernel of the natural map from K₀(P^<∞,0) onto K₀(P^<∞).

Proof. Let us suppose that P^<∞(Λ) is of finite type and let M be in P^<∞(Λ). According to the preceding lemma we only need to prove that [M] − [ ˆM] is in AR. So we need to write [M]−[ ˆM] as a linear combination of the elements r_C. By Prop.

8 this amounts to prove that [M]−[ ˆM] = X

C

(< [C],[M]−[ ˆM] > / < [C], r_C >)r_C. So we write

x = [ ˆM] +X

C

(< [C],[M]−[ ˆM]> / < [C], r_C >)r_C, and prove that x = [M].

¿¿From the definition of ˆM and the properties of the r_C’s it follows that < P_i, x >=< P_i,[M] > for all i = 1,· · · , n and

< [C], x >=< [C],[M] > for all indecomposable nonprojective

C in P^<∞(Λ). If follows then from Prop. 7 that x = [M], as

desired.

Let us assume now that AR = ES, so that each [M]−[ ˆM] is a linear combination of the r_C’s. Then the coefficient of each r_C is a nonzero multiple of < [C],[M] − [ ˆM] >, as we proved in Prop. 8. Hence for each M there is only a finite number of nonisomorphic indecomposable modules C in P^<∞(Λ) such that < [C],[M] − [ ˆM] >6= 0. Let us denote by U the set of isomorphism classes of elements in ind P^<∞(Λ) that satisfy this

condition for at least one of the modules ˜Ii or ˜Ii/Ai. Then U is a finite set. According to the definition, C 6∈ U if and only if

< [C],[˜Ii] >=< [C],[ˆ˜Ii] > and< [C],[˜Ii/Ai]>=< [C],ˆ[˜Ii/Ai] >

for all i = 1,· · · , n.

We claim now that if an indecomposable moduleC ofP^<∞(Λ) is not in U then C is projective. To prove this we apply Λ(C,−) to the exact sequence 0 → A_i → ˜I_i → ˜I_i/A_i → 0, so that [ˆ˜Ii] = [Ai] + [ˆ˜Ii/Ai]. Then, using that C is not in U, we obtain that the sequence 0 → Λ(C,A_i) →Λ(C,˜I_i) → Λ(C,˜ ˆ

i/AI)_i → 0 is exact. This shows that Ext¹_Λ(C,A_i) = 0 for each i. With a standard induction argument on the P^<∞(Λ)-length of M it follows that that Ext¹_Λ(C, M) = 0 for each M in P^<∞(Λ). So C is projective, proving our claim.

Finally, we see that what we proved amounts to say that the set of isomorphism classes of indecomposable modules in

P^<∞(Λ) consists of the finite set U and the n indecomposable

projective modules. Thus P^<∞(Λ) is of finite type, and the first part of the theorem is proved. The remaining statements follow now from Prop. 8.

Remark 1. . We finish this section observing that in our situa- tion the classes of indecomposable projective modules also com- prise a free generating set for K₀(P^<∞). Therefore if we write every indecomposable projective P_i in the basis of the A⁰_js, then we have an invertible matrix in the set of the integers.

Remark 2.

Let Λ be a finite dimensional algebra. Let {Γ_i}_i∈I be the set of components of its Auslander Reiten quiver. Let us de- note also by Γ_i the full subcategory of mod-Λ whose objects are

the modules in Γi. Then the Γi’s, clearly, have almost split se- quences and the almost split sequences in Γi are the almost split sequences of mod-Λ whose terms are in Γ_i. It is not hard to see

that K₀(P^<∞,0)/AR) ∼= `

K_o(Γ_i,0)/AR(Γ_i).

Moreover, for each component of the right hand side of the sum above we have a map to the integers induced by the length.

Therefore we get that rank(K₀(P^<∞,0)/AR) ≥#I.

We would like to observe that there are counter examples to the property in Auslander-Butler theorem, in the sense that there exist subcategories with almost split sequences and of infi- nite type, such that the defining relations of their Grothendieck group are given by the group AR.

Let us take Λ to be a hereditary algebra and let P₀ denote the subcategory defined by the preprojective component of the Auslander-Reiten quiver of Λ. Then one sees that the set of projective modules forms a generating set for K₀(P₀,0)/AR.

We prove this by induction in the following way. Let C ∈ P₀, C 6∈ AR be such that the number n such that DT rⁿC is projective is minimal, and such that the projective in the orbit of C is of minimal length.

Let

0 →DtrC →X

i

B_i →C →0

be an almost split sequence. Then, DT rC is in AR, but (as we can assume), B₁ is not. Therefore, for j = 1,· · · , n−1, DT r^jC is not projective, implying that there is an arrow from DT rⁿB₁ to DT rⁿC, a contradiction to the minimality of the length of the projective DT rⁿC.

From the natural maps

Ko(P₀,0) →K0(P^<∞,0) →K0(P^<∞)

it follows that K₀(P^<∞,0)/AR is isomorphic to K₀(P^<∞).

Clearly, if Λ is of infinite representation type, so is P₀ and we have the desired example.

Observe that we can get Auslander-Butler’s theorem for hered- itary algebras out of these simple remarks.

In a similar way we can show the following:

Proposition 9. . If Λ is a finite dimensional hereditary algebra such that the classes of the projective modules form a generating set for

(K₀(P^<∞,0)/AR)

then Λ is of finite representation type.

Proof In this case it is clear that the set of classes of projec- tives forms a generating set for K0(P^<∞), so the natural map

(K₀(P^<∞,0)/AR) → K₀(P^<∞) is an isomorphism and we con-

clude that the only component on the Auslander Reiten quiver is the component which contains the projectives, therefore Λ is of finite representation type. .

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