Representation theory of some infinite-dimensional algebras arising in continuously controlled algebra and topology

arXiv:math/0310334v1 [math.RT] 21 Oct 2003 REPRESENTATION THEORY OF SOME INFINITE-DIMENSIONAL ALGEBRAS ARISING IN CONTINUOUSLY CONTROLLED ALGEBRA AND TOPOLOGY FERNANDO MURO Contents 1. Introduction 1 1.1. Notation and conventions 3 2. Ringoids and modules 4 3. The algebras R(E) and related additive categories 5 3.1. The special case card Efinite 7 4. Countably presented R-modules as finitely presented RCFM(R)-modules 8 5. A review on pro-categories 10 6. Numerical invariants of finitely presented k(n)-modules 10 7. Classification of finitely presented RCFM(k)-modules 12 8. Representations of the n-subspace quiver 22 9. Finitely presented k(n)-modules and finite-dimensional n-subspaces 24 10. Classification of finitely presented k(n)-modules 29 Appendix A. Some computations of Ext1 k(n)groups 30 References 32 1. Introduction Suppose that the discrete set IN0of non-negative integers is embedded IN0⊂X in a compact metrizable space X, and let E= IN′ 0⊂Xbe the derived set (i. e. points of Xwhich contain infinitely many points of IN0in any neighborhood). Consider the set R(E) of IN0×IN0matrices (aij )i,j∈IN0with entries in a (unital associative) ring Rsuch that if {in}n∈IN0,{jn}n∈IN0⊂IN0are sequences convergent in Xto different points then the vector (ainjn)n∈IN0is almost all zero. This set is an R-algebra with the usual matrix operations. Any compact metrizable space can arise as Ein this way. In fact the isomorphism class of the algebra R(E) only depends on E. These algebras are Morita equivalent to some additive categories of free R-modules continuously controlled at infinity by Eappearing in the literature. These categories play an important role in many areas such as controlled homotopy theory, proper homotopy theory, C∗-algebra theory, K-theory and L-theory, see for example [15], [5], [9], [2] and [1]. The elementary properties of the algebras R(E) have been studied by Baues- Quintero ([2]) for R=ZZ the integers. If Eis zero-dimensional, R(E) is a particular case of the rings considered by Farrell-Wagoner ([7]). When E=∗is a singleton R(E) = RCFM(R) is the well-known algebra of row-column-finite (or locally finite) 1 2 FERNANDO MURO matrices over R. This algebra has been studied from a purely ring-theoretical point of view, see for example [17]. It was also used by Wagoner ([18]) to construct deloopings in algebraic K-theory. In this paper we concentrate on the representation theory of the algebras k(E), where kis any field. Representation theory considers the decomposition problem in a small additive category A. A solution to this problem consists of a set of objects (which we call elementary objects) and of a set of isomorphisms (elementary isomorphisms) between finite direct sums of elementary objects. These sets must satisfy the following properties: any object in Ais isomorphic to a finite direct sum of elementary ones, and any isomorphism relation between two such direct sums can be derived from the elementary isomorphisms. Notice that this is exactly a presentation of the abelian monoid Iso(A) of isomorphisms classes of objects in A. The trivial solution is taking all objects as elementary objects and all isomorphisms as elementary isomorphisms, however one is often interested in solutions minimizing the cardinal of the sets of elementary objects and isomorphisms. We say that Ahas finite representation type if there exists a finite set of elementary objects, or equivalently Iso(A) is finitely presented. The representation type of Ais wild if a solution to the decomposition problem in Awould yield a solution to the decomposition problem in the category of finite-dimensional modules over a polynomial k-algebra in two non-commuting variables. Otherwise Ahas tame representation type. If Ahas wild representation type the word problem for finitely presented groups, which is undecidable, can be embedded in the decomposition problem in A, hence one can not expect to get satisfactory solutions in this case. The representation type of an algebra Ais that of the category fp(A) of finitely presented (right) A-modules. One of the main results of this paper is the following theorem, where we compute the representation type of the algebra k(E) in terms of the cardinal of E, without restrictions on the ground field k. Theorem 1.1. The representation type of k(E)is card Etype <4finite = 4 tame >4wild In the finite and tame cases we construct explicit presentations of Iso(fp(k(E))). Moreover, for Efinite, we prove that there are presentations of Iso(fp(k(E))) with a finite number of elementary isomorphisms and we compute them. These presentations satisfy the next properties. Theorem 1.2. If card Eis finite there are solutions to the decomposition problem in fp(k(E)) with the next cardinals of elementary modules and isomorphisms card Emodules isomorphisms 1 6 6 2 12 12 3 21 18 ≥4≥ ℵ06 card E REPRESENTATION THEORY OF SOME INFINITE-DIMENSIONAL ALGEBRAS. . . 3 There are two key steps in the proof of these results. The first is to solve the decomposition problem for finitely presented RCFM(k)-modules. The second is to relate the decomposition problem in fp(k(E)) when card E=nis finite to the decomposition problems in fp(RCFM(k)) and in the category of finite-dimensional n-subspaces. We shall use the pro-category of pro-vector spaces and the inverse limit functor to construct invariants detecting isomorphism types of finitely presented k(E)- modules. Moreover, we shall not usually work directly with the categories of finitely presented k(E)-modules but with the equivalent categories of finitely presented modules over certain small additive categories. This setting allows more flexibility and technical proofs become less complicated than if we use k(E)-modules. In order to facilitate the reading we now describe the contents of this paper. In the next section we briefly recall from [12] the basic tools of ringoid theory that we need. Afterwards, in Section 3, we introduce the ringoids which are Morita equivalent to the algebras R(E) and establish their basic properties. For this we use the approach in [2], generalizing some results in this reference for R=ZZ to arbitrary rings. We put emphasis on the case Efinite because we shall always work under this assumption (even for the proof of Theorem 1.1, see Remark 3.10). In Section 4 we construct an embedding of the decomposition problem for countably presented R- modules into the decomposition problem