Generic modules over artin algebras

Henning Krause

Generic modules were introduced by Crawley-Boevey in order to provide a better understanding of finite dimensional algebras of tame representation type. In fact he has shown that the generic modules correspond to the one-parameter families of indecomposable finite dimensional modules over a tame algebra [5]. The Second Brauer-Thrall Conjecture provides another reason to study generic modules because the existence of a generic module over an artin R-algebra Λ (R/ rad R an infinite field) implies that Λ has strongly unbounded representation type, i.e. there are infinitely many pairwise non-isomorphic Λ-modules of length n for infinitely many n ∈ ℕ [6].

The aim of this paper is further to develop the analysis of existence and properties of generic modules. Our approach depends to a large extent on the embedding of a module category into a bigger functor category. These general concepts are explained in the first two sections. We continue in Section 3 with a new characterization of the pure-injective modules which occur as the source of a minimal left almost split morphism. This is of interest in our context because generic modules are pure-injective. Next we consider indecomposable endofinite modules. Recall that a module is endofinite if it is of finite length when regarded in the natural way as a module over its endomorphism ring. Changing the original definition slightly, we say that a module is generic if it is indecomposable endofinite but not finitely presented. Section 4 is devoted to several characterizations of generic modules in order to justify the choice of the non-finitely presented modules as the generic objects. We prove them for dualizing rings, i.e. a class of rings which includes noetherian algebras and artinian PI-rings. Existence results for generic modules over dualizing rings follow in Section 5. Several results in this paper depend on the fact that a functor f: Mod(Γ) → Mod(Λ), which commutes with direct limits and products, preserves certain finiteness conditions. For example, if a Γ-module M is endofinite, then f(M) is endofinite. If, in addition, End_{Γ}(M) is a PI-ring, then End_{Λ}(N) is a PI-ring for every indecomposable direct summand N of f(M). This material is collected in Sections 6 and 7. In Section 8 we introduce an effective method to construct generic modules over artin algebras from so-called generalized tubes. The special case of a tube in the Auslander-Reiten quiver is discussed in the following section. We obtain as a consequence a parametrization of the stable tubes by indecomposable Σ-pure-injective modules. These results are combined with Crawley-Boevey’s work on tame algebras. It is shown that for a tame algebra Λ and a fixed n ∈ ℕ almost all n-dimensional indecomposable Λ-modules belong to a union ⋃_{G} ind_{G}(Λ) of naturally defined sets such that each ind_{G}(Λ) corresponds to a generic module G. Moreover we present a new characterization of tameness in terms of homogeneous tubes in the Auslander-Reiten quiver. We continue in Section 10 with a discussion of representation embeddings. A pure representation embedding Mod(Γ) → Mod(Λ) induces a well-behaved map between the corresponding Ziegler spectra, and we apply this to algebras of wild representation type. The final section is devoted to the Krull dimension of the category of finitely presented modules over an artin algebra. We obtain a characterization of those algebras having dimension 1 which allows one to deduce that this cannot occur for a finite dimensional algebra over an algebraically closed field.

I would like to thank A. King, C.M. Ringel and M. Schmidmeier for several useful discussions.

The author is supported by a DFG Habilitandenstipendium.

1. Filtrations and dimensions for abelian categories

Let C be an abelian category. Recall that a full subcategory S of C is a Serre subcategory provided that for every exact sequence 0 → X′ → X → X′′ → 0 in C the object X belongs to S iff X′ and X′′ belong to S. For every Serre subcategory S of C one can form the quotient category C/S as follows. The objects of C/S are those of C and Hom_{C/S}(X, Y) = ⟶ lim Hom_{C}(X′, Y/Y′) with X′ ⊆ X, Y′ ⊆ Y and X/X′, Y′ ∈ S. Again the category C/S is abelian and there is canonically defined the quotient functor q: C → C/S such that q(X) = X; it is exact with Ker(q) = S [8]. Here the kernel Ker(f) of a functor f: C → D is, by definition, the full subcategory of all objects X such that f(X) = 0. If f is exact, then Ker(f) is automatically a Serre subcategory and there is induced an exact faithful functor C/ Ker(f) → D. A Serre subcategory is called localizing provided that the corresponding quotient functor admits a right adjoint. For each object X in C we denote by L(X) the lattice of its subobjects in C; it is a modular lattice with 0 and 1.

Now suppose that L is a class of modular lattices which is closed under intervals, quotients and isomorphisms. An object X is called an L-object of C if there exists a finite filtration

0 = X_{0} ⊆ X_{1} ⊆ … ⊆ X_{n} = X

of X with L(X_{i}/X_{i−1}) ∈ L for all i ∈ {1, …, n}. Note that the L-objects form a Serre subcategory of C. The L-filtration (C_{α})_{α} of C is defined recursively as follows.

– C_{−1} = 0;

– if α is an ordinal of the form α = β + 1, then C_{α} is the the Serre subcategory of all objects in C which become L-objects in C/C_{β};

– if α is a limit ordinal, then C_{α} is the smallest Serre subcategory containing all C_{γ} with γ < α.

We denote by C_{∞} the smallest Serre subcategory containing all C_{α} and define the L-dimension of C as follows. If C_{∞} = C, then the least ordinal α with C_{α} = C is denoted by dim_{L} C. Otherwise let dim_{L} C = ∞.

The L-filtration of an abelian category has the following analogue for a modular lattice which was introduced by Prest [21]. Suppose that L is a modular latice with 0 and 1. Given a pair of elements x, y ∈ L we define x ∼_{L} y, or simply x ∼ y, if there exists a finite filtration

x ∧ y = z_{0} ≤ z_{1} ≤ … ≤ z_{n} = x ∨ y

with [z_{i−1}, z_{i}] ∈ L for all i ∈ {1, …, n}. Note that ∼ defines a congruence relation on L. Therefore the set of equivalence classes L/∼ carries again the structure of a modular lattice and the canonical map L → L/∼ is a morphism. The L-cofiltration (L_{α})_{α} of L is defined recursively as follows.

– L_{−1} = L;

– if α is an ordinal of the form α = β + 1, then L_{α} = L_{β}/∼;

– if α is a limit ordinal, then L_{α} = ⟶ lim_{γ<α} L_{γ}. We denote by L_{∞} the colimit of this cofiltration and define the L-dimension of L as follows. If L_{∞} = 0, then the least ordinal α with L_{α} = 0 is denoted by dim_{L} L. Otherwise let dim_{L} L = ∞.

We collect some elementary facts. Let (C_{α})_{α} be the L-filtration of an abelian category C and denote for each α by q_{α}: C → C/C_{α} the quotient functor. Also let (L_{α})_{α} be the L-cofiltration of a modular lattice L and denote for each α by π_{α}: L → L_{α} the canonical map.

Lemma 1.1 Let X ∈ C.

(1) L(q_{α}(X)) ≅ L(X)_{α} for every ordinal α.

(2) If there is no proper Serre subcategory of C containing X, then dim_{L} C = dim_{L} L(X).

(3) If dim_{L} L < ∞, then dim_{L} L = β + 1 for some ordinal β.

Proof : (1) follows directly from the definitions.

(2) is a consequence of (1) since C_{α} = C iff q_{α}(X) = 0.

(3) Let I_{α} = {x ∈ L | π_{α}(x) = 0} for each ordinal α and note that I_{α} = ⋃_{γ<α} I_{γ} for a limit ordinal α. The assertion follows from this fact since L_{α} = 0 iff 1 ∈ I_{α}.

Now let art (noeth) be the class of modular lattices having the descending (ascending) chain condition and let fin = art ∩ noeth. Note that in [21] dim_{fin} L is called the m-dimension of L where the letter m stands for minimal. We assume for the rest of this section that L is one of those three classes. Recall that a dense chain in a lattice L is a sublattice C ≠ 0 having the property that for every pair x < y in C there is some z ∈ C with x < z < y. Note that having a dense chain is equivalent to having a sublattice isomorphic to {m·2^{−n} ∈ [0, 1] | m ∈ ℤ, n ∈ ℕ} with its usual ordering. The following lemma is well-known.

Lemma 1.2 dim_{L} L = ∞ if and only if there is a dense chain in L.

Proof : Let π: L → L_{∞} be the canonical map and suppose first L_{∞} ≠ 0. By definition, L_{∞} is a dense chain in L_{∞}. Therefore we can construct inductively a dense chain isomorphic to {m · 2^{−n} ∈ [0, 1] | m ∈ ℤ, n ∈ ℕ} in L, since for any pair x < y in L with π(x) < π(y) there is some z ∈ L with π(x) < π(z) < π(z) and therefore x < z′ < y for z′ = (x ∨ z) ∧ y. Now suppose that there is a dense chain in L, say between x and y. Using induction one shows that π(x) ≠ π(y) and therefore dim_{L} L = ∞.

Lemma 1.3 Let f: D → C be a faithful exact functor between abelian categories. Then dim_{L} D ≤ dim_{L} C.

Proof : The L-filtration (C_{α})_{α} of C induces a filtration (f^{−1}(C_{α}))_{α} of D and it is easily checked that f^{−1}(C_{α}) ⊆ D_{α} for every α, where (D_{α})_{α} denotes the L-filtration of C.

2. The functor category

Let Λ be a ring. We denote by Mod(Λ) the category of (right) Λ-modules and by mod(Λ) the full subcategory of finitely presented Λ-modules. Let D(Λ) = (mod(Λ^{op}), Ab) be the category of additive functors mod(Λ^{op}) → Ab into the category of abelian groups. The fully faithful functor

Mod(Λ) ⟶ D(Λ), M ↦ M ⊗_{Λ} −

will play an important role in our considerations. An exact sequence 0 → L → M → N → 0 in Mod(Λ) is said to be pure-exact if its image

0 ⟶ L ⊗_{Λ} − ⟶ M ⊗_{Λ} − ⟶ N ⊗_{Λ} − ⟶ 0

under this functor is exact and M ∈ Mod(Λ) is pure-injective if M ⊗_{Λ} − is injective. Note that any injective object in D(Λ) is isomorphic to M ⊗_{Λ}− for some module M. We shall also use the full subcategory C(Λ) = fp(mod(Λ^{op}), Ab) of finitely presented functors in D(Λ). Observe that C(Λ) is an abelian subcategory, equivalently that finitely generated subfunctors of finitely presented functors again are finitely presented. For details and proofs of the basic properties of the category D(Λ) we refer to Crawley-Boevey’s exposition [6].

Now fix a Λ-module M and let Γ = End_{Λ}(M)^{op}. Identifying End(M ⊗_{Λ} −)^{op} in D(Λ) with Γ we obtain a functor

h_{M}: C(Λ) ⟶ Mod(Γ), X ↦ Hom(X, M ⊗_{Λ} −)

which is exact. We denote by S_{M} the kernel of h_{M} which is a Serre subcategory of C(Λ). The endocategory E_{M} of M is, by definition, the image of the induced functor C(Λ)/S_{M} → Mod(Γ) and we shall tacitly assume that this subcategory of Mod(Γ) is closed under isomorphisms. The endocategory of M seems to be closely related to M^{eq+} which occurs in the model theoretic literature.

