REPRESENTATION TYPE AND STABLE EQUIVALENCE OF MORITA TYPE FOR FINITE DIMENSIONAL ALGEBRAS

HENNING KRAUSE

In this note we show that two finite dimensional algebras have the same representation type if they are stably equivalent of Morita type.

Stable equivalences of Morita type were introduced for blocks of group algebras by Broué [2], see also [7]. The concept is motivated by a result of Rickard. In [10], he proved that any derived equivalence between finite dimensional self-injective algebras Λ and Γ induces a stable equivalence of a particulary nice form. More precisely, he constructs bimodules _{Λ}B_{Γ} and _{Γ}C_{Λ} such that the corresponding tensor functors induce mutually inverse equivalences between the stable categories \underline{mod} Λ and \underline{mod} Γ of finite dimensional modules for Λ and Γ. Stable equivalences of this form are said to be of Morita type. They occur frequently, for instance in the theory of blocks of group algebras. Other examples are obtained from two finite dimensional algebras Λ and Γ which are tilted from each other. The corresponding trivial extensions T Λ and T Γ are derived equivalent self-injective algebras [9], and therefore stably equivalent of Morita type. In fact, Assem and de la Peña proved in [1] that T Λ and T Γ have the same representation type. It seems to be an interesting project to understand the precise relation between the geometric approach of de la Peña [8] which is used in [1], and the approach presented in this paper.

The representation type of a finite dimensional algebra Λ is traditionally defined using the concept of a continuous one-parameter family in the category mod Λ of finite dimensional Λ-modules. For instance, Λ is of tame representation type if for every n ∈ ℕ only finitely many such families are needed to parametrize all indecomposable Λ-modules of dimension n. An alternative approach uses so-called generic modules. This was suggested by Crawley-Boevey and he established for tame algebras a correspondence between continuous one-parameter families and generic modules [3]. In [6], we used generic modules to show that a stable equivalence \underline{mod} Λ → \underline{mod} Γ induces a bijection between continuous one-parameter families in mod Λ and mod Γ. However, without any extra assumption on the stable equivalence it remains an open question how the dimensions of the modules in mod Λ and mod Γ are related. In this paper we settle the problem for stable equivalences which are induced by an appropriate functor mod Λ → mod Γ. Stable equivalences of Morita type are of this form, and therefore we can prove that two finite dimesional algebras have the same representation type if they are stably equivalent of Morita type.

I would like to thank Thorsten Holm for drawing my attention to the representation type problem for derived equivalent algebras.

We now begin with some notation and recall briefly the definitions which are needed in this paper. Let Λ be an associative ring with identity. Denote by Mod Λ the category of (right) Λ-modules and let mod Λ be the full subcategory of finitely presented Λ-modules. A sequence of morphisms 0 → L → M → N → 0 in Mod Λ is pure-exact if the induced sequence 0 → Hom_{Λ}(X, L) → Hom_{Λ}(X, M) → Hom_{Λ}(X, N) → 0 is exact for every X in mod Λ. In this case the morphism L → M is called a pure monomorphism, and L is pure-injective if every pure monomorphism L → M splits. We shall always assume that every finitely presented Λ-module is pure-injective. For example, every artin algebra has this property. We denote by Pinj Λ the full subcategory of pure-injective Λ-modules and the Ziegler spectrum Zsp Λ is the set of isomorphism classes of indecomposable pure-injective Λ-modules. A Λ-module M is endofinite if the length ℓ_{end}(M) of M, when regarded in the natural way as an End_{Λ}(M)^{op}-module, is finite. Note that every endofinite module is pure-injective. An endofinite module is called generic if it is indecomposable but not finitely presented [3]. Given a Λ-module X we denote by Add X the smallest full subcategory of Mod Λ which contains X and is closed under taking arbitrary coproducts and direct summands. The stable category \underline{Mod}_{X} Λ has the same objects as Mod Λ but \underline{Hom}_{X}(M, N) is the group of all Λ-morphisms M → N modulo the subgroup of those which factor through an object in Add X. For example, one obtains the usual stable module category \underline{Mod} Λ by taking X = Λ. Analogously, the categories \underline{Pinj}_{X} Λ and \underline{mod}_{X} Λ are defined.

We shall need the following lemma.

Lemma. Let X be endofinite. Then every Λ-module M has a decomposition M = M_{X} ∐ X_{M} such that X_{M} is a maximal direct summand of M which belongs to Add X.

Proof. The union of a chain of pure submodules is again a pure submodule. Also, Add X is closed under taking direct limits and every module in Add X is pureinjective [5, Corollary 9.8]. Thus, by Zorn’s lemma there is a maximal pure submodule X_{M} of M contained in Add X which is a direct summand.

The next proposition collects the essential properties of a stable equivalence \underline{mod}_{X} Λ → \underline{mod}_{Y} Γ which is induced by some functor Mod Λ → Mod Γ.

Proposition. Let f: Mod Λ → Mod Γ be a functor commuting with direct limits and products. Suppose there are endofinite modules X in mod Λ and Y in mod Γ such that f induces an equivalence \underline{mod}_{X} Λ → \underline{mod}_{Y} Γ. Then the following holds:

(1) f induces an equivalence \underline{Pinj}_{X} Λ → \underline{Pinj}_{Y} Γ.

(