for finitely presented R(E)-modules, where Eis any non-empty compact metrizable space. Section 5 contains basic facts about pro-categories. In Section 6 we construct some invariants of the isomorphism class of a finitely presented k(E)-module, Efinite. These invariants are used in Section 7 to classify finitely presented k(E)-modules when E=∗is just one point and k(E) = RCFM(k). In particular we prove that this k-algebra has finite representation type. The classification theorem (Theorem 7.1) is derived from several technical lemmas. In Section 8 we recall the definition and representation theory of the n-subspace quiver. We also define the class of rigid n-subspaces, which plays an important role in what follows. We show that all but 3nindecomposable representations of the n-subspace quiver are rigid n-subspaces. In Section 9 we relate the representation theories of both k(E) and the n-subspace quiver, where nis the cardinal of E. The properties of this relation are established through many technical results which lead us to complete the proofs of Theorems 1.1 and 1.2 in Section 10. In this last section we compute the structure of the monoid Iso(fp(k(E))) (Theorem 10.1) and give a classification theorem for finitely presented k(E)-modules (Corollary 10.5) for Efinite. This classification theorem explicitly describes the (finite) set of elementary isomorphisms, and also the set of elementary objects when Ehas less than 5 points. We include an Appendix with some computations of Ext1groups of finitely presented k(E)-modules, Efinite. These computations will be applied to proper homotopy theory in a forthcoming paper ([13]). 1.1. Notation and conventions. In this paper all rings and algebras are associative with unit. We use bold letters Cfor categories, Rfor an arbitrary (non-commutative) ring, ZZ for the ring of integers, and kfor fields. As usual IN = {1,2,3,...}is the set of natural numbers, and IN0= IN ∪ {0}the free abelian monoid with one generator. Capital sans serif letters Aare names of matrices with entries in some ring. Here all matrices are square matrices indexed by IN0, and the entry of a matrix A corresponding to the subindexes i, j ∈IN0is denoted by aij, it is A= (aij )i,j∈IN0. 4 FERNANDO MURO The identity matrix is denoted by I, it is defined as iii = 1 (i∈IN0) and iij = 0 for i6=j. The entries of the transposed matrix At= (at ij)i,j∈IN0of Aare at ij =aji (i, j ∈IN0). Vectors are denoted by (vi)n i=1 or (vn)n∈IN0provided they have a finite or an infinite countable number of entries. We regard vectors as column matrices, hence matrices act on vectors on the left. 2. Ringoids and modules Aringoid Ris a category whose morphism sets HomR(X, Y ) are abelian groups in such a way that composition is bilinear. The endomorphism set EndR(X) = HomR(X, X) of an object Xhas a ring structure with product given by composition of morphisms. Conversely any ring Ris identified with the ringoid with a single object whose endomorphism set is R. An additive category is a ringoid with finite biproducts (direct sums). In this section we recall basic facts about modules over a small ringoid. Our main reference for this subject is [12]. An additive functor between ringoids is a functor which induces homomorphisms between morphism sets. Let Ab be the category of abelian groups. A right-R-module Mis an additive functor M:Rop →Ab. Morphisms of right- R-modules are natural transformations, and the category of right-R-modules is denoted by mod(R) whenever Ris small. Left-R-modules are the same thing as right-Rop-modules, where Rop is the opposite category, so every statement about right-modules has a convenient translation to left-modules. From now on every module is a right-module unless we state the contrary. There is a Yoneda full inclusion of categories R⊂mod(R) which sends an object Xin Rto the associated contravariant representable functor HomR(−, X). These R-modules are said to be finitely generated free. They are projective by Yoneda’s lemma. An R-module Mis finitely presented (f. p.) if it is the cokernel of a morphism between two finite direct sums of finitely generated free R-modules. The cokernel of a morphism between f. p. modules is also f. p. In particular direct summands of f. p. modules are f. p. We write fp(R)⊂mod(R) for the full subcategory of f. p. R- modules. If R=Ais an additive category, then a f. p. A-module Mis in fact the cokernel in mod(A) of a morphism ϕ:X1→X0in A. One can readily check that if N= Coker[ψ:Y1→Y0] is another f. p. A-module, any morphism τ:M→Nis represented by a morphism τ0:X0→Y0such that there exists τ1:X1→Y1with τ0ϕ=ψτ1. Another morphism τ′ 0: : X0→Y0represents τif and only if there exists η:X0→Y1with τ0+ψη =τ′ 0. More precisely, let pair(A) be the additive category whose objects are morphisms in A, and morphisms τ= (τ1, τ0): ϕ→ψ are commutative squares X1 ϕ// τ1  X0 τ0  Y1 ψ//Y0 There is an obvious functor Coker: pair(A)→fp(A) given by taking cokernels. We define in pair(A) the natural equivalence relation ∼with τ∼τ′if there exists η:X0→Y1satisfying τ0+ψη =τ′ 0. Proposition 2.1. The functor Coker factors through the quotient category pair(A)/∼ and the induced functor Coker: pair(A)/∼ → fp(A)is an equivalence of categories. REPRESENTATION THEORY OF SOME INFINITE-DIMENSIONAL ALGEBRAS. . . 5 Any additive functor F:R→Sbetween small ringoids induces two “change of coefficients” additive functors F∗:mod(S)→mod(R) and F∗:mod(R)→ mod(S). The first one is exact and sends an S-module Mto the composite F∗M= MF. The second one is left-adjoint to F∗(F∗is the left additive Kan extension along F, see [12] 6) and hence right-exact. Moreover, the next diagram commutes (2.A) RF// Yoneda  S Yoneda  mod(R)F∗//mod(S) In addition if Fis full and faithful then so is F∗, and in this case F∗F∗is naturally equivalent to the identity, see [3] 3.4.1. This follows from the fact that any R-module admits a projective resolution by (arbitrary) direct sums of finitely generated free ones. The functor F∗restricts to the full subcategories of f. p. modules. If we identify the endomorphism ring EndR(X) of an object Xwith the full subcategory of Rwhose unique object is Xthe change of coefficients F∗induced by the inclusion F: EndR(X)⊂Ris the evaluation functor evX=F∗:mod(R)−→ mod(EndR(X)): M7→ M(X). The next proposition is an useful criterion to detect when a ringoid is Morita equivalent to a ring. It is a consequence of [12] 8.1. Proposition 2.2. If every object in Ris a retract of Xthen the evaluation functor evXis an additive equivalence of