We consider now subobjects of M. Denote by L(M) = L(M_{Λ}) the lattice of Λ-submodules of M. Recall that an additve subgroup of M is of finite definition if it arises as the kernel of a morphism M → M ⊗_{Λ} X, m ↦ m ⊗ x for some finitely presented Λ^{op}-module X, and some element x ∈ X. Note that a subgroup of finite definition is a Γ-submodule of M. Moreover, the set of all these subgroups is a sublattice of L(M_{Γ}). Following the notation in the preceding section we denote by dim_{L} M the L-dimension of the lattice of subgoups of finite definition. The following lemma collects the essential properties of E_{M}.

Lemma 2.1 (1) The endocategory E_{M} of M is abelian and the inclusion E_{M} → Mod(Γ) is exact.

(2) The assignment X ↦ Hom(X, M ⊗_{Λ} −) induces an equivalence (C(Λ)/S_{M})^{op} → E_{M}.

(3) The Γ-module M belongs to E_{M} and the subobjects of M in E_{M} are precisely the subgroups of finite definition.

(4) Every object in E_{M} is a subquotient of a finite coproduct of copies of M.

Proof : (1) and (2) follow immediately from the definition of E_{M}.

(3) We observe first that M belongs to E_{M} since M ≅ Hom(Hom_{Λ^{op}}(Λ, −), M ⊗_{Λ} −) by Yoneda’s lemma. Now suppose that U is subobject of M in E_{M}. Thus there is a an epimorphism Hom_{Λ^{op}}(Λ, −) → F in C(Λ) such that h_{M}(F) = U. Choose a presentation Hom_{Λ^{op}}(X, −) → Hom_{Λ^{op}}(Λ, −) → F → 0 and apply Yoneda’s lemma to obtain U as kernel of a morphism M → M ⊗_{Λ} X of the right form. Conversely, suppose that U = Ker(M ⊗_{Λ} φ) for some φ: Λ → X in mod(Λ^{op}). It follows that U = h_{M}(F) for F = Coker(Hom_{Λ^{op}}(φ, −)).

(4) The assertion follows from the fact that every object in C(Λ) is a subquotient of a finite coproduct of copies of Hom_{Λ^{op}}(Λ, −).

A full subcategory X of Mod(Λ) is called definable if there is a Serre subcategory S of C(Λ) such that a Λ-module M belongs to X iff Hom(S, M ⊗_{Λ}−) = 0. Note that the modules in X form the models of the theory determined by S [28, 21]. A definable subcategory is automatically closed under direct limits, products and pure submodules, and it can be shown that this property characterizes a definable subcategory [19, Corollary 4.6]. Let X be a class of Λ-modules. We shall frequently identify X = M if X consists of a single object M. We denote by S_{X} the Serre subcategory of all objects X ∈ C(Λ) satisfying Hom(X, M ⊗_{Λ} −) = 0 for all M in X and we call the definable subcategory corresponding to S_{X} generated by X.

Finally, let us recall from [28] that the Ziegler spectrum Zsp(Λ) of Λ is the set of isomorphism classes of indecomposable pure-injective Λ-modules. The following description of the topology on Zsp(Λ) is due to Herzog [11].

Proposition 2.2 The assignment

S ↦ {M ∈ Zsp(Λ) | Hom(S, M ⊗_{Λ} −) = 0}

defines a bijection between the Serre subcategories of C(Λ) and the closed subsets of Zsp(Λ). Moreover, for any object X ∈ C(Λ) the set {M ∈ Zsp(Λ) | Hom(X, M ⊗_{Λ} −) = 0} is closed.

Proof : See [15, Theorem 4.2].

Given a class X of modules we denote by \overline{X} the closed subset corresponding to S_{X} and call \overline{X} the Ziegler closure of X.

We end this section with some examples which illustrate the preceding definitions. Recall that a ring Λ is right coherent if every finitely generated submodule of a finitely presented Λ-module is finitely presented. Furthermore a Λ-module M is fp-injective if Ext_{Λ}^{1}(X, M) = 0 for every finitely presented X. The following characterization of a right coherent ring is a variation of a model theoretic result due to Eklof and Sabbagh [7].

Example 2.3 Λ is right coherent iff the fp-injective Λ-modules form a definable subcategory. In this case the indecomposable injective Λ-modules form a closed subset of Zsp(Λ). The corresponding Serre subcategory S of C(Λ) is the kernel of the functor C(Λ) → mod(Λ), F ↦ F (Λ), which induces an equivalence between C(Λ)/S and mod(Λ).

Proof : It is well-known that Λ is right coherent iff the full subcategory fpinj(Λ) of fp-injective Λ-modules is closed under direct limits [27]. Moreover fpinj(Λ) is automatically closed under pure submodules and products. It follows that Λ is right coherent iff fpinj(Λ) is definable. An fp-injective Λ module is pure-injective iff it is injective and therefore fpinj(Λ) ∩ Zsp(Λ) is precisely the set of indecomposable injective Λ-modules. We conclude that this set is closed if Λ is right coherent. Given a functor F ∈ C(Λ) there is an exact sequence

0 ⟶ F ⟶ X ⊗_{Λ} − ⟶^{φ⊗_{Λ}−} Y ⊗_{Λ} −

with X, Y ∈ mod(Λ). Thus F (Λ) = 0 iff φ is a monomorphism and therefore S_{fpinj(Λ)} = {F ∈ C(Λ) | F (Λ) = 0}. We leave it to the reader to verify that the evaluation

D(Λ) = (mod(Λ^{op}), Ab) ⟶ (Λ^{op}, Ab) = Mod(Λ), F ↦ F (Λ)

induces an equivalence C(Λ)/S → mod(Λ).

Suppose now that Λ is right noetherian. Thus every finitely generated Λ-module is finitely presented and therefore Baer’s criterion implies that every fp-injective Λ-module is injective. We proceed with a description of the closed subsets of the injective modules in Zsp(Λ), assuming that Λ is commutative. Denote by Spec(Λ) the prime spectrum of Λ. We shall identify every prime ideal 𝔭 ∈ Spec(Λ) with the injective envelope E(Λ/𝔭) ∈ Zsp(Λ). Recall that every localizing subcategory T of Mod(Λ) is completely described by the corresponding Gabriel topology F = {𝔞 ⊆ Λ | Λ/𝔞 ∈ T }.

Example 2.4 (1) The assignment F ↦ Spec(Λ) \ F defines a bijection between the Gabriel topologies on Λ and the closed subsets of Spec(Λ) ⊆ Zsp(Λ).

(2) If M is an injective cogenerator for Mod(Λ), then dim_{fin} M equals dim_{fin} L(Λ) which is the Krull dimension of Λ.

Proof : (1) The assignment T ↦ T ∩ mod(Λ) defines a bijection between the localizing subcategories of Mod(Λ) and the Serre subcategories of mod(Λ) [8, III, Proposition 4.10]. But the Serre subcategories of mod(Λ) are in bijective correspondence to the Serre subcategories of C(Λ) containing S_{fpinj(Λ)} by the preceding example. The assertion now follows from Proposition 2.2.

(2) We have dim_{fin} M = dim_{fin} mod(Λ) since S_{M} = S_{fpinj(Λ)}. Moreover dim_{fin} mod(Λ) = dim_{fin} L(Λ) by Lemma 1.1 and this equals the Krull dimension of Λ in the sense of Gabriel [8, III, Corollary 4.2].

3. Almost split sequences for pure-injective modules.

We characterize the indecomposable pure-injective modules which occur as a source of an almost split sequence. This property will play a significant role in our discussion of indecomposable endofinite modules. We need to recall the following terminology. A morphism φ: M → N is left almost split if it is not a split monomorphism, and any morphism M → X which is not a split monomorphism factors through φ. Analogously, a right almost split morphism is defined. An exact sequence 0 → M →^{φ} N →^{ψ} L → 0 is almost split if φ is left almost split and ψ is right almost split. A morphism φ: M → N is left minimal if any endomorphism ψ of N satisfying φ = ψ∘φ is an automorphism. Note that a morphism φ: M → N is minimal left almost split if and only if one of the following two cases occurs [2, II, Proposition 4.4]:

(1) M is injective with simple socle and φ is isomorphic to M → M/ soc M.

(2) There is an almost split sequence 0 → M →^{φ} N →^{ψ} L → 0.

Now let M be a finitely presented Λ-module with local endomorphism ring Γ = End_{Λ}(M)^{op}. Then we denote by DM the Λ^{op}-module Hom_{Γ}(M, I) where I is an injective envelope of the unique simple Γ-module. Note that DM has automatically a local endomorphism ring. If M is pure-injective and DM is finitely presented, then DDM ≅ M [29, Lemma 5]. We are now in a position to state the main result of this section.

Theorem 3.1 Let Λ be a semi-perfect ring. Then the following are equivalent for a Λ-module M.

(1) M is pure-injective and occurs as a source of a minimal left almost split morphism.

(2) M ≅ DN for some finitely presented Λ^{op}-module N having a local endomorphism ring.

The proof is based on an existence result for almost split sequences which is due to Auslander. Note that Auslander’s result also gives an explicit relation between the module N occuring in (2) and the cokernel of the minimal left almost split morphism starting in M in case M is non-injective. We need the following definition. If X is a finitely presented Λ-module without non-zero projective direct summand and Q →^{φ} P → X → 0 is a minimal projective presentation, then the transpose Tr X is the finitely presented Λ^{op}-module which is the cokernel of Hom_{Λ}(φ, Λ). Note that

Hom_{Λ}(P, Λ) ⟶^{Hom_{Λ}(φ,Λ)} Hom_{Λ}(Q, Λ) ⟶ Tr X ⟶ 0

is again a minimal projective presentation. Therefore End_{Λ}(X) is local iff End_{Λ^{op}}(Tr X) is local.

Proposition 3.2 Let X be a finitely presented non-projective Λ-module with local endomorphism ring. Then there exists an almost split sequence 0 → D Tr X → E → X → 0.

Proof : See [2, II, Proposition 5.1].

The following lemma is well-known.

Lemma 3.3 A sequence 0 → M → N → L → 0 of Λ-modules is pure-exact iff the induced sequence 0 → Hom_{Λ}(X, M) → Hom_{Λ}(X, N) → Hom_{Λ}(X, L) → 0 is exact for every finitely presented Λ-module X.

Proof : Use the isomorphism \underline{Hom}_{Λ}(X, −) ≅ Tor_{1}^{Λ}(Tr X, −) [2, I, Proposition 3.2] where \underline{Hom} denotes the goup of morphisms modulo the subgroup of those which factor through a projective.

Proof of Theorem 3.1 : (1) ⇒ (2) Suppose first that M is injective and let S = soc M. Using the fact that Λ is semi-perfect we find an indecomposable projective Λ^{op}-module N such that S ⊗_{Λ} N ≠ 0. The isomorphism

Hom_{Λ}(−, Hom_{Γ}(N, I)) ≅ Hom_{Γ}(− ⊗_{Λ} N, I)

shows that DN = Hom_{Γ}(N, I) is injective with Hom_{Λ}(S, DN) ≠ 0. Thus DN is an injective envelope of S and therefore isomorphic to M. Now suppose that 0 → M →^{φ} E →^{ψ} L → 0 is an almost split sequence. The sequence is not pure-exact since M is pure-injective and therefore Hom_{Λ}(X, ψ) is not an epimorphism for some finitely presented X. The minimality of φ now implies that L is isomorphic to a direct summand of X and therefore finitely presented. Moreover, End_{Λ}(L) is local since ψ is right almost split. We can apply the above proposition and infer that M ≅ D Tr L.