abelian categories which restricts to an equivalence between the full subcategories of finitely presented modules. In a more categorical language ringoids are defined as categories enriched over the monoidal category of abelian groups with the usual tensor product, compare [4] 6.2. For any ring Rone can consider the monoidal category of R-R-bimodules with the R-tensor product and define an R-ringoid as a category enriched over it. This is the same as an R-category in the sense of [12] when Ris commutative. If Ris an R-ringoid the endomorphism ring EndR(X) of an object Xis in fact an R-algebra. In this case R-modules and morphisms between them take values in the category of (right) R-modules in a natural way. 3. The algebras R(E)and related additive categories Given a ring Rand a set Awe write RhAifor the free R-module with basis set A. Free R-modules are R-R-bimodules, hence the additive category of free R-modules and right-R-module homomorphisms is an R-ringoid. The carrier of an element x∈RhAiis the (finite) set carr(x)⊂Asuch that z∈carr(x) if zappears with a non-trivial coefficient in the linear expansion of x. For every non-empty compact metrizable space Ethere exists another one X containing Esuch that the complement Y=X−Eis dense in X. The triple ¯ T= (X, Y, E) can always be chosen to be a tree-like space in the sense of [2] III.1.1. But one can also take Xto be the (unreduced) cone over E,X=CE = E×[0,1]/E × {1}. Here we identify Ewith E× {0}inside the cone CE. Afree ¯ T-controlled R-module RhAiαis a free R-module RhAitogether with a function α:A→Y, called height function, such that α−1(K) is finite for every compact subspace K⊂Y. The set Ais necessarily countable and the derived 6 FERNANDO MURO set of α(A) in Xsatisfies α(A)′⊂E. This derived set is the support of RhAiα. Controlled homomorphisms ϕ:RhAiα→RhBiβare homomorphisms between the underlying R-modules such that for every x∈Eand every neighborhood Uof xin Xthere exists another neighborhood V⊂Uof xin Xsuch that if a∈Asatisfies α(a)∈Vthen β(carr(ϕ(a))) ⊂U. The category MR(¯ T) of free ¯ T-controlled R- modules and controlled homomorphisms is a small additive category. Moreover, it is an R-ringoid. The sum and R-actions on morphism sets are given by those of the underlying free R-module homomorphisms, and the direct sum of two objects is RhAiα⊕RhBiβ=RhA⊔Bi(α,β), where A⊔Bis the disjoin union of sets and (α, β): A⊔B→Yis defined as αover Aand βover B. Remark 3.1.The category MR(¯ T) is defined in [2] III.4.7 for ¯ Ta tree-like space. However, as it is pointed out in the Remark after that definition, it is equivalent to the category B(X, E;R) in [5]. In particular MR(¯ T) only depends on Eup to equivalence of categories preserving supports of objects (in fact equivalence of R-ringoids), see 1.23 and 1.24 in [5]. The next proposition shows that free ¯ T-controlled R-modules are classified by the underlying R-module and the support. Proposition 3.2. Two free ¯ T-controlled R-modules RhAiα,RhBiβare isomorphic if and only if the next two conditions are satisfied: (1) The underlying R-modules are isomorphic RhAi ≃ RhBi, (2) both have the same support α(A)′=β(B)′. If the supports are non-empty then condition (1) is automatically satisfied. Furthermore, any compact subset K⊂Eis the support of some free ¯ T-controlled R-module. In the proof of this proposition we shall use the following Lemma 3.3. Given an injective controlled homomorphism ϕ:RhAiα→RhBiβwe have that α(A)′⊂β(B)′. Proof. For any e∈α(A)′we can take a sequence {an}n∈IN ⊂Awith lim n→∞ α(an) = e. Since ϕis injective carr(ϕ(an)) is non-empty for every n∈IN so we can take elements bn∈carr(ϕ(an)). By definition of controlled homomorphism lim n→∞ β(bn) = e, hence e∈β(B)′and the inclusion holds.  Proof of (3.2). The case R=ZZ and ¯ Ta tree-like space follows from [2] III.4.8 and III.4.16. In general condition (1) is necessary since an isomorphism of free ¯ T-controlled R-modules is also an isomorphism between the underlying R-modules. Moreover, condition (2) is necessary by (3.3). By Remark 3.1 it is enough to make the proof for tree-like spaces. In the rest of the proof we shall suppose that ¯ Tis tree-like. If α(A)′=α(B)′=∅then Aand Bare both finite and any isomorphism RhAi ≃ RhBiis a controlled isomorphism RhAiα≃RhBiβ. If α(A)′=α(B)′6=∅then A and Bare infinite countable, so ZZhAi ≃ ZZhBi. Since the proposition holds for R=ZZ there is a controlled isomorphism ϕ:ZZhAiα≃ZZhBiβ. Now one can check that ϕ⊗R:RhAiα≃RhBiβis an isomorphism of free ¯ T-controlled R-modules. Finally, given K⊂Ecompact, if ZZhCiγis a free ¯ T-controlled ZZ-module whose support is Kthen the support of RhCiγis Kas well, because it only depends on γ. The proof is now complete.  REPRESENTATION THEORY OF SOME INFINITE-DIMENSIONAL ALGEBRAS. . . 7 The next result follows directly from Remark 3.1, Proposition 3.2 and the definition of controlled homomorphisms. Proposition 3.4. Up to isomorphism, the endomorphism algebra of a free ¯ T- controlled R-module with support Eonly depends on E. Moreover, it is isomorphic to R(E). The last isomorphism of this proposition is given by the fact that the basis of a free ¯ T-controlled R-module RhAiαwith support Emust be infinite countable, and hence it can be identified with the non-negative integers A= IN0. Moreover, we can suppose that αis the inclusion of a discrete subspace α:A⊂Y, changing A by α(A) if necessary. Now we are in the same situation as in the beginning of the introduction. We also derive from (3.4) that the isomorphism class of the R-algebra R(E) only depends on E, as we claimed in the introduction. Proposition 3.5. Every free ¯ T-controlled R-module is a retract of any object whose support is E. Proof. Recall from (3.2) that all objects with support Eare isomorphic. By (3.1) it is enough to check the proposition for ¯ Ta tree-like space. For R=ZZ and ¯ Ttreelike this proposition is contained in the proof of [2] V.3.4. The result for arbitrary rings follows from the special case R=ZZ. More precisely, given a free ¯ T-controlled R-module RhAiαif the support of RhBiβis Ethen ZZhAiαis a retract of ZZhBiβ (the supports only depend on the height functions) hence we obtain a retraction of RhBiβonto RhAiαby tensoring by R. As a consequence of this proposition we get by (2.2) the following equivalence of categories which will be used from now on as an identification. Corollary 3.6. The evaluation functor in a free ¯ T-controlled R-module with support Einduces an additive equivalence of abelian categories mod(MR(¯ T)) ≃mod(R(E)) which restricts to another one fp(MR(¯ T)) ≃fp(R(E)). Remark 3.7.If R≃Rop, in particular if Ris commutative, the transposition of matrices and an explicit isomorphism R≃Rop induce isomorphisms of R-ringoids MR(¯ T)≃MR(¯ T)op (preserving objects) and R-algebras R(E)≃R(E)op, compare [5], so in this case right-modules over MR(¯ T) or R(E) are the same as left-modules. In the following proposition we compute the dimension of the k-algebra k(E), k any field. Proposition 3.8. dim k(E) = 2ℵ0. Proof. The k-vector space of all IN0×IN0matrices is the direct product of IN0×IN0 copies of k, and it is known that dim ΠIN0×IN0k= 2ℵ0, hence dim k(E)≤2ℵ0. Moreover, for any element (an)n∈IN0∈QIN0kthe diagonal matrix (bij )i,j∈IN0with bnn =anbelongs to k(E), therefore k(E) has a vector subspace isomorphic to QIN0k, and dim QIN0k= 2ℵ0as well, so the equality of the statement holds.  3.1. The special case card Efinite. If card Eis finite, since Eis metrizable, it must have the discrete topology, so Eis the discrete set nwith n= card Epoints. For this particular space we can take a tree-like space ¯ Tn= ( ˆ Tn, Tn,n) where Tnis a locally compact tree with nFreudenthal ends and ˆ Tnis the Freudenthal compactification of Tn, see [2] III.1.3. Moreover, if T0 nis the vertex set of Tnand 8 FERNANDO MURO δ:T0 n⊂Tnis the inclusion, the support of RhT0 niδis n, in particular R(n) is the endomorphism ring of this object. Let us fix the following particular tree Tn: the vertex set of Tnis T0 n={v0} ∪ v1 m,...vn mm≥1, and there are edges joining v0with vi 1and vi mwith vi m+1 (1 ≤i≤n, m ≥1). The additive category MR(¯ Tn) is equivalent to the full subcategory of objects RhAiα such that α(A)⊂T0 n, compare the proof of [2] V.3.4. From now on we shall always work in this subcategory, and we denote it by MR(¯ Tn) as well. We are going to give an alternative description for controlled homomorphisms in MR(¯ Tn). For this we define the following sets for any height function α:A→ T0 n⊂Tn(1 ≤i≤n, j ≥1) Ai j=[ l≥j α−1(vi l) A morphism ϕ:RhBiβ→RhAiαin MR(¯ Tn) is controlled if and only if for every m≥1 there exists M≥1 such that ϕ(Bi M)⊂RhAi mifor any 1 ≤i≤n. We shall omit the superindex iwhen n= 1. Moreover, for the next sections we fix the following notation (m≥0) mA=α−1(v0)∪ [ 1≤i≤n[ l≤m α−1(vi l) ,mAi j=mA∩Ai j. Remark 3.9.If n= 1, T1= [0,+∞) is the half-line and T0 1= IN0the non-negative integers. Moreover R(1) is the R-algebra RCFM(R) of IN0×IN0matrices with entries in Rsuch that every row and every column has a finite number of non-zero entries (row-column-finite matrices), compare [2] V.3.8. Remark 3.10.If Eis any compact metrizable space with at least npoints we can fully include MR(¯ Tn) into MR(¯ T). For this we only need to take ndisjoint sequences vi mm≥1(1 ≤i≤n) contained in Yconverging in Xto ndifferent points belonging to E, and an additional point v0∈Yout of the sequences. Now we identify MR(¯ Tn) with the full subcategory of MR(¯ T) given by objects RhAiα with α(A)⊂ {v0} ∪ v1 m,...vn mm≥1. If we call Fto this full inclusion, we get another one F∗:mod(MR(¯ Tn)) →mod(MR(¯ T)) together with a retraction up to natural equivalence F∗:mod(MR(¯ T)) →mod(MR(¯ Tn)). Moreover, the first functor F∗restricts to the full subcategories of f. p. modules, see Section 2, hence the decomposition problem for f. p. R(n)-modules is included in the decomposition problem for f. p. R(E)-modules, in particular we only need to prove Theorem 1.1 for card Efinite. 4. Countably presented R-modules as finitely presented RCFM(R)-modules There is a full exact inclusion of abelian categories i:mod(R)−→ mod(RCFM(R)) defined by iM= HomR(RhIN0i, M). The ring RCFM(R) acts on iMby endomorphisms of RhIN0i. REPRESENTATION THEORY OF SOME INFINITE-DIMENSIONAL ALGEBRAS. . . 9 Let f:MR(¯ T1)→mod(R) be the forgetful functor which sends a free ¯ T1- controlled R-module to its underlying R-module. The RCFM(R)-module iMcan be regarded as the functor iM= HomR(f, M): MR(¯ T1)op →Ab. Proposition 4.1. The functor ihas an exact left-adjoint rsuch that ri is naturally equivalent to the identity functor. Moreover, rcan be chosen to be the evaluation functor in a free ¯ T1-controlled R-module with one generator. Proof. Let Rheiφbe a free ¯ T1-controlled R-module whose basis is a singleton {e} (all these objects are isomorphic in MR(¯ T1) by (3.2)). The endomorphism ring of this object is R, hence the evaluation functor evRheiφtakes values in the category of R-modules. Let us see that the exact functor evRheiφis left-adjoint to i. A leftadjoint for iexists and can be constructed by left additive Kan extension, see [12] 6, so we have just to check that HomMR(¯ T1)(RhAiα,iM) = HomR(evRheiφRhAiα, M) for any free ¯ T1-controlled R-module RhAiαin a natural way. This follows from the obvious natural identification evRheiφRhAiα=RhAiand Yoneda’s lemma.  Corollary 4.2. If Mis an R-module and Nan RCFM(R)-module then there are natural isomorphisms (n≥0) Extn R(E)(N,iM)≃Extn R(rN, M). In particular if R=kis a field the RCFM(k)-modules iMare all injective. An R-module is countably presented provided it is the cokernel of a morphism between free R-modules with countable basis. Obviously the cokernel of a morphism between countably presented R-modules is countably presented as well. In particular direct summands of countably presented R-modules are countably presented. Proposition 4.3. The functor isends countably presented R-modules to finitely presented RCFM(R)-modules. In the proof of this proposition we shall use the row-column-finite matrices A and Bdefined by •ai+1,i = 1 (i∈IN0) and aij = 0 in other cases, •bn(n+1) 2+i, (n−1)n 2+i= 1 for any n > 0 and 0 ≤i < n, and bij = 0 otherwise. And we regard RCFM(R) as the endomorphism R-algebra of the free ¯ T1-controlled R-module RhIN0iδwhere δ: IN0⊂[0,+∞) is the inclusion, see Remark 3.9 and Proposition 3.4. Proof of (4.3). Since iis exact it will be enough to check the proposition for the countably presented R-modules Rand RhIN0i. Recall that HomMR(¯ T1)(RhIN0iδ,iM) = HomR(RhIN0i, M) for any R-module M. The RCFM(R)-modules iRand iRhIN0i are the cokernels of (I−A) and (I−B) respectively. The natural projections onto the cokernel are given by the homomorphisms RhIN0i → Rand RhIN0i → RhIN0i defined on generators by n7→ 1 (n≥0) and n(n+1) 2+i7→ i(n≥i≥0) respectively.  As a consequence of Propositions 4.1 and 4.3, and Remark 3.10 we get the next Corollary 4.4. The representation problem for countably generated R-modules is contained into the representation problem for finitely presented R(E)-modules, where Eis any non-empty compact metrizable space. 