(2) ⇒ (1) Let M ≅ DN. Suppose first that N is projective. The isomorphism used in the first part of the proof shows that M is injective. Moreover, if the Λ-module Hom_{Λ^{op}}(N, Λ) is the projective cover of the simple module S, then the isomorphism

S ⊗_{Λ} N ≅ Hom_{Λ}(Hom_{Λ^{op}}(N, Λ), S)

combined with the above isomorphism shows that there is a non-zero morphism S → DN. The cokernel morphism then gives a minimal left almost split morphism with source DN. If N is non-projective, then Tr N is non-projective and the above proposition yields an almost split sequence with source DN since Tr Tr N ≅ N. Moreover, the sequence is not pure-exact since the end term is finitely presented, and therefore DN is pure-injective.

4. Endofinite modules over dualizing rings

Given a Λ-module M we denote by ℓ_{end}(M) the length of M when regarded in the natural way as a Γ = End_{Λ}(M)^{op}-module. The module is called endofinite if its endolength is finite. Any endofinite module is automatically Σ-pure-injective since it satisfies the descending chain condition on subgroups of finite definition. Recall that M is Σ-pure-injective if any coproduct of copies of M is pure-injective. We call a module generic if it is endofinite and indecomposable but not finitely presented. Now suppose that M is endofinite and also finitely presented indecomposable. Then DM denotes the Λ^{op}-module Hom_{Γ}(M, I) where I is an injective envelope of the unique simple Γ-module. Note that DM is automatically indecomposable and endofinite. Following [18] the ring Λ is said to be right dualizing if it satisfies the following condition:

(D) If M is a finitely presented indecomposable endofinite Λ-module, then its dual DM is a finitely presented indecomposable endofinite Λ^{op}-module.

If Λ and Λ^{op} both satisfy (D), then we call Λ dualizing. Important examples of dualizing rings are noetherian algebras [5] and artinian PI-rings [26]. Recall that Λ is a noetherian algebra if the centre Z(Λ) is noetherian and Λ is a finitely generated Z(Λ)-module. Noetherian algebras over complete local rings are examples of Krull-Schmidt rings [2]. A ring Λ is Krull-Schmidt if every finitely presented Λ-module decomposes into a coproduct of indecomposable modules having local endomorphism rings. Observe that this notion is right-left symmetric since Λ needs to be semi-perfect. Of course every artinian ring is Krull-Schmidt.

Our aim in this section is a characterization of the endofinite modules which are not generic. We shall use the following description of the injective envelopes of simple functors in D(Λ) which is due to Crawley-Boevey [6].

Proposition 4.1 For an indecomposable pure-injective Λ-module M the following are equivalent.

(1) M ⊗_{Λ} − is the injective envelope of a simple functor.

(2) M occurs as a source of a left almost split morphism.

(3) M ≅ Hom_{Γ}(N, I) with N some finitely presented Λ^{op}-module, Γ = End_{Λ^{op}}(N)^{op} and I the injective envelope of a simple Γ-module.

Proof : See [6, Theorem 2.3].

Having determined in the preceding section the indecomposable pure-injective modules which occur as a source of a minimal left almost split morphism, we observe now that there is no difference between the existence of a left almost split and a minimal left almost split morphism if Λ is a Krull-Schmidt ring.

Proposition 4.2 Let Λ be a Krull-Schmidt ring. Then the following are equivalent for an indecomposable pure-injective Λ-module M.

(1) M occurs as a source of a left almost split morphism.

(2) M occurs as a source of a minimal left almost split morphism.

Proof : Suppose that M is the source of a left almost split morphism. Then M ≅ Hom_{Γ}(N, I) for some N ∈ mod(Λ^{op}) by the preceding lemma. In fact M ⊗_{Λ} − is an injective envelope of a simple factor of Hom_{Λ^{op}}(N, −) which needs to be isomorphic to a factor of Hom_{Λ^{op}}(N′, −) for some indecomposable direct summand N′ of N by our assumption on the ring. The assertion now follows from Theorem 3.1 since M ≅ DN′.

We call an indecomposable pure-injective Λ module M isolated if {M} is an open set in Zsp(Λ). The next lemma collects the basic properties of isolated modules; related ideas can be found in [10].

Lemma 4.3 Let M be an indecomposable pure-injective Λ-module.

(1) If M is isolated, then M ⊗_{Λ} − is the injective envelope of a simple functor.

(2) If M is finitely presented and M ⊗_{Λ} − is the injective envelope of a simple functor, then M is isolated.

Proof : (1) The module M is isolated if there is a non-zero X ∈ C(Λ) such that Hom(X, N ⊗_{Λ} −) = 0 for every N ∈ Zsp(Λ) different from M. Choosing a maximal subobject U ⊆ X in D(Λ) we infer that M ⊗_{Λ} − is an injective envelope of X/U.

(2) A simple subfunctor S of M ⊗_{Λ} − is obviously finitely generated and therefore also finitely presented since M ⊗_{Λ} − is finitely presented and C(Λ) is abelian. Thus {M} = {N ∈ Zsp(Λ) | Hom(S, N ⊗_{Λ} −) ≠ 0} since an injective envelope is unique up to isomorphism, and it follows from Proposition 2.2 that {M} is open.

Lemma 4.4 Let M be a finitely presented indecomposable pure-injective Λ-module and suppose that DM is finitely presented. Then M ⊗_{Λ} − is the injective envelope of a finitely presented and simple functor in D(Λ), and therefore M is isolated.

Proof : The assumption implies that M ≅ DDM by [29, Lemma 5]. Thus M ⊗_{Λ} − is the injective envelope of a simple functor, say S, by Proposition 4.1. The arguments used in the preceding lemma show that S is finitely presented and that M is isolated.

Lemma 4.5 A non-zero module M is endofinite if and only if dim_{fin} M = 0.

Proof : See [6, Proposition 4.1].

Lemma 4.6 Let M be an indecomposable endofinite Λ-module. If M occurs as a source of an almost split morphism, then M ≅ DN for some finitely presented indecomposable endofinite Λ^{op}-module N.

Proof : It follows from Proposition 4.1 that M ⊗_{Λ} − is the injective envelope of a simple functor S and M ≅ Hom_{Γ}(L, I) with L some finitely presented Λ^{op}-module, Γ = End_{Λ^{op}}(L)^{op} and I the injective envelope of a simple Γ-module. More precisely, there is an epimorphism π: Hom_{Λ^{op}}(L, −) → S and the kernel of π_{L}: Hom_{Λ^{op}}(L, L) → S(L) is the maximal ideal such that I is the injective envelope of Γ/ Ker(π_{L}). We shall identify End_{Γ}(I)^{op} with Σ = End_{Λ}(M)^{op} via φ ↦ Hom_{Γ}(L, φ). Taking an injective envelope J of I ∐ Σ/ rad Σ in Mod(Σ) we define P = Hom_{Σ}(M, J). The module J is an injective cogenerator for Σ and therefore Hom_{Σ}(−, J) induces an anti-isomorphism between the lattice of subgroups of finite definition of M and P. This follows from the well-known isomorphism

X ⊗_{Λ} Hom_{Σ}(M, J) ⟶ Hom_{Σ}(Hom_{Λ}(X, M), J)

for all X ∈ mod(Λ) which is functorial in X [4, VI, Proposition 5.2], if one uses the alternative description of a subgroup of finite definition of M as Im(Hom_{Λ}(φ, M)) for some morphism φ: Λ → X in mod(Λ). Thus P is endofinite by the preceding lemma and there is a decomposition P = ∐_{i∈K} P_{i} into indecomposable endofinite Λ^{op}-modules [6, Proposition 4.5]. We claim that M ≅ DP_{i} for some i. To this end consider the morphism α: L → P defined by α(x)(φ) = φ(x). The following commutative diagram shows that M ⊗_{Λ} α is a monomorphism.

Hom_{Γ}(L, I) ⊗_{Λ} L ⟶ Hom_{Γ}(Hom_{Λ^{op}}(L, L), I) ⟶ I
↓ M⊗_{Λ}α ↓
M ⊗_{Λ} Hom_{Σ}(M, J) ⟶ Hom_{Σ}(Hom_{Λ}(M, M), J) ⟶ J

In fact, the horizontal morphisms are the obvious evaluation maps and the upper ones are isomorphisms since L is finitely presented. We consider now the composition

Hom_{Λ^{op}}(P, −) ⟶^{Hom_{Λ^{op}}(α,−)} Hom_{Λ^{op}}(L, −) ⟶^{π} S.

Applying Hom(−, M ⊗_{Λ} −) it follows by Yoneda’s lemma that π∘ Hom_{Λ^{op}}(α, −) ≠ 0. Thus for some i ∈ K the corresponding morphism α_{i}: L → P_{i} induces an epimorphism

Hom_{Λ^{op}}(P_{i}, −) ⟶^{Hom_{Λ^{op}}(α_{i},−)} Hom_{Λ^{op}}(L, −) ⟶^{π} S.

Using again Yoneda’s lemma, we see that there is a morphism β: P_{i} → L such that

π = π∘ Hom_{Λ^{op}}(α_{i}, −)∘ Hom_{Λ^{op}}(β, −).

The composition γ = α_{i}∘β is an automorphism of P_{i} since

π∘ Hom_{Λ^{op}}(α_{i}, −) = π∘ Hom_{Λ^{op}}(α_{i}, −)∘ Hom_{Λ^{op}}(γ, −)

and End_{Λ^{op}}(P_{i}) is local. Thus P_{i} is isomorphic to a direct summand of L and therefore finitely presented. Moreover M ≅ DP_{i} and this finishes the proof.

We collect our findings and obtain the following characterization of endofinite modules which are not generic. This generalizes a result of Crawley-Boevey for noetherian algebras [6, Proposition 7.7] and includes also a result of Schmidmeier [26, II, Satz 5]. However, their methods are quite different.

Theorem 4.7 Let Λ be a dualizing ring. Then the following are equivalent for an indecomposable endofinite Λ-module M.

(1) M is finitely presented.

(2) M is isolated.

(3) M occurs as a source of a left almost split morphism.

(4) M ≅ DN for some finitely presented indecomposable endofinite Λ^{op}-module N.

Proof : (1) ⇒ (2) follows from Lemma 4.4 since Λ is right dualizing.

(2) ⇒ (3) This part needs no assumption on the ring. It follows from Lemma 4.3 and Proposition 4.1.

(3) ⇒ (4) This part needs no assumption on the ring. It follows from Lemma 4.6.

(4) ⇒ (1) follows from the assumption that Λ is left dualizing.

The assumption on the ring to be dualizing is necessary in the preceding theorem. In fact, one obtains the following characterization of dualizing rings.

Corollary 4.8 For a ring Λ the following are equivalent.

(1) Λ is dualizing.

(2) Every indecomposable endofinite Λ- or Λ^{op}-module is finitely presented if and only if it is isolated.

(3) Every indecomposable endofinite Λ- or Λ^{op}-module is finitely presented if and only if it occurs as a source of a left almost split morphism.

Proof : (1) ⇒ (2), (3) follows from the preceding theorem.

(3) ⇒ (1) Let M be a finitely presented indecomposable and endofinite Λ-module. We want to show that DM is finitely presented. It follows from Lemma 4.6 that M ≅ DN for some finitely presented indecomposable endofinite Λ^{op}-module N. Thus DM ≅ DDN ≅ N, and therefore DM is finitely presented. We conclude that Λ is right dualizing and the dual argument shows that Λ is also left dualizing.

(2) ⇒ (1) Adapt the argument of (3) ⇒ (1).

5. Existence of generic modules

This section is devoted to existence results for endofinite modules. Our first criterion needs no assumption on the ring. It also follows from Lemma 8.4 in [28].

Proposition 5.1 Let M be a non-zero Λ-module and suppose that M has no dense chain of subgroups of finite definition. Then there exists an endofinite Λ-module in the Ziegler closure of M.

Proof : Let C = C(Λ)/S_{M}. The assumption on M implies that dim_{fin} C = dim_{fin} M = β + 1 for some ordinal β by Lemma 2.1 and Lemma 1.1. Let (C_{α})_{α} be the fin-filtration of C and denote by q the composition of the quotient functors C(Λ) → C(Λ)/S_{M} and C → C/C_{β}. The kernel S = Ker(q) is a proper Serre subcategory of C(Λ) and the bijective correspondence between Serre subcategories and closed subsets of Zsp(Λ) allows to choose N ∈ Zsp(Λ) such that Hom(S, N ⊗_{Λ} −) = 0. We have S_{M} ⊆ S ⊆ S_{N}. Clearly, the inclusion implies that N belongs to the definable subcategory generated by M. The second inclusion implies that N is endofinite. In fact there is an induced quotient functor C/C_{β} → C(Λ)/S_{N} and therefore

dim_{fin} N = dim_{fin} C(Λ)/S_{N} ≤ dim_{fin} C/C_{β} = 0.

We conclude from Lemma 4.5 that N is endofinite.

Our next aim is to show that under certain additional assumptions the endofinite module constructed above is a generic module. We need some lemmas.

Lemma 5.2 Let M and N be Λ-modules. Suppose that N is finitely presented indecomposable pure-injective and that DN is finitely presented. Then the following are equivalent.

(1) N is isomorphic to a direct summand of M.

(2) N belongs to the Ziegler closure of M.

Proof : One direction is trivial. Therefore suppose (2). It follows from Lemma 4.4 that N ⊗_{Λ}− has a finitely presented and simple socle, say S, and (2) implies Hom(S, M ⊗_{Λ}−) ≠ 0. A non-zero morphism S → M ⊗_{Λ} − extends to a morphism φ: N ⊗_{Λ} − → M ⊗_{Λ} − since Hom(−, M ⊗_{Λ} −) is exact on C(Λ) and N ⊗_{Λ} − is finitely presented. Moreover, φ is a monomorphism since N ⊗_{Λ} − is the injective envelope of S. We conclude that φ induces a split monomorphism N → M since N is pure-injective, and therefore (1) holds.

We need to study the lattice of subgroups of finite definition of a Λ-module M. To this end consider again the exact functor

h_{M}: C(Λ) ⟶ Mod(Γ), X ↦ Hom(X, M ⊗_{Λ} −)

where Γ = End_{Λ}(M)^{op} and let F = Hom_{Λ^{op}}(Λ, −). A chain

0 = F_{0} ⊆ F_{1} ⊆ … ⊆ F_{n} = F

of subobjects in C(Λ) is denoted by (F_{i})_{0≤i≤n}.

Lemma 5.3 M is endofinite of endolength at most n iff for every chain (F_{i})_{0≤i≤n+1} there exists i ∈ {0, …, n} such that h_{M}(F_{i+1}/F_{i}) = 0.

Proof : We use the fact that all Γ-submodules are of finite definition if M is endofinite [6, Proposition 4.1]. The assertion now follows with Lemma 4.5 since h_{M} induces an epimorphism from the lattice of subobjects of F onto the lattice of subgroups of finite definition of M by Lemma 2.1.

The following is due to Herzog [11].

Proposition 5.4 For every n ∈ ℕ the modules of endolength at most n form a closed subset of the Ziegler spectrum.

Proof : For every chain φ = (F_{i})_{0≤i≤n+1} we define U_{φ,i} = {M ∈ Zsp(Λ) | h_{M}(F_{i+1}/F_{i}) = 0} for all i and U_{φ} = ⋃_{i=0}^{n} U_{φ,i}. These are closed subsets of Zsp(Λ) and therefore also the intersection U = ⋂_{φ} U_{φ} is closed where φ runs through all chains φ = (F_{i})_{0≤i≤n+1}. But U is precisely the set of all modules of endolength at most n by the preceding lemma, and this finishes the proof.

We shall also use the compactness of Zsp(Λ) which is due to Ziegler [28].

Proposition 5.5 Zsp(Λ) is a quasi-compact topological space.

Proof : The proof is based on our description of the Ziegler topology in Proposition 2.2. Let (S_{i})_{i∈I} be a family of Serre subcategories. Then ⋃_{i∈I} S_{i} = C(Λ) iff F = Hom_{Λ^{op}}(Λ, −) ∈ ⋃_{i∈I} S_{i} since every finitely presented functor is a subquotient of F^{n} for some natural n. But the latter condition is equivalent to F ∈ ⋃_{i∈J} S_{i} for some finite J ⊆ I since F ∈ ⋃_{i∈I} S_{i} iff there exists a finite chain (F_{j})_{0≤j≤n} with elements i_{j} ∈ I such that F_{j+1}/F_{j} ∈ S_{i_{j}} for all j.

We are now in a position to prove the following main result. Note that for artin algebras the second criterion is due to Crawley-Boevey [6].

Theorem 5.6 Let Λ be a right dualizing ring. Then there exists a generic Λ-module provided that one of the following conditions holds.

(1) There is a non-zero Λ-module having no finitely presented indecomposable endofinite direct summand and no dense chain of subgroups of finite definition.

(2) For some natural number n there is an infinite number of finitely presented indecomposable Λ-modules of endolength at most n.

Proof : (1) It is shown in Proposition 5.1 that there exists an indecomposable endofinite module N in the Ziegler closure of M. To show that N is not finitely presented we assume the converse. Then N is isomorphic to a direct summand of M by Lemma 5.2 since Λ is right dualizing. This contradicts condition (1) and therefore N is a generic module.

(2) We follow an idea of Herzog. Suppose that U = {M ∈ Zsp(Λ) | ℓ_{end}(M) ≤ n} is infinite. It is a quasi-compact space according to the preceding propositions and therefore {M} is not open for some M ∈ U. We conclude from Lemma 4.4 that U contains a generic module.

Our next aim is to prove a refinement of the preceding theorem. It depends on an analysis of injective envelopes in certain quotient categories of D(Λ).

Lemma 5.7 Let S be a Serre subcategory of C(Λ) and denote by S^{→} the full subcategory of D(Λ) formed by all direct limits of objects in S.

(1) S^{→} is a localizing subcategory of D(Λ).

(2) The quotient functor q: D(Λ) → D(Λ)/S^{→} induces a fully faithful and exact functor

C(Λ)/S → D(Λ)/S^{→}.

(3) Let X ∈ C(Λ) and suppose that q(X) is simple in C(Λ)/S. Then q(X) is also simple

in D(Λ)/S^{→}.

(4) The quotient functor q sends an injective object M ∈ D(Λ) satisfying Hom(S, M) = 0 to an injective object and induces an isomorphism Hom(X, M) → Hom(q(X), q(M)) for every X ∈ D(Λ).

Proof : (1) See [15, Theorem 2.8].

(2) We need to show that S^{→} ∩ C(Λ) = S. Then the kernel of the canonical functor

C(Λ) ⟶ D(Λ) ⟶^{q} D(Λ)/S^{→}

is S and there is induced a faithful exact functor C(Λ)/S → D(Λ)/S^{→}. Now let X = ⟶ lim X_{i} ∈ S^{→} ∩ C(Λ) with X_{i} ∈ S for all i. Denote by Y_{i} the image of X_{i} → X and observe that Y_{i} ∈ S for all i. Clearly, X = ∑ Y_{i} and therefore X = Y_{j} for some j since X is finitely generated. Thus X ∈ S. For the fact that the induced functor is full we refer to [15, Theorem 2.6].

(3) We shall view C(Λ)/S as an abelian subcategory of D(Λ)/S^{→} which is possible by (2). Now suppose that X is simple in C(Λ)/S and let U be a proper subobject of q(X) in D(Λ)/S^{→}. We claim that U = 0. We find a subobject Y ⊆ X such that q(Y) = U and write Y = ∑ Y_{i} as sum of finitely presented subobjects. It follows that U = ∑ q(Y_{i}) = 0 since every q(Y_{i}) is a proper subobject of q(X) in C(Λ)/S.

(4) These are well-known facts which can be found in [8].

Lemma 5.8 Let S be a Serre subcategory of C(Λ) and let X ∈ C(Λ) be simple in C(Λ)/S. If M_{1} and M_{2} is a pair of indecomposable injective objects in D(Λ) satisfying Hom(S, M_{i}) = 0 and Hom(X, M_{i}) ≠ 0 for i = 1, 2, then M_{1} ≅ M_{2}.

Proof : It follows from (3) and (4) in the preceding lemma that the quotient functor q: D(Λ) → D(Λ)/S^{→} sends M_{1} and M_{2} to injective envelopes of q(X). Therefore M_{1} ≅ M_{2}.

Lemma 5.9 Let S be a Serre subcategory of C(Λ) and denote by X the class of Λ-modules M satisfying Hom(S, M ⊗_{Λ} −) = 0. Then the following are equivalent.

(1) dim_{fin} C(Λ)/S = 0

(2) X is a class of endofinite modules and X ≠ 0.

(3) The Ziegler closure \overline{X} is a finite non-empty set of endofinite modules.

Proof : (1) ⇒ (2), (3) If M ∈ X, then S ⊆ S_{M} and therefore

dim_{fin} M = dim_{fin} C(Λ)/S_{M} ≤ dim_{fin} C(Λ)/S = 0.

It follows from Lemma 4.5 that M is endofinite. The Ziegler closure of X is non-empty by Proposition 2.2 since S is a proper Serre subcategory of C(Λ). To show that \overline{X} is finite observe first that there are only finitely many simple objects in C(Λ)/S since every simple object must occur as a composition factor of Hom_{Λ^{op}}(Λ, −) in C(Λ)/S. It is clear that for every M ∈ \overline{X} there is X ∈ C(Λ) which is simple in C(Λ)/S, such that Hom(X, M ⊗_{Λ}−) ≠ 0. Using the preceding lemma we conclude that \overline{X} is finite.

(3) ⇒ (1) Taking M = ∏_{N∈\overline{X}} N we have S = S_{M} by Proposition 2.2. The module M is endofinite and therefore dim_{fin} C(Λ)/S = dim_{fin} M = 0 by Lemma 4.5.

(2) ⇒ (1) Adapt the argument of (3) ⇒ (1).

An indecomposable pure-injective Λ-module M is called simply reflexive if the endocategory of M contains a simple object. It is not hard to see that M is simply reflexive iff there is φ: X → Y in mod(Λ) such that the cokernel of Hom_{Λ}(φ, M) is a simple End_{Λ}(M)^{op}-module [19]. Note that M is simply reflexive iff M has a minimal pair in the terminology of [28]. Clearly, dim_{fin} M < ∞ implies that M is simply reflexive; in particular a Σ-pure-injective module is simply reflexive. The following property will be needed later.

Lemma 5.10 Let M be indecomposable pure-injective with Γ = End_{Λ}(M)^{op}. If X ∈ E_{M} is simple, then Γ/ rad Γ ≅ End(X).

Proof : Let S = S_{M}. We apply Lemma 5.7 as follows. The quotient functor D(Λ) → D(Λ)/S^{→} sends M ⊗_{Λ} − to an injective envelope N of the simple object f(X) where f is the fully faithful functor

E_{M}^{op} ⟶^{∼} C(Λ)/S ⟶ D(Λ)/S^{→}.

We have therefore

Γ/ rad Γ ≅ (End(N)/ rad End(N))^{op} ≅ End(f(X))^{op} ≅ End(X).

Theorem 5.11 Let M be an indecomposable pure-injective Λ-module. Then dim_{fin} M = 1 if and only if M is simply reflexive and there is a finite set {N_{1}, …, N_{n}} of endofinite modules such that {M, N_{1}, …, N_{n}} is the Ziegler closure of M. If Λ is right dualizing, then every N_{i} is a generic module.

Proof : Let C = C(Λ)/S_{M} and denote by S the kernel of the canonical functor C(Λ) → C/C_{0}. We have dim_{fin} M = dim_{fin} C = 1 iff dim_{fin} C(Λ)/S = dim_{fin} C/C_{0} = 0. By definition

\overline{M} = \{N ∈ Zsp(Λ) | Hom(S_{M}, N ⊗_{Λ} −) = 0\}

and the first part of the assumption follows immediately from Lemma 5.9 once we have shown that \overline{M} \ \{M\} = U for

U = {N ∈ Zsp(Λ) | Hom(S, N ⊗_{Λ} −) = 0}.

Moreover, in this case every module in U is generic by Lemma 5.2 if Λ is right dualizing. To show that \overline{M} \ \{M\} = U observe first that \overline{M} \ U is non-empty since C_{0} ≠ 0 by our assumption on M. Now suppose that N ∈ \overline{M} \ U. It follows that Hom(X, N ⊗_{Λ} −) ≠ 0 for some object X ∈ C(Λ) which becomes a non-zero object of finite length in C, and we may even suppose that X is simple in C. On the other hand Hom(X, M ⊗_{Λ} −) ≠ 0 and therefore M = N by Lemma 5.8. This finishes the proof.

6. Functors between module categories

The purpose of this section is to collect all facts about functors between module categories which are needed in the rest of this paper.

Proposition 6.1 Let f: Mod(Γ) → Mod(Λ) be a functor which commutes with direct limits and products, and denote by F: Mod(Γ) → Ab its composition with the forgetful functor.

(1) f preserves pure-exactness and pure-injectivity.

(2) If X is a definable subcategory of Mod(Λ), then the full subcategory f^{−1}(X) of all Γ-modules M with f(M) ∈ X is a definable subcategory of Mod(Γ).

(3) If X is a class of Γ-modules, then the Ziegler closure \overline{f(X)} equals the set of indecomposable direct summands of modules in f(\overline{X}).

(4) If f(M) is indecomposable for every M ∈ Zsp(Γ), then f induces a continuous and closed map Zsp(Γ) → Zsp(Λ).

(5) The canonical morphism End_{Γ}(M) → End_{Λ}(f(M)) induces a faithful exact functor E_{f(M)} → E_{M} for all M ∈ Mod(Γ).

(6) dim_{fin} f(M) ≤ dim_{fin} M for all M ∈ Mod(Γ).

(7) If f induces an injective map Zsp(Γ) → Zsp(Λ), then dim_{fin} f(M) = dim_{fin} M for all M ∈ Mod(Γ) satisfying dim_{fin} M < ∞.

(8) There is a presentation Hom_{Γ}(Y, −) → Hom_{Γ}(X, −) → F → 0 with X, Y ∈ mod(Γ).

(9) There exists c ∈ ℕ such that ℓ_{end}(f(M)) ≤ c · ℓ_{end}(M) for all M ∈ Mod(Γ).

Proof : (1) See [13, Proposition 8.61] or [19, Theorem 7.8].

(2) This follows from the fact that a full subcategory of modules is definable iff it is closed under direct limits, products and pure submodules [19, Corollary 4.6].

(3) See [19, Theorem 7.8].

(4) f induces a map Zsp(Γ) → Zsp(Λ) by (1). It is continuous by (2) and closed by (3).

(5) See [19, Theorem 8.4].

(6) follows immediately from (5) with Lemma 1.3 and Lemma 2.1.

(7) If dim_{fin} M < ∞, then dim_{fin} M equals the so-called Cantor-Bendixson rank of the space \overline{M} by [19, Corollary 12.12]. The assumption on f implies that f induces a homeomorphism \overline{M} → \overline{f(M)} by (3) and (4). We conclude that dim_{fin} f(M) = dim_{fin} M.

(8) See [17, Corollary 12.2].

(9) is a direct consequence of (8). In fact, choosing an epimorphism Γ^{n} → X in mod(Γ) one takes c = n. Then c · ℓ_{end}(M) bounds the length of the End_{Γ}(M)^{op}-module F (M) and therefore also the endolength of f(M) since the End_{Γ}(M)^{op}-module structure on F (M) is induced by that of End_{Λ}(f(M))^{op} via the canonical homomorphism End_{Γ}(M) → End_{Λ}(f(M)).

7. The PI-property

Recall that a ring Γ is a polynomial identity ring, or simply a PI-ring, if there exists a monic polynomial p ∈ ℤ⟨X_{1}, …, X_{n}⟩ such that p(x_{1}, …, x_{n}) = 0 for all x_{i} ∈ Γ. In our context this property leads to an interesting finiteness condition if one considers the endomorphism ring Γ of an endofinite module. The radical of Γ is nilpotent and therefore Γ is a PI-ring iff Γ/ rad Γ is a PI-ring. This fact explains the following definition. Let C be a skeletally small abelian category with dim_{fin} C < ∞. We say that C has the PI-property if for every Serre subcategory S of C the endomorphism ring End(X) is a PI-ring for every simple object X ∈ C/S. A Λ-module M with dim_{fin} M < ∞ is said to have the PI-property if the endocategory E_{M} has the PI-property. The aim of this section is to prove the following result.

Theorem 7.1 Let M be a Λ-module with dim_{fin} M < ∞.

(1) M has the PI-property if and only if End_{Λ}(N)/ rad End_{Λ}(N) is a PI-ring for every module N in the Ziegler closure of M.

(2) Let f: Mod(Λ) → Mod(Γ) be a functor commuting with direct limits and products. If M has the PI-property, then f(M) has the PI-property.

Let C be a skeletally small abelian category and denote by A = Lex(C^{op}, Ab) the category of left exact functors C^{op} → Ab. The category A is a locally coherent Grothendieck category. Moreover if C is a length category, i.e. every object in C is of finite length, then the Yoneda functor C → A, X → Hom(−, X) induces an equivalence between C and the full subcategory of finite length objects in A.

Lemma 7.2 Let X be an object of finite length in any abelian category. If End(S) is a PI-ring for every composition factor S of X, then End(X) is a PI-ring.

Proof : Let S_{1}, …, S_{n} be the simples which occur as composition factors of X and denote by C the abelian category of all finite length objects with composition factors in {S_{1}, …, S_{n}}. We identify C with the full subcategory of finite length objects in A = Lex(C^{op}, Ab). Let I be the injective envelope of ∐_{i=1}^{n} S_{i} in A and let Λ = End(I)^{op}. The functor

h: C ⟶ Mod(Λ), Y ↦ Hom(Y, I)

is fully faithful and h(Y) is a finite length module for every Y ∈ C. Now let 𝔞 = ann h(X) and Γ = Λ/𝔞. The assumption implies that Λ/ rad Λ ≅ ∏_{i=1}^{n} End(S_{i})^{op} is a PI-ring. Therefore also Γ/ rad Γ is a PI-ring and we conclude that Γ is a PI-ring since rad Γ is nilpotent by Nakayama’s lemma. We conclude that End(X) ≅ End_{Γ}(h(X))^{op} is a PI-ring since h(X) is a finitely generated Γ-module [20, Corollary 13.4.9].

Lemma 7.3 Let C be a skeletally small abelian category and suppose there is a filtration (C_{α})_{α≥−1} of Serre subcategories such that

(i) C_{−1} = 0 and ⋃_{α} C_{α} = C;

(ii) if α = β + 1, then C_{α}/C_{β} is a length category; (iii) if α is a limit ordinal, then C_{α} = ⋃_{γ<α} C_{γ}.

Then for every indecomposable injective object M in A = Lex(C^{op}, Ab) there is α and a simple object X ∈ C_{α+1}/C_{α} ⊆ C/C_{α} such that End(M)/ rad End(M) ≅ End(X).

Proof : We shall identify C with a full subcategory of A via the Yoneda functor. Denote for every α by A_{α} = C_{α}^{→} the full subcategory of direct limits ⟶ lim X_{i} with X_{i} ∈ C_{α} for all i. These are localizing subcategories of A and we obtain quotient functors q_{α}: A → A/A_{α} [15, Theorem 2.8]. Now let M be an indecomposable injective object in A. It follows from (i) that there is a minimal β such that Hom(Y, M) ≠ 0 for some Y ∈ C_{β} and (iii) implies that β = α + 1. We apply Lemma 5.7 as follows. The quotient functor q_{α} sends M to an indecomposable injective object and Hom(q_{α}(Y), q_{α}(M)) ≠ 0. The object X = q_{α}(Y) is of finite length by (ii) and we may suppose that X is simple. Thus q_{α}(M) is an injective envelope of X and therefore

End(M)/ rad End(M) ≅ End(q_{α}(M))/ rad End(q_{α}(M)) ≅ End(X).

Lemma 7.4 Let C be a skeletally small abelian category with dim_{fin} C < ∞. Then C has the PI-property iff End(M)/ rad End(M) is a PI-ring for every indecomposable injective object M in A = Lex(C^{op}, Ab).

Proof : One direction follows immediately from the preceding lemma if one uses the finfiltration of C. To show the converse let X ∈ C/S be a simple object. Using the same argument as above we identify X with a simple object in A/S^{→}. Let I be an injective envelope of X and denote by M the image under the section functor A/S^{→} → A. Then End(X) ≅ End(M)/ rad End(M) and therefore C has the PI-property.

Lemma 7.5 Let f: D → C be a faithful exact functor between skeletally small abelian categories and suppose that dim_{fin} C < ∞. If C has the PI-property, then D has the PI-property.

Proof : We need to show that End(M)/ rad End(M) is a PI-ring for every indecomposable injective object M ∈ Lex(D^{op}, Ab) by the preceding lemma. To this end consider the finfiltration (C_{α})_{α} of C and let D_{α} = f^{−1}(C_{α}) for all α. We obtain a filtration (D_{α})_{α} of D which satisfies the assumption (i) - (iii) in Lemma 7.3. The functor f induces for every α a faithful exact functor f_{α}: D_{α+1}/D_{α} → C_{α+1}/C_{α} which sends a simple object X to a finite length object Y. Thus End(X) is identified with a subring of End(Y) and is therefore a PI-ring by our assumption on C and Lemma 7.2. We can now apply Lemmas 7.3 and 7.4 and conclude that D has the PI-property.

Proof of Theorem 7.1 : (1) Suppose that M has the PI-property and let N ∈ \overline{M}. The endocategory E_{N} is a quotient of E_{M} and has therefore the PI-property. It follows from Lemma 5.10 that End_{Λ}(N)/ rad End_{Λ}(N) is a PI-ring. The converse follows similarly from Lemma 7.4 using the fact that the equivalence D(Λ) → Lex(C(Λ)^{op}, Ab), X ↦ Hom(−, X)|_{C(Λ)}, induces an equivalence D(Λ)/S_{M}^{→} → Lex(E_{M}, Ab).

(2) follows from Proposition 6.1 and Lemma 7.5.

8. Generalized tubes

A collection T = (M_{i}, φ_{i}, ψ_{i})_{i∈ℕ_{0}} of Λ-modules M_{i} and morphisms φ_{i}: M_{i+1} → M_{i} and ψ_{i}: M_{i} → M_{i+1} is called a generalized tube if M_{0} = 0 and for every i ∈ ℕ the diagram

M_{i} ⟶^{φ_{i−1}} M_{i−1}
↓ψ_{i}  ↓ψ_{i−1}
M_{i+1} ⟶^{φ_{i}} M_{i}

is a pullback and a pushout. We say that a Λ-module N belongs to T if N ≅ M_{i} for some i and we shall often simplify the notation by writing T = (M_{i})_{i∈ℕ}. A generalized tube induces the following directed system of exact sequences which is indexed by i ∈ ℕ.

0 ⟶ M_{1} ⟶ M_{i} ⟶^{φ_{i−1}} M_{i−1} ⟶ 0
∥  ↓ψ_{i}  ↓ψ_{i−1}
0 ⟶ M_{1} ⟶ M_{i+1} ⟶^{φ_{i}} M_{i} ⟶ 0

Taking direct limits gives an exact sequence 0 → M_{1} → M →^{φ} M → 0. We have also the following inverse system of exact sequences.

0 ⟶ M_{i} ⟶^{ψ_{i}} M_{i+1} ⟶ M_{1} ⟶ 0
↓φ_{i−1}  ↓φ_{i}  ∥
0 ⟶ M_{i−1} ⟶^{ψ_{i−1}} M_{i} ⟶ M_{1} ⟶ 0

Taking inverse limits gives an exact sequence 0 → M̂ →^{ψ} M̂ → M_{1} → 0. This gives rise to the following directed system of exact sequences.

0 ⟶ M̂ ⟶^{ψ^{i}} M̂ ⟶ M_{i} ⟶ 0
∥  ↓ψ  ↓ψ_{i}
0 ⟶ M̂ ⟶^{ψ^{i+1}} M̂ ⟶ M_{i+1} ⟶ 0

Taking direct limits gives an exact sequence 0 → M̂ → E → M → 0 which we call the universal exact sequence corresponding to T .

Lemma 8.1 There is, up to isomorphism, a bijective correspondence between generalized tubes and pairs (M, φ) consisting of a Λ-module M and an epimorphism φ: M → M such that M = ⟶ lim_{i∈ℕ} Ker(φ^{i}).

Proof : We have already constructed a pair (M, φ) from a generalized tube. Given a pair (M, φ) one constructs a generalized tube T = (M_{i}, φ_{i}, ψ_{i})_{i∈ℕ_{0}} as follows. Put M_{i} = Ker(φ^{i}) and denote by ψ_{i}: M_{i} → M_{i+1} the inclusion. The morphism φ_{i}: M_{i+1} → M_{i} is obtained by restricting φ to M_{i+1}. It is easily checked that both constructions are mutually inverse.

Our aim is to show that the module M corresponding to a generalized tube is Σ-pure-injective under suitable assumptions. Let us mention before that M is never endofinite if it is indecomposable.

Lemma 8.2 Let M be an indecomposable endofinite Λ-module. Then every epimorphism (monomorphism) φ ∈ End_{Λ}(M) is an isomorphism.

Proof : The radical of End_{Λ}(M) is nilpotent by Nakayama’s lemma. It follows that φ is invertible since φ^{n} ≠ 0 for every n ∈ ℕ.

The following result is based on Ringel’s use of the shift endomorphism in [24].

Proposition 8.3 Let Λ be an R-algebra over some commutative artinian ring R and let M be a Λ-module. Let φ: M → M be an epimorphism and let M_{i} = Ker(φ^{i}). If M = ⟶ lim Ker(φ^{i}) and M_{1} is of finite length over R, then the following holds.

(1) M is artinian over Γ = End_{Λ}(M)^{op} and dim_{fin} L(M_{Γ}) ≤ 1.

(2) M is Σ-pure-injective and dim_{fin} M ≤ 1.

(3) If infinitely many of the M_{i} are indecomposable, then M has no non-zero direct summand which is finitely generated.

The proof of the proposition requires some preparations.

Lemma 8.4 Let 0 → M′ → M → M′′ → 0 be an exact sequence of Λ-modules. Then dim_{fin} L(M) = max(dim_{fin} L(M′), dim_{fin} L(M′′)).

Proof : Clear.

We discuss now a special case of the preceding proposition. Let Λ = R[T] be the polynomial ring in one indeterminate and let M = ∐_{i∈ℕ} x_{i}R be a free R-module with basis (x_{i})_{i∈ℕ}. Defining x_{i}T = x_{i−1} for i > 1 and x_{1}T = 0 induces a Λ-module structure on M.

Lemma 8.5 The Λ-module M is artinian and dim_{fin} L(M) = 1.

Proof : We prove the lemma by induction on the length n of the R-module R. If n = 1, then R is a field and the assertion follows from the fact that every proper submodule of M is of the form ∐_{i=1}^{r} x_{i}R for some r ∈ ℕ. Now suppose n > 1 and choose a simple ideal 𝔞 of R. We consider the submodule M′ = ∐_{i∈ℕ} x_{i}𝔞 of M. The assertion holds by induction for the Λ-modules M′ and M/M′ since M′ ≅ ∐_{i∈ℕ} x_{i}(R/ ann 𝔞) and M/M′ ≅ ∐_{i∈ℕ} x_{i}(R/𝔞). Applying the preceding lemma we conclude that M is artinian and dim_{fin} L(M) = 1.

Proof of Proposition 8.3 : (1) The ring homomorphism R[T] → Γ, T ↦ φ induces a R[T]- module structure on M. Therefore it suffices to show that M is artinian over R[T] and that dim_{fin} L(M) ≤ 1 over R[T]. To this end choose elements m_{11}, …, m_{1n} which generate M_{1} over R. For i > 1 choose inductively elements m_{i1}, …, m_{in} in M_{i} such that φ(m_{ij}) = m_{(i−1)j} for every j. Now let N_{j} = ∐_{i∈ℕ} x_{ij}R be a copy of the R[T]-module ∐_{i∈ℕ} x_{i}R for every j ∈ {1, …, n}. The assignment x_{ij} ↦ m_{ij} induces an epimorphism ∐_{j=1}^{n} N_{j} → M of R[T]- modules by our construction. The assertion is now a consequence of the preceding lemmas.

(2) It is well-known that M is Σ-pure-injective iff M has the descending chain condition on subgroups of finite definition. But every subgroup of finite definition is a Γ-submodule, and therefore (2) is a direct consequence of (1).

(3) Suppose that N is a finitely generated direct summand of M. Then N ⊆ M_{n} for some n and therefore N is a direct summand of M_{i} for every i ≥ n. If infinitely many of the M_{i} are indecomposable, then N = 0 and this finishes the proof of the proposition.

We now combine Proposition 8.3 with a result from the preceding section and obtain the following consequence.

Corollary 8.6 Let Λ be an artin algebra and suppose that T = (M_{i})_{i∈ℕ} is a generalized tube of finitely presented Λ-modules. Suppose also that every morphism M_{i} → M_{i+1} in T belongs to the radical of mod(Λ). Then there is a generic module in the Ziegler closure of M and therefore also in the Ziegler closure of T .

Proof : Recall that a morphism φ: X → Y belongs to the radical of mod(Λ) iff for every indecomposable Z and every pair ρ: Z → X and σ: Y → Z the composition σ∘φ∘ρ is not an isomorphism. The assumption on each M_{i} → M_{i+1} implies that the module M has no finitely presented indecomposable direct summand. Also, M has no dense chain of subgroups of finite definition by Proposition 8.3. It follows from Theorem 5.6 that there is a generic module N in the Ziegler closure of M.

Given a generalized tube T and the direct limit M = ⟶ lim M_{i} of modules M_{i} belonging to T , it is natural to ask for an explicit method to construct the generic modules corresponding to M. The first approach simply uses the fact that the generic modules occur in the Ziegler closure of the Σ-pure-injective module M. We give a description of the Ziegler closure of M which implies that a generic module corresponding to T occurs, up to isomorphism, as a direct summand of a product of copies of M. This is also a consequence of Corollary 9.33 in [21] which is equivalent to the second assertion of the following proposition.

Proposition 8.7 A Λ-module M is Σ-pure-injective if and only if every product of copies of M decomposes into a coproduct of copies of modules which belong to the Ziegler closure \overline{M} of M. In this case an indecomposable Λ-module N belongs to \overline{M} if and only if N is isomorphic to a direct summand of a product of copies of M.

Proof : It is well-known that M is Σ-pure-injective iff every product of copies of M decomposes into a coproduct of indecomposable modules. These modules belong to the definable subcategory generated by M since this category is closed under products and direct summands. But \overline{M} = \{N ∈ Zsp(Λ) | Hom(S_{M}, N ⊗_{Λ} −) = 0\} and therefore every indecomposable direct summand of ∏_{I} M belongs to \overline{M}. The last assertion follows from the fact that the quotient functor q: D(Λ) → D(Λ)/S_{M}^{→} sends M ⊗_{Λ} − to an injective cogenerator of the quotient category [19, Lemma 9.9]. This implies that for every N ∈ \overline{M} there is a pure monomorphism N → ∏_{I} M for some I by Lemma 5.7. We conclude that N is a direct summand.

We present now another approach to the above question. For the rest of this section let Λ be a k-algebra over a field k. Let T = (M_{i})_{i∈ℕ} be a generalized tube of finite dimensional Λ-modules and let 0 → M̂ → E → M → 0 be the corresponding universal exact sequence. Denote by d the k-dimension of M_{1}. Ringel has recently shown that for every product P = ∏_{I} M the module Q = ⟶ lim(∏_{I} M_{i}) is a pure submodule such that P/Q is of endolength at most d [25]. The module P/Q is non-zero if I is infinite, and P/Q decomposes into generic modules if M has no non-zero finitely presented direct summand. However, we use now a different construction which is based on the module M̂. Let k[[T]] be the ring of formal power series and denote by k((T)) its quotient field. Note that k[[T]] ≅ ⟵ lim k[[T]]/(T^{i}) and k((T)) is obtained by forming the direct limit over the directed system

k[[T]] ⟶^{·T} k[[T]] ⟶^{·T} ….

In other words, 0 → k[[T]] → k((T)) → k((T))/k[[T]] → 0 is the universal exact sequence corresponding to the tube (k[[T]]/(T^{i}))_{i∈ℕ}. The following lemma is a consequence of these observations.

Lemma 8.8 (1) M̂ is a k[[T]]-Λ-bimodule which is free over k[[T]] of rank d.

(2) T ≅ (k[[T]]/(T^{i}) ⊗_{k[[T]]} M̂)_{i∈ℕ} and the universal exact sequence is isomorphic to

0 ⟶ k[[T]] ⊗_{k[[T]]} M̂ ⟶ k((T)) ⊗_{k[[T]]} M̂ ⟶ k((T))/k[[T]] ⊗_{k[[T]]} M̂ ⟶ 0.

Proof : Adapt the proof of Proposition 8.3.

We need the following facts about complete discrete valuation rings.

Lemma 8.9 Let R be a commutative complete discrete valuation ring with maximal ideal 𝔪 and quotient field K.

(1) Zsp(R) = {R/𝔪^{i} | i ∈ ℕ} ∪ {R, K/R, K}.

(2) dim_{fin} M ≤ 1 for every M ∈ Zsp(R).

(3) Every R-module has the P I-property.

Proof : (1) See [28, Lemma 5.1].

(2) dim_{fin} M = 0 if M is endofinite by Lemma 4.5. Otherwise M = R or M = K/R. We have dim_{fin} R = dim_{fin} L(R) = 1 and dim_{fin} K/R = 1 by Proposition 8.3.

(3) For every M ∈ Zsp(R) the ring End_{R}(M)/ rad End_{R}(M) is either k or k((T)). The assertion therefore follows from Theorem 7.1.

Consider now the functor − ⊗_{k[[T]]} M̂: Mod(k[[T]]) → Mod(Λ). We apply our general results about functors commuting with direct limits and products, using the fact that k[[T]] is a complete discrete valuation ring.

Theorem 8.10 Let T = (M_{i})_{i∈ℕ} be a generalized tube of finite dimensional Λ-modules and let 0 → M̂ → E → M → 0 be the corresponding universal exact sequence. Let N be a Λ-module in the Ziegler closure of T .

(1) E is endofinite of endolength at most the k-dimension of M_{1}.

(2) N is either finite dimensional or a direct summand of M, M̂ or E.

(3) dim_{fin} N ≤ 1 and End_{Λ}(N)/ rad End_{Λ}(N) is a PI-ring.

(4) If N is endofinite, then End_{Λ}(N) is a PI-ring.

(5) A module in the Ziegler closure of M is a direct summand of M or E.

(6) A module in the Ziegler closure of M̂ is a direct summand of M̂ or E.

Proof : Most parts follow immediately from Proposition 6.1, Theorem 7.1 and the preceding lemmas. For (1) observe that E ≅ k((T)) ⊗_{k[[T]]} M̂ is a k((T))-Λ-bimodule. Therefore the endolength of E is bounded by the length over k((T)) which equals the k-dimension of M_{1}.

(4) follows from (3) since End_{Λ}(N) is nilpotent if N is endofinite. Finally, (5) and (6) follow from the fact that

\overline{k[[T]]} = \{k[[T]], k((T))\} and \overline{k((T))/k[[T]]} = \{k((T))/k[[T]], k((T))\}

which can be deduced from Theorem 5.11.

9. Tubes in the Auslander-Reiten quiver

Let Λ be an artin algebra. Following [23] a component C of the Auslander-Reiten quiver of Λ is called quasi-serial if

(i) C contains neither projective nor injective modules;

(ii) if 0 → X → ∐_{i=1}^{k} Y_{i} → Z → 0 is an almost split sequence of modules in C, then k ≤ 2 and k = 2 with ℓ(Y_{1}) ≤ ℓ(Y_{2}) implies that ℓ(Y_{1}) < ℓ(X) < ℓ(Y_{2}). The component C is called a stable tube if there exists a natural n such that (D Tr)^{n}M ≅ M for some M in C. The rank of C is the minimal n having this property and a stable tube of rank 1 is called homogeneous. An indecomposable module M belonging to a quasi-serial component C determines, up to isomorphism, uniquely a sequence of irreducible monomorphisms M_{i} → M_{i+1}, i ∈ ℕ, such that

(i) M_{i} belongs to C for all i and M = M_{j} for some j;

(ii) M_{1} is quasi-simple, i.e. there is no irreducible monomorphism ending in M_{1}.

Theorem 9.1 Let T be a stable tube and suppose that M_{1} → M_{2} → … is a sequence of irreducible monomorphisms starting in a quasi-simple module M_{1} belonging to T . Denote by M = ⟶ lim M_{i} the corresponding direct limit.

(1) M is an indecomposable module which depends, up to isomorphism, only on M_{1} (2) Let N_{1} be a second quasi-simple module and denote by N = ⟶ lim N_{i} the corresponding

direct limit. Then M ≅ N if and only if M_{1} ≅ N_{1}.

(3) M is Σ-pure-injective but not endofinite. There is a finite set {G_{1}, …, G_{n}} of generic modules such that {M, G_{1}, …, G_{n}} is the Ziegler closure of M.

Proof : (1) and (2) are given by [14, Theorem 8]. To prove (3) denote by r the rank of T . Composing the irreducible morphisms between the modules belonging to T in the appropriate way we obtain a generalized tube T′ = (M_{ri})_{i∈ℕ}. Therefore the module M is Σ-pure-injective by Proposition 8.3 but not endofinite by Lemma 8.2. Finally, the description of the Ziegler closure of M follows from Theorem 5.11.

The author is indebted to S. Liu for pointing out the following application of Theorem 9.1.

Corollary 9.2 Let Λ be a connected artin algebra of infinite representation type. Suppose there is a finitely generated indecomposable Λ-module M such that D Tr M ≅ M. Then there exists a generic Λ-module.

Proof : M belongs to a homogeneous tube by a result of Hoshino [12, Theorem 1].

For the rest of this section let Λ be a finite dimensional algebra over a field k. Our next result is a refinement of Theorem 9.1.

Corollary 9.3 Let T be a homogeneous tube and let d be the k-dimension of the quasi-simple module in T . Then there are modules M and N with dim_{fin} M = dim_{fin} N = 1, and there is a finite set {G_{1}, …, G_{n}} of generic modules such that T ∪ {M, N, G_{1}, …, G_{n}} is the Ziegler closure of T and ∑_{i=1}^{n} ℓ_{end}(G_{i}) ≤ d.

Proof : Let T = (M_{i})_{i∈ℕ} and denote by 0 → M̂ → E → M → 0 the universal exact sequence corresponding to T . We apply now Theorem 8.10 as follows. The Ziegler closure of M consists of M and a finite set {G_{1}, …, G_{n}} of generic modules which are direct summands of E and therefore ∑_{i=1}^{n} ℓ_{end}(G_{i}) = ℓ_{end}(E) ≤ d. Now let (N_{i})_{i∈ℕ} be the homogeneous tube of Λ^{op}-modules obtained from T by applying the duality Hom_{k}(−, k). We have

M̂ = ⟵ lim M_{i} ≅ ⟵ lim Hom_{k}(N_{i}, k) ≅ Hom_{k}(⟶ lim N_{i}, k)

and obtain therefore an endolength preserving bijection X ↦ DX between the Ziegler closure of ⟶ lim N_{i} and M̂ from [16]. It follows that every module in the Ziegler closure of T is either endofinite, M or N = D ⟶ lim N_{i}.

We call a generalized tube T = (M_{i})_{i∈ℕ} generic if there exists a generic module G such that {M_{i} | i ∈ ℕ} ∪ {⟶ lim M_{i}, ⟵ lim M_{i}, G} is the Ziegler closure of T . Note that ⟶ lim M_{i} and ⟵ lim M_{i} cannot be endofinite by Lemma 8.2. The following is a consequence of Crawley-Boevey’s work on generic modules over tame algebras [5].

Theorem 9.4 Let Λ be a finite dimensional algebra over an algebraically closed field k and suppose that Λ is of tame representation type. Then for every d ∈ ℕ almost all d-dimensional indecomposable Λ-modules belong to a homogeneous tube which is generic.

To prove this theorem we need some information about the Ziegler spectrum of a Dedekind domain.

Lemma 9.5 Let R be a Dedekind domain. Then for every prime ideal 𝔭 ≠ 0 the corresponding tube (R/𝔭^{i})_{i∈ℕ} is generic. The unique generic R-module is the quotient field of R.

Proof : See [28, Example 9.5].

Proof of Theorem 9.4 : Crawley-Boevey has shown that for each generic Λ-module G there is an R_{G}-Λ-bimodule M_{G} having the following properties [5, p. 242].

(i) R_{G} is a finitely generated localization of k[T] and M_{G} is free over R_{G}. The rank of M_{G} equals the endolength of G and if K_{G} is the quotient field of R_{G}, then G ≅ K_{G} ⊗_{R_{G}} M_{G}.

(ii) The functor − ⊗_{R_{G}} M_{G}: Mod(R_{G}) → Mod(Λ) preserves isomorphism classes, indecomposability and Auslander-Reiten sequences.

(iii) For each d ∈ ℕ almost all indecomposable Λ-modules of dimension d arise as R_{G}/𝔭^{n} ⊗_{R_{G}} M_{G} for some generic module G, some prime ideal 𝔭 and some n ∈ ℕ. The assertion of the theorem is a direct consequence of this result, Proposition 6.1 and the above lemma. In fact, each functor − ⊗_{R_{G}} M_{G} commutes with direct limits and products since M_{G} is a finitely presented R_{G}-module. Moreover one uses the fact that any localization of k[T] is a Dedekind domain. Given a generic Λ-module G and a prime ideal 0 ≠ 𝔭 ⊆ R_{G}, the image of the tube (R_{G}/𝔭^{i})_{i∈ℕ} under the functor − ⊗_{R_{G}} M_{G} is a homogeneous tube and G ≅ K_{G} ⊗_{R_{G}} M_{G} is the unique generic module in its Ziegler closure.

Let Λ be a finite dimensional algebra over an algebraically closed field and suppose that Λ is of tame representation type. Given a generic Λ-module G we denote by ind_{G}(Λ) the set of isomorphism classes of finite dimensional indecomposable Λ-modules belonging to a generic tube such that G is the unique generic module in its Ziegler closure. With this definition we obtain the following consequence of the above theorem.

Corollary 9.6 For every d ∈ ℕ almost all d-dimensional indecomposable Λ-modules belong to ⋃_{G} ind_{G}(Λ).

Corollary 9.7 Let d ∈ ℕ. Then a Λ-module M lies in the Ziegler closure of all d-dimensional indecomposable Λ-modules if and only if M is indecomposable of dimension d or generic of endolength d · n^{−1} for some n ∈ ℕ.

Proof : Almost all d-dimensional indecomposable Λ-modules are of the form R_{G}/𝔭^{n} ⊗_{R_{G}} M_{G} for some generic Λ-module G of endolength d · n^{−1} and some prime ideal 𝔭. For fixed d ∈ ℕ there are only finitely many generic modules involved [5, Theorem 4.4] and therefore the assertion follows from Theorem 6.1 and the fact that {R_{G}/𝔭^{n} | 𝔭 ≠ 0 prime } ∪ {K_{G}} is the Ziegler closure of {R_{G}/𝔭^{n} | 𝔭 ≠ 0 prime }, for instance by Proposition 5.4 and Theorem 5.6.

Having shown for an algebra of tame representation type that all generic Λ-modules lie in the Ziegler closure of a homogeneous tube, it is an obvious question to ask whether this property characterizes tameness. We have the following answer.

Corollary 9.8 The following are equivalent for a finite dimensional algebra Λ over an algebraically closed field.

(1) Λ has tame representation type.

(2) Every generic Λ-module occurs as the unique generic module in the Ziegler closure of a homogeneous tube.

(3) Every indecomposable endofinite Λ-module occurs in the Ziegler closure of a generalized tube of finite dimensional modules.

Proof : We have already shown that (1) implies (2) and it is clear that (2) implies (3). Therefore suppose (3). It follows from Theorem 8.10 that End_{Λ}(G) is a PI-ring for every generic Λ-module G. On the other hand, Crawley-Boevey has shown that a representation embedding Mod(k⟨X, Y ⟩) → Mod(Λ) sends the universal skew field of fractions of k⟨X, Y ⟩ to a generic module G such that End_{Λ}(G) is not a PI-ring [5, Theorem 4.4]. It follows that Λ is not wild and therefore tame by Drozd’s Tame and Wild Theorem.

10. Pure representation embeddings

Let Λ and Γ be a pair of rings. In this section we compare the corresponding module categories using certain functors between Mod(Γ) and Mod(Λ). Recall that a functor f: Mod(Γ) → Mod(Λ) is said to be a representation embedding if

(i) f is exact and commutes with coproducts and products;

(ii) f sends indecomposable modules to indecomposable modules;

(iii) f sends non-isomorphic modules to non-isomorphic modules. The following weaker conditions seem to be appropriate for a discussion of pure-injective modules. We call a functor f: Mod(Γ) → Mod(Λ) a pure representation embedding if

(i’) f commutes with direct limits and products;

(ii’) f sends indecomposable modules in Zsp(Γ) to indecomposable modules; (iii’) f sends non-isomorphic modules in Zsp(Γ) to non-isomorphic modules.

Note that a pure representation embedding preserves pure-exactness and pure-injectivity. It has been shown by Prest that any representation embedding induces a homeomorphism onto a closed subset between the Ziegler spectra [22]. Using different methods we can replace representation embeddings by pure representation embeddings and obtain the following characterization of wild representation type.

Theorem 10.1 If Λ is a finite dimensional algebra over an algebraically closed field k, then the following are equivalent.

(1) Λ has wild representation type, i.e. there is a k-linear representation embedding Mod(k⟨X, Y ⟩) → Mod(Λ).

(2) U_{Λ,n} = {M ∈ Zsp(Λ) |M is not isolated and ℓ_{end}(M) ≤ n} is infinite for some n ∈ ℕ.

(3) There is a pure representation embedding Mod(k⟨X, Y ⟩) → Mod(Λ).

(4) For every finitely generated k-algebra Γ there is a homeomorphism f: Zsp(Γ) → Zsp(Λ) onto a closed subset of Zsp(Λ) and c ∈ ℕ such that ℓ_{end}(f(M)) ≤ c · ℓ_{end}(M) for all M ∈ Zsp(Γ).

Proof : (1) ⇔ (2) It has been shown by Crawley-Boevey that Λ has wild representation type iff there are infinitely many generic Λ-modules of endolength n for some n ∈ ℕ [5, Theorem 4.4]. The assertion follows from this and the fact that a module M ∈ Zsp(Λ) is generic iff it is not isolated by Theorem 4.7.

(1) ⇒ (3) Every representation embedding is pure.

(3) ⇒ (4) If Γ = k⟨X_{1}, …, X_{n}⟩/I, then the composition

Mod(Γ) ⟶^{g} Mod(k⟨X_{1}, …, X_{n}⟩) ⟶^{h} Mod(k⟨X, Y ⟩)

of the restriction functor g with the functor h used by Brenner [3] is a fully faithful functor which commutes with direct limits and products. In particular it preserves indecomposability and isomorphism classes. Therefore the composition of h∘g with the pure representation embedding Mod(k⟨X, Y ⟩) → Mod(Λ) is again a pure representation embedding. The assertion now follows from Proposition 6.1.

(4) ⇒ (2) Choose a finite dimensional algebra Γ of wild representation type and a map f: Zsp(Γ) → Zsp(Λ) with c ∈ ℕ. Using (2) we know that U_{Γ,n} is infinite for some n ∈ ℕ. The properties of f and c imply that f(U_{Γ,n}) is an infinite subset of U_{Λ,c·n} and this proves (2).

We proceed now with an interesting numerical property of algebras having wild representation type.

Proposition 10.2 Let Λ be any ring and suppose there is a pure representation embedding Mod(k⟨X, Y ⟩) → Mod(Λ). Then for every n ∈ ℕ there exists M ∈ Zsp(Λ) such that dim_{fin} M = n.

Proof : Let Γ be the localization of k[X_{1}, …, X_{n}] at 𝔭 = (X_{1}, …, X_{n}) and denote by I the injective envelope of Γ/ rad Γ. Then dim_{fin} I = n by Example 2.4 since Γ has Krull dimension n. The argument used in the preceding proof shows that there is a pure representation embedding f: Mod(Γ) → Mod(Λ) and therefore dim_{fin} f(I) = n by Proposition 6.1.

We end this section with a characterization of tame hereditary algebras.

Theorem 10.3 For a finite dimensional hereditary algebra Λ of infinite representation type the following are equivalent.

(1) Λ is tame.

(2) End_{Λ}(M)/ rad End_{Λ}(M) is a PI-ring for every M ∈ Zsp(Λ).

(3) dim_{fin} M ≤ 1 for every M ∈ Zsp(Λ).

Proof : Suppose that Λ is tame. It follows, for example from Geigle’s work [9], that every infinite dimensional indecomposable pure-injective Λ-module M lies in the Ziegler closure of a stable tube. Thus End_{Λ}(M)/ rad End_{Λ}(M) is a PI-ring and dim_{fin} M ≤ 1 by Theorem 8.10. If Λ is not tame, then there exists a representation embedding Mod(k⟨X, Y ⟩) → Mod(Λ) by [6, Theorem 8.4]. The argument used in the proof of Corollary 9.8 shows that (2) fails and the preceding proposition shows that (3) fails.

11. Krull dimension one

In this section we discuss the Krull dimension of the category D(Λ) in the sense of Gabriel. We refer to [8] for the precise definition of the Krull dimension Kdim D(Λ). Denote by ind(Λ) the set of finitely presented modules in Zsp(Λ).

Proposition 11.1 For an artin algebra Λ the following are equivalent.

(1) Kdim D(Λ) = 1.

(2) dim_{fin} C(Λ) = 1.

(2) Every module having no finitely presented indecomposable direct summand is endofinite and there exists a non-zero module with this property.

(3) Zsp(Λ) \ ind(Λ) is a finite non-empty set of endofinite modules.

We collect some well-known facts which are needed in the proof of this proposition.

Lemma 11.2 Let Λ be an artin algebra. Then the simple functors in D(Λ) are finitely presented, and a Λ-module M has a finitely presented indecomposable direct summand iff Hom(S, M ⊗_{Λ} −) ≠ 0 for some simple functor S.

Proof : Observe first that if M ∈ ind(Λ), then the socle of M ⊗_{Λ} − is finitely presented and simple by Lemma 4.4. Now suppose that S ∈ D(Λ) is simple and let Hom_{Λ^{op}}(X, −) → S be an epimorphism with indecomposable X. The well-known duality between C(Λ) and C(Λ^{op}) sends the finitely presented socle of − ⊗_{Λ} X to a finitely presented and simple quotient of Hom_{Λ^{op}}(X, −) which is isomorphic to S since End_{Λ^{op}}(X) is local. The second part of the assertion now follows with the argument used in the proof of Lemma 5.2.

Proof of Proposition 11.1 : (1) ⇔ (2) Denote by D(Λ)_{0} the smallest localizing subcategory of D(Λ) containing all objects of finite length, and let C(Λ)_{0} be the Serre subcategory of all objects of finite length in C(Λ). Every object of finite length in D(Λ) is finitely presented according to the preceding lemma, and therefore D(Λ)_{0} = C(Λ)_{0}^{→} by Lemma 5.7. Moreover, the full subcategory of finitely presented objects in D(Λ)/D(Λ)_{0} is equivalent to C(Λ)/C(Λ)_{0} by [15, Theorem 2.6]. Now, by definition, Kdim D(Λ) = 1 if D(Λ)/D(Λ)_{0} is locally finite but non-zero. This condition is equivalent to dim_{fin} C(Λ)/C(Λ)_{0} = 0. We conclude that (1) and (2) are equivalent.

(2) ⇔ (3) ⇔ (4) Let S = C(Λ)_{0}. The preceding lemma implies that a Λ-module M has no finitely presented indecomposable direct summand iff Hom(S, M ⊗_{Λ} −) = 0. The assertion is therefore an immediate consequence of Lemma 5.9 since dim_{fin} C(Λ) = 1 iff dim_{fin} C(Λ)/S = 0.

Corollary 11.3 Let Λ be an artin algebra and suppose that T = (M_{i})_{i∈ℕ} is a generalized tube of finitely presented Λ-modules. If M = ⟶ lim M_{i} is indecomposable, then Kdim D(Λ) > 1.

Proof : The module M is not endofinite by Lemma 8.2.

The following consequence gives an answer to a problem posed in the book of Jensen and Lenzing [13].

Corollary 11.4 Let Λ be a finite dimensional algebra over an algebraically closed field k. Then Kdim D(Λ) ≠ 1.

Proof : If Λ is of finite representation type, then Kdim D(Λ) = 0 by a result of Auslander [1, Theorem 3.1]. Now suppose that Λ is of infinite type. We use Drozd’s Tame and Wild Theorem. If Λ is wild, then a k-linear representation embedding Mod(k[T]) → Mod(Λ) produces generalized tubes of the above form. Otherwise Λ is tame, and the validity of the Second Brauer-Thrall Conjecture guarantees the existence of homogeneous tubes in the Auslander-Reiten quiver of Λ by the result of Crawley-Boevey discussed in Theorem 9.4.

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Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

E-mail address: henning@mathematik.uni-bielefeld.de