16 FERNANDO MURO Proof. Since Vn 0is a finite-dimensional vector space it is artinian and the decreasing sequence Vn 0⊃Vn 1⊃ · · · ⊃ Vn m⊃Vn m+1 ⊃ · · · stabilizes, it is, there exists Mn∈IN0 such that Vn m=Vn Mnfor every m≥Mn, in particular Vn Mn=Vn 0∩V∞. If we choose for every n∈IN0the minimum Mnsatisfying this condition then Mn≤Mn+1 since Vn Mn+1 =Vn Mn+1 ∩Vn+1 Mn+1 =Vn 0∩VMn+1 ∩Vn+1 0∩V∞=Vn 0∩V∞=Vn Mn. Notice that (1) is trivial for m≥Mnsince Vn m=Vn p=Vn Mnand al ij ; 1 ≤l≤rij = ∅whenever p≥m≥Mn,i≤nand j≥Mn. Therefore the unique elements (n, m, p) for which we still have to check (1) lie in the set S={(n, m, p) ; n∈IN0,0≤m≤Mn, p ≥m}. Let us order this set in the following way (n, m, p)≤(n′, m′, p′)⇔           n < n′ or n=n′and m > m′ or n=n′, m =m′and p≤p′. One readily checks that this is a well order on S, since the second coordinate has an upper bound (depending on the first one). The minimum of Sis (0, M0, M0), moreover if m < Mnthe element (n, m, m) is the least upper bound of {(n, m + 1, p) ; p > m}, and given n > 0 the element (n, Mn, Mn) is the least upper bound of the set {(n−1, m, p) ; m≤Mn−1, p ≥m}. Any other element in Sis a successor. We have already checked (1) for the elements (n, Mn, p)∈S, moreover, it is trivial for (n, m, m)∈S, hence (1) holds for the minimum and all limit elements in S. A generic successor in Shas the form (n, m, p + 1) for some p≥m. Notice that we have already check (1) for some successors as well, namely for those with m=Mn. We are now going to proceed by induction, it is, we shall prove (1) for every successor in Swith m < Mnsupposing that (1) holds for all the strictly lower elements. We are going to distinguish three cases: For (0, m, p + 1) (1) follows from the exactness of the sequence V0 p V0 p+1 ֒→V0 m V0 p+1 ։V0 m V0 p , the equality V−1 0= 0 and the inequality (0, m, p)<(0, m, p+1), and the inequality (0, p + 1, p)<(0, m, p + 1) if p < M0or the equalities V0 p=V0 p+1 =V0 M0and al 0p; 1 ≤l≤r0p=∅if p≥M0. For (n, m, m + 1) with n > 0(1) is a consequence of the exactness of the sequence Vn−1 m+Vn m+1 Vn m+1 ֒→Vn m Vn m+1 ։Vn m Vn−1 m+Vn m+1 , the obvious isomorphism Vn−1 m Vn−1 m+1 ≃Vn−1 m+Vn m+1 Vn m+1 , the inequality (n−1, m, m + 1) <(n, m, m + 1) if m < Mn−1, or the equalities Vn−1 m=Vn−1 m+1 =Vn−1 Mn−1and {ain ; 1 ≤l≤rin}=∅if i≤n−1 and m≥Mn−1. REPRESENTATION THEORY OF SOME INFINITE-DIMENSIONAL ALGEBRAS. . . 17 For (n, m, p + 1) with n > 0 and p > m (1) follows from the exactness of the sequence Vn p Vn p+1 ֒→Vn m Vn p+1 ։Vn m Vn p , the inequality (n, m, p)<(n, m, p+1), and the inequality (n, p, p+1) <(n, m, p+1) if p < Mnor the equalities Vn p=Vn p+1 =Vn Mnand al ip ; 1 ≤l≤rip=∅if i≤n and p≥Mn. Once we have seen that (1) holds, (2) is a consequence of (1) for p=Mn, the isomorphism (Vn m+V∞)/V∞≃Vn m/V n Mn, and the fact that al ij ; 1 ≤l≤rij =∅ is the empty set for i≤nand j≥Mn. Finally (3) follows from (2) and the equality Vm=Sn∈IN0Vn m. The next proposition is an interesting consequence of the previous lemma. It does not hold in general when the ground ring is not a field, compare [1]. Proposition 7.14. The image of a morphism between finitely generated free Mk(¯ T1)- modules is finitely generated free. Proof. Let ϕ:khBiβ→khAiαbe a morphism in Mk(¯ T1). If we define the k-vector spaces V0=ϕ(khBi), Vn=ϕ(khBni)⊂khAi(n∈IN) and Vn 0=ϕ(khnBi) (n≥0) we can apply Lemma 7.13. Moreover, with the notation of that lemma V∞= 0 since for every n≥1 there exists Nn≥1 such that VNn⊂khAniand ∩n≥1khAni= 0. We define the set B=al ij ;i, j ∈IN0,1≤l≤rij and the function β:B→ IN0⊂[0,+∞) by β(al nm) = m. By (7.13) (3) the set Bis a basis of V0and Bma basis of Vm(m≥1) since V∞= 0. The function βis a height function because the cardinal of β−1(m) is dim Vm/Vm+1 <ℵ0(m∈IN0). Moreover, the inclusion khBi=ϕ(khBi)⊂khAiand the projection khBi։ϕ(khBi) = khBi give rise to controlled homomorphisms khBiβ֒→khAiαand khBiβ։khBiβwhich are an Mk(¯ T1)-module monomorphism and epimorphism respectively and their composition is ϕ, hence khBiβtogether with these morphisms is the image of ϕ. Corollary 7.15. Any finitely presented Mk(¯ T1)-module is the cokernel of a monomorphism between finitely generated free Mk(¯ T1)-modules. Corollary 7.16. Finitely presented R-modules have projective dimension ≤1. Corollary 7.17. We have Ext1(M,Cd) = 0 for any f. p. R-module Mand d∈IN∞. Proof. By (7.16) the functor Ext1(M,−) is right-exact, hence the corollary follows from (7.10) and (7.12).  Now we begin with the lemmas which prove Theorem 7.2. Lemma 7.18. Given any f. p. R-module M, there exists another f. p. R-module Nwith λN= 0 such that M≃AλM⊕N. Proof. Suppose that M= Coker ϕfor some ϕ:khBiβ→khAiαin Mk(¯ T1). Let us consider the decreasing sequence of k-vector spaces given by V0=khAi/ϕ(khBi) and Vn=khAni+ϕ(khBi) ϕ(khBi), n ∈IN. 18 FERNANDO MURO The vector space V0is the union of the following sequence of finite-dimensional k-vector spaces (n∈IN0) Vn 0=khnAi+ϕ(khBi) ϕ(khBi). If al nm ; 1 ≤l≤rnm⊂Vn mis a set as in (7.13) we can suppose that al nm = el nm +ϕ(khBi) for some el nm ∈khnAmi, here we use the next obvious isomorphism khnAmi khnAmi ∩ ϕ(khBi)≃khnAmi+ϕ(khBi) ϕ(khBi)=Vn m. We consider the set C=alnm nm +V∞;n, m ∈IN0,1≤lnm ≤rnmand the function γ:C→IN0⊂[0,+∞) with γ(alnm nm +V∞) = m. This function is a height function, since the set γ−1(m) = alnm nm +V∞;n∈IN0,1≤lnm ≤rnmis bijective with a basis of Vm/Vm+1 by (7.13) (3), and we have the following surjection and isomorphisms khAmi khAm+1i։khAmi khAm+1i+ [ϕ(khBi)∩khAmi]≃khAmi+ϕ(khBi) khAm+1i+ϕ(khBi)≃Vm Vm+1 , and dim khAmi/khAm+1i= card α−1(m)<ℵ0. The underlying k-vector space of khCiγis V0/V∞, moreover, the natural projection (a) khAi։khAi T n≥1 [khAni+ϕ(khBi)] ≃V0 V∞ =khCi give rise to a ¯ T1-controlled homomorphism υ0:khAiα→khCiγwith υ0ϕ= 0, hence υ0induces a morphism υ:M→khCiγ. Furthermore, the section V0/V∞֒→khAi which sends al nm +V∞to el nm determines another ¯ T1-controlled homomorphism τ0:khCiγ→khAiαwith υ0τ0= 1, in particular if τ:khCiγ→Mis the morphism induced by τ0we have that υτ = 1, hence M≃khCiγ⊕Nwhere N= Coker τ. Notice that the morphism (ϕ, τ0): khBiβ⊕khCiγ→khAiαis a finite presentation of N, and by (a) we have the following equality and inclusions for every m≥1 khAi=\ n≥1 [khAni+ϕ(khBi)] ⊕τ(khCi)⊂khAmi+ϕ(khBi) + τ(khCi)⊂khAi, therefore λN= 0. Observe that by (7.6) and (7.7) khCiγis isomorphic to the free ¯ T1-controlled k-vector space which corresponds to Rprovided λM=∞, and to the direct sum of λMcopies of Aotherwise.  Lemma 7.19. Given a f. p. R-module Mwith λM= 0, there exists another f. p. R-module Nwith λN=µN= 0 such that M≃BµM⊕N. Proof. Suppose that Mis the cokernel of ϕ:khBiβ→khAiαin Mk(¯ T1). Since λM= 0 we have that khAi=khAmi+ϕ(khBi) for every m∈IN. Let V•be the inverse system indexed by IN ×IN given by Vmn =khAmi khAmi ∩ ϕ(khBni), REPRESENTATION THEORY OF SOME INFINITE-DIMENSIONAL ALGEBRAS. . . 19 and bonding homomorphisms induced by the obvious inclusions of vector spaces. There are inclusions Uϕ mn ⊂Vmn with quotients khAmi khAmi ∩ ϕ(khBi)≃khAmi+ϕ(khBi) ϕ(khBi)=khAi ϕ(khBi). This determines a short exact sequence in the pro-category of pro-vector spaces (a) Uϕ •֒→V•։khAi ϕ(khBi). Here we regard khAi/ϕ(khBi) as the inverse system indexed by a singleton. The vector space Uϕ mn is always finite dimensional, because it is contained in ϕ(khBi)/ϕ(khBni)≃ϕ(khn−1Bi) and n−1Bis a finite set. Since finite-dimensional vector spaces are artinian it is easy to see that Uϕ •satisfies the Mittag-Leffler property, in particular lim1Uϕ •= 0 and hence by (5.B) Ext1(khAi/ϕ(khBi), Uϕ •) = 0, so the sequence (a) admits a splitting s:khAi/ϕ(khBi)֒→V•. This splitting is given by splittings smn :khAi/ϕ(khBi)֒→Vmn of the natural projections Vmn ։ khAi/ϕ(khBi) which are compatible with the bonding homomorphisms of V•. Let ˜ Cbe a basis of khAi/ϕ(khBi). This basis is either finite ˜ C={b1, . . . bµM} if µM∈IN0, or infinite countable ˜ C={bn}n∈IN0if µM=∞. Moreover, since ϕ is controlled there exists an increasing sequence of natural numbers {ln}n≥1with ϕ(khBlni)⊂khAni. We choose elements bn−1 m∈khAniand yn−1 m∈khBlnisuch that bn−1 m+ϕ(khBlni) = sn,ln(bm)∈Vn,ln=khAni/ϕ(khBlni) and ϕ(yn−1 m) = bn m−bn−1 mfor every n∈IN and min the corresponding range. Furthermore, we define the sets nC⊂khAn+1iand Cin the following way: nC={bn 1, . . . bn µM} and C=`n∈IN0 nCif µM∈IN0, and nC={bn 0, . . . bn n} ∪ {bm m;m > n}and C=Sn∈IN0 nC={bn m;n≥m≥0}if µM=∞. Let γ:C→IN0⊂[0,+∞) be the height function given by γ(bn m) = nand ψthe endomorphism of khCiγgiven by ψ(bn m) = bn+1 m−bn m. One readily checks that Cokerψ=i(khAi/ϕ(khBi)) and the natural projection khCiγ։i(khAi/ϕ(khBi)) is given by the homomorphism p1:khCi → khAi/ϕ(khBi) = kh˜ Cidefined by p1(bn m) = bm. For this one uses the finite presentations constructed in the proof of (4.3) and, if µM=∞, the bijection IN0≈Cwhich sends m∈IN0, with n(n−1) 2≤m < (n+1)n 2for some n∈IN0, to bn−1 m−n(n−1) 2 . Moreover, by (7.11) Coker ψ=BµM. The homomorphism τ0:khCi → khAiinduced by the inclusions nC⊂khAn+1i ⊂ khAidetermines a controlled homomorphism τ0:khCiγ→khAiα. Moreover, the homomorphism τ1:khCi → khBigiven by τ1(bn m) = yn mdefines a controlled homomorphism τ1:khCiγ→khBiβwith ϕτ1=τ0ψ, hence τ0gives rise to a Mk(¯ T1)- module morphism τ:BµM→M. Let us check that τis a monomorphism. Given a free ¯ T1-controlled vector space khDiφYoneda’s lemma yields a natural identification HomR(khDiφ,i(khAi/ϕ(khBi))) = Homk(khDi, khAi/ϕ(khBi)). This identification carries a morphism υ:khDiφ→ i(khAi/ϕ(khBi)) represented by υ0:khDiφ→khCiγto the vector space homomorphism p1υ0. If p2:khAi։khAi/ϕ(khBi) is the natural projection then p2ϕ= 0 and p1=p2τ0. Moreover τυ = 0 if and only if τ0υ0=ϕη for some controlled homomorphism η:khDiφ→khBiβ, so in this case p1υ0=p2τ0υ0=p2ϕη = 0, it is, υ= 0, therefore τis a monomorphism. 20 FERNANDO MURO By (7.12) if N= Coker τthen M≃BµM⊕N. The morphism (ϕ, τ0): khBiβ⊕ khCiγ→khAiαis a finite presentation of Nand by construction ϕ(khBi) + τ0(khCi) = khAi, it is λN=µN= 0.  Lemma 7.20. Let Mbe a f. p. R-module with λM=µM= 0 then there exists another one Nwith λN=µN=νN= 0 such that M≃CνM⊕N. In the proof of this lemma we shall use the next one. Lemma 7.21. Let {dn}n∈IN be an increasing sequence of integers with lim n→∞ dn= ∞. Consider the set A={(n, m) ; n∈IN, m ≤dn} ⊂ IN ×IN, the height function α:A→IN0⊂[0,+∞)with α(n, m) = n, and the endomorphism ϕof khAiαwith ϕ(n, m) = (n, m)if n= 1 or n > 1and m > dn−1, and ϕ(n, m) = (n, m)−(n−1, m) otherwise. Then Coker ϕis isomorphic to C∞. Proof. Consider the infinite countable subsets A1={(1, m) ; 1 ≤m≤d1} ∪ {(n, m) ; n > 1, dn−1< m ≤dn} ⊂ A, A2=A−A1,B1={(n+ 1)n/2}n∈IN0⊂IN0and B2= IN0−B1. The lexicographic order from the left on Ais a well order without limit elements, since the second coordinate of an element (n, m)∈Ais bounded by dn, hence the restriction of this order to the subsets A1and A2induces enumerations A1={en 1}n∈IN0,A2= e2 nn∈IN0. Similarly the usual order in IN0induces enumerations in the subsets B1={fn 1}n∈IN0and B2={fn 2}n∈IN0. Now the theorem follows from the bijection IN0≈Awhich sends fn ito en i(i= 1,2; n∈IN0).  Proof of (7.20). If M= Coker[ϕ:khBiβ→khAiα] the equalities λM=µM= 0 are equivalent to ϕ(khBi) = khAi. Let φ: lim Uϕ •→Uϕ •be the canonical promorphism. This pro-morphism is given by vector space homomorphisms φmn : lim Uϕ •→ Uϕ mn compatible with the bonding homomorphisms of Uϕ •. Since ϕis controlled there is an increasing sequence of natural numbers {mn}n≥1such that ϕ(khBmni)⊂ khAni. If νM∈IN0and {a1,...aνM}is a basis of lim Uϕ •we define nC=an 1,...an νM⊂ khAnias a set such that φn,mn(ai) = an i+ϕ(khBmni) (1 ≤i≤νM), and choose elements yn i∈khBmn−1iif n > 1 and y1 i∈khBiwith an i−an−1 i=ϕ(yn i) (n > 1) and a1 i=ϕ(y1 i). If νM=∞we take nC={an i}dn i=1 ⊂khAnisuch that {an i+ϕ(khBmni)}is a basis of φn,mn(lim Uϕ •), here we use that Uϕ •is an inverse system of finite-dimensional vector spaces, compare the proof of (7.19). The bonding homomorphisms of Uϕ •induce surjections φn+1,mn+1 (lim Uϕ •)։φn,mn(lim Uϕ •), hence dn≤dn+1 and we can suppose without loose of generality that there exist yn i∈khBmn−1i(n > 1) and y1 i∈khBisuch that an i−an−1 i=ϕ(yn i) if n > 1 and i≤dn−1, and an i=ϕ(yn i) if n > 1 and dn−1< i ≤dnor n= 1 and i≤d1. We define the height function γ:C=`n≥1nC→IN0⊂[0,+∞) as γ(an i) = n, and the controlled homomorphisms τ0:khCiγ→khAiα,τ1:khCiγ→khBiβ, ψ:khCiγ→khCiγby τ0(an i) = an i,τ1(an i) = yn i, and ψ(an i) = an i−an−1 iif n > 1, and νM∈IN0or νM=∞and i≤dn−1, and ψ(an i) = an iotherwise. One can readily check, by using the bijection IN ≈IN0:n7→ n−1 if νM∈IN0or (7.21) if νM=∞, that Cokerψ≃CνM. Moreover, (τ1, τ0): ψ→ϕis a morphism in pair(Mk(¯ T1)) which induces an R-module morphism τ:CνM→M. In order to check that τis a monomorphism of Mk(¯ T1)-modules we are going to prove that φis a monomorphism of pro-vector spaces if νM∈IN0. In this case Ker φ REPRESENTATION THEORY OF SOME INFINITE-DIMENSIONAL ALGEBRAS. . . 21 is an inverse system of finite-dimensional vector spaces, in particular it satisfies the Mittag-Leffler property. If we apply the left-exact functor lim to the exact sequence of pro-vector spaces Ker φ ֒→lim Uϕ • φ −→ Uϕ • we get another one lim Ker φ ֒→lim lim Uϕ • = −→ lim Uϕ •, so lim Ker φ= 0 and hence Ker φ= 0 by [11] II.6.2 Lemma 2, therefore φis a monomorphism. In particular there exists N∈IN big enough such that φn,mnis an injective homomorphism for every n≥N. We set N= 1 if νM=∞. Now it is easy to see that the injection ψn:khCn+1i֒→khCnigiven by the restriction of ψ is the kernel of the next composition whenever n≥N khCniτ0 →khAni։khAni ϕ(khBmni)=Uϕ n,mn. Any morphism υ:khDiχ→CνMis represented by a controlled homomorphism υ0:khDiχ→khCiγ. Suppose that τυ = 0. This means that there exists another controlled homomorphism η:khDiχ→khBiβwith τ0υ0=ϕη. By the alternative characterization of controlled homomorphisms given in Subsection 3.1 we see that there exists an increasing sequence of natural numbers {pn}n≥1such that υ0(khDpni)⊂khCniand η(khDpni)⊂khBmni, hence if n≥Nthen there exists a unique homomorphism σn:khDpni → khCn+1isuch that ψnσn:khDpni → khCni is the restriction of υ0. If σ′:khpN−1Di → khCiis any homomorphism such that ψσ′coincides with the restriction of υ0to khpN−1Diwe define the controlled homomorphism σ:khDiχ→khCiγby σ(d) = σn(d) if d∈pn+1−1Dpn(n≥N) and σ(d) = σ′(d) if d∈pN−1D. This controlled homomorphism satisfies ψσ =υ0 therefore υ= 0 and hence τis a monomorphism. Since τis a monomorphism if we define N= Coker τwe get by (7.17) that M≃CνM⊕N. Now one can check that λN=µN=νN= 0 by using that N= Coker[(ϕ, τ0): khBiβ⊕khCiγ→khAiα].  Lemma 7.22. If Mis a f. p. R-module with λM=µM=νM= 0 then M= 0. Proof. If M= Coker[ϕ:khBiβ→khAiα] the conditions of the statement are equivalent to ϕ(khBi) = khAiand lim Uϕ •= 0. In the proof of (7.19) we checked that Uϕ •satisfies the Mittag-Leffler property, hence Uϕ •= 0 is a trivial pro-vector space by [11] II.6.2 Lemma 2. This means that if {mn}n≥1is an increasing sequence such that ϕ(khBmni)⊂khAni(see Subsection 3.1) then there exists another increasing sequence {pn}n≥1such that the next bonding homomorphisms are trivial Uϕ pn+1,mpn+1 =khApn+1 i ϕ(khBmpn+1 i) 0 −→ khApni ϕ(khBmpni)=Uϕ pn,mpn. It is, khApn+1 i ⊂ ϕ(khBmpni). Hence we can define a controlled homomorphism ψ:khAiα→khBiβsending a∈pn+2−1Apn+1 (n≥1) to any element b∈Bmpnsuch that ϕ(b) = a, and if a∈p2Awe take any ψ(a) = b∈khBisuch that ϕ(b) = a. This morphism satisfies ϕψ = 1 hence ϕis an epimorphism and M= Coker ϕ= 0.  22 FERNANDO MURO 8. Representations of the n-subspace quiver The n-subspace quiver Qnis the next directed graph /.-,()*+ 0 /.-,()*+ 1 '' OOOOOOOOOOOOOO /.-,()*+ 2  ? ? ? ? ? ? ? ? /.-,()*+ n ww o o o o o o o o o o o o o o Fixed any field k, a representation Vof Qnis a diagram of k-vector spaces indexed by Qn, it is, n+ 1 vector spaces V0, V1,...Vntogether with homomorphisms Vi→V0(1 ≤i≤n). Morphisms of representations are commutative diagrams. The category repQnof representations of Qnis an abelian category. It is equivalent to the category of kQn-modules, where kQnis the path algebra of Qn, whose dimension is dim kQn= 2n+ 1. A representation is said to be finite-dimensional provided Viis a finite-dimensional vector space for every 0 ≤i≤n. Finitely presented (or equivalently finite-dimensional) kQn-modules correspond to finitedimensional representations under this equivalence and indecomposable representations are finite-dimensional, hence by the classical Krull-Schmidt theorem the monoid Iso(repfin Qn) of isomorphisms classes of finite-dimensional representations of Qnis the free abelian monoid generated by the (isomorphism classes of) indecomposable representations. The representation type of the quiver Qnis that of its path algebra. An n-subspace Vis a representation of Qnsuch that the homomorphisms Vi→V0 are inclusions of subspaces Vi⊂V0(1 ≤i≤n). The category subn(resp. subfin n) of (finite-dimensional) n-subspaces is a full additive subcategory of repQn(resp. repfin Qn). In fact direct summands in repQnof n-subspaces are also n-subspaces, hence Proposition 8.1. Iso(subfin n)is the free abelian monoid generated by the isomorphism classes of indecomposable n-subspaces. Up to isomorphism there are just nindecomposable representations of Qnwhich are not n-subspaces, namely those with Vi=kfor some 1 ≤i≤nand Vj= 0 if j6=i, therefore Proposition 8.2. The representation type of subfin nis the same as the n-subspace quiver. We say that an n-subspace Vis rigid provided Vi⊂Pj6=0,i Vj(1 ≤i≤n) and V0=Pn i=1 Vi. As before, the category subrig n(resp. subfr n) of (finite-dimensional) rigid n-subspaces is an additive (small) subcategory of subn(resp. subfin n) and direct summands of rigid n-subspaces are also rigid, so Proposition 8.3. Iso(subfr n)is the free abelian monoid generated by the isomorphism classes of indecomposable rigid n-subspaces. There is an additive “rigidification” functor subn−→ subrig n:V7→ Vrig given by Vrig i=Vi∩Pj6=0,i Vj(1 ≤i≤n) and Vrig 0=Pn i=1 Vrig i, which is rightadjoint to the inclusion subrig n⊂subnand preserves finite-dimensional objects. REPRESENTATION THEORY OF SOME INFINITE-DIMENSIONAL ALGEBRAS. . . 23 The unit of this adjunction is the obvious natural inclusion Vrig ⊂V, which is an equality if and only if Vis already rigid. In order to determine indecomposable rigid n-subspaces we consider the full inclusions of additive categories Fi:sub1→subn(1 ≤i≤n) sending a 1-subspace Wto the n-subspace FiW=iWwith iW0=W0,iWi=W1and iWj= 0 otherwise. Proposition 8.4. The natural inclusion Vrig ⊂Vadmits a (not natural) retraction in subn. More precisely, there exist 1-subspaces Vi(1 ≤i≤n)and an isomorphism V≃(Ln i=1 FiVi)⊕Vrig such that the natural inclusion Vrig ⊂Vcorresponds to the inclusion of the direct summand. Proof. By using the definition of Vrig we see that there is a short exact sequence of vector spaces n M i=1 Vi Vrig i ֒→V0 Vrig 0 ։V0 Pn i=1 Vi . We define the 1-subspaces Vi(1 ≤i≤n) as V1 0=V0 Pn i=1 Vi⊕V1 Vrig 1 ,V1 1=V1 Vrig 1 and Vi 0=Vi 1=Vi Vrig i if 1 < i ≤n. Now the isomorphism of the statement follows from the former exact sequence.  By this proposition an indecomposable n-subspace is rigid unless it is isomorphic to FiVfor some indecomposable 1-subspace Vand 1 ≤i≤n. It is known that such Vmust be either k→kor 0 →k, so there are just 2nindecomposable n-subspaces which are not rigid, and 3nindecomposable representations of Qnwhich are not rigid n-subspaces, in particular Proposition 8.5. The category subfr nhas the same representation type as the nsubspace quiver. Remark 8.6.The representation type of the n-subspace quiver is well-known. It is finite for n < 4, tame for n= 4 and wild if n > 4, see [8] and [14]. In [8] the finite sets of indecomposable representations of Qnare described for n < 4 hence discarding the 3nindecomposable representations previously described which are not rigid n-subspaces we get the next result. Proposition 8.7. The following are complete lists of (representatives of the isomorphism classes of) indecomposable rigid n-subspaces for n < 4 •n= 1, none, •n= 2,V(2,1) = (k→k←k), •n= 3, V(3,1) =  k ↓ k→k←0 , V (3,2) =  0 ↓ k→k←k , V(3,3) =  k ↓ 0→k←k , V (3,4) =  k ↓ k→k←k , V(3,5) =  khx+yi ↓ khxi → khx, yi ← khyi . 24 FERNANDO MURO Remark 8.8.In [14] there is a (not finite) list of indecomposable representations of Q4. We do not include the list here because it is quite tedious to describe, however the interested reader can easily find and remove the 12 indecomposable representations of Q4which are not rigid 4-subspaces, obtaining in this way a complete list of indecomposable rigid 4-subspaces . 9. Finitely presented k(n)-modules and finite-dimensional n-subspaces Given an n-subspace Vwe define MV:Mk(¯ Tn)op →Ab as the additive functor which sends an object khAiαto the vector subspace MV(khAiα)⊂Homk(khAi, V0) formed by the homomorphisms φ:khAi → V0such that there exists M≥1 depending on φsatisfying φ(Ai M)⊂Vi(1 ≤i≤n). This construction defines an exact full inclusion of additive categories (9.A) M:subn→mod(Mk(¯ Tn)). Proposition 9.1. If Vis a finite-dimensional n-subspace then the Mk(¯ Tn)-module MVis finitely presented. Proof. Let {w1,...wd}be a basis of V0,wi 1,...wi dia basis of Vi(1 ≤i≤n), and φi:Vi→V0the inclusion. We define the sets D=mwi 1,...mwi di; 1 ≤i≤n, m ≥1 and C=D⊔ {w1,...wd}, and the height functions γ:C→T0 nand δ:D→T0 n with γ(wj) = v0(1 ≤j≤d) and γ(mwi j) = δ(mwi j) = vi m(1 ≤i≤n, 1≤ j≤di, m ≥1). Let ρ:khDiδ→khCiγbe the controlled homomorphism defined as ρ(mwi j) = mwi j−m−1wi jif m > 1 and ρ(1wi j) = 1wi j−φi(wi j) otherwise, and p:khCiγ→MVthe Mk(¯ Tn)-module morphism determined by the k-vector space homomorphism p0:khCi → V0with p0(wj) = wjand p0(mwi j) = φi(wi j). Here we use that Hom(khCiγ,MV) = MV(khCiγ) by Yoneda’s lemma. Now it is immediate to check that MV= Coker ρand pis the natural projection.  One can check by using the finite presentation constructed in the proof of the former proposition that Corollary 9.2. If Vis a finite-dimensional rigid n-subspace then Φn([MV]) = 0. By (9.1) the additive functor in (9.A) restricts to a functor M:subfin n→fp(Mk(¯ Tn)). Now we are going to construct a functor in the opposite direction. For this if ϕ:khBiβ→khAiαis a morphism in Mk(¯ Tn) we define the k-vector spaces (1 ≤i≤n) Wϕ i=Tm≥1nkhAi mi+ϕ(khBi)∩hPj6=ikhAj mi+ϕ(khBi)io ϕ(khBi). Proposition 9.3. The vector space Wϕ iis finite-dimensional (1 ≤i≤n). This proposition is an immediate consequence of the next Lemma 9.4. For any m≥1 dim khAi mi+ϕ(khBi)∩hPj6=ikhAj mi+ϕ(khBi)i ϕ(khBi)<ℵ0. REPRESENTATION THEORY OF SOME INFINITE-DIMENSIONAL ALGEBRAS. . . 25 Proof. One readily checks that khAi mi+ϕ(khBi)∩ X j6=i khAj mi+ϕ(khBi) = khAi mi ∩  X j6=i khAj mi+ϕ(khBi) +ϕ(khBi), therefore (a) khAi mi+ϕ(khBi)∩hPj6=ikhAj mi+ϕ(khBi)i ϕ(khBi)≃ khAi mi ∩ hPj6=ikhAj mi+ϕ(khBi)i khAi mi ∩ ϕ(khBi). Since ϕis a controlled homomorphism there exists M≥1 such that ϕ(Bi M)⊂ khAi mi(1 ≤i≤n), hence (b) X j6=i khAj mi+ϕ(khBi) = X j6=i khAj mi+ϕ(khM−1Bi) + ϕ(khBi Mi). The set M−1Bis finite, hence there exists N≥0 big enough with ϕ(M−1B)⊂ khNAi. Let us check that the next homomorphism induced by the inclusion NAi m⊂ Ai mis an isomorphism (c) khNAi mi ∩ hPj6=ikhAj mi+ϕ(khBi)i khNAi mi ∩ ϕ(khBi)→ khAi mi ∩ hPj6=ikhAj mi+ϕ(khBi)i khAi mi ∩ ϕ(khBi). The injectivity is obvious. Now by (b) an arbitrary element in the range of (c) is represented by an element ai∈khAi misuch that there are aj∈khAj mi(j6=i), a′ i∈khAi mi ∩ ϕ(khBi), b∈ϕ(khM−1Bi), and bi∈ϕ(khBi Mi) such that X j6=i aj+b+bi=ai+a′ i, but ai+a′ i−bi∈khAi mi, (⊕n j=1khAj mi)∩khNAi=⊕n j=1khNAj mi, and Pj6=iaj− (ai+a′ i−bi) = b∈khNAi, therefore aj, ai+a′ i−bi∈khNAi(j6=i), and a′ i−bi∈khAi mi ∩ ϕhBi, so ai+a′ i−birepresents the same element as aiin the range of (c), hence the homomorphism (c) is surjective. Now the proposition follows from the isomorphisms (a) and (c), and the finiteness of the set NAi m. By (9.3) the vector space Wϕ 0=Pn i=1 Wϕ itogether with the subspaces Wϕ 1,...Wϕ n define a finite dimensional n-subspace Wϕ. Proposition 9.5. There is an additive functor S:fp(Mk(¯ Tn)) →subfin nwhich sends M= Coker ϕto SM=Wϕ. Proof. By using the alternative description of controlled homomorphisms in Mk(¯ Tn) given in Subsection 3.1 one can easily check hat the correspondence ϕ7→ Wϕis a functor from pair(Mk(¯ Tn)) to the category of n-subspaces. Furthermore, this functor factors through the natural equivalence relation ∼, therefore the proposition follows by (2.1).  32 FERNANDO MURO by (A.A). Moreover, since Bis a direct summand of B∞by (7.1) then (5) also follows.  By (7.1), (7.12) and (7.17) the Ext1 Rgroup of any other pair of elementary f. p. R-modules is zero, hence the first extension groups are now completely computed for f. p. R-modules. Proposition A.4. For any pair of finite-dimensional n-subspaces there is a natural isomorphism Ext1 kQn(V, W)≃Ext1 k(n)(MV , MW). Proof. Projective representations of Qnare (arbitrary) direct sums of the following n+ 1 indecomposable n-subspaces, F1(0 →k),Fi(k→k) (1 ≤i≤n). Since Mis an exact full inclusion of categories and any finite-dimensional representation of Qnadmits a length-one projective resolution by finite-dimensional projective representations it is enough to check that (1) Ext1 k(n)(MF1(0 →k),MF1(0 →k)) = 0, (2) Ext1 k(n)(MF1(0 →k),MFi(k→k)) = 0 (1 ≤i≤n), (3) Ext1 k(n)(MFi(k→k),MF1(0 →k)) = 0 (1 ≤i≤n), (4) Ext1 k(n)(MFi(k→k),MFj(k→k)) = 0 (1 ≤i, j ≤n). The resolution constructed in the proof of (9.1) shows that MF1(0 →k) is a projective k(n)-module isomorphic to a 1-dimensional ¯ Tn-controlled k-vector space, hence (1) and (2) hold. Moreover, one can easily check (3) and (4) by using the definition of Min (9.A) and the resolutions of MFi(k→k) = 0 (1 ≤i≤n) in the proof of (9.1).  References 1. R. Ayala, M. C´ardenas, F. Muro, and A. Quintero, An elementary approach to projective dimension in proper homotopy theory, To appear in Comm. in Alg. 2. H.-J. Baues and A. 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Universidad de Sevilla. E-mail address:[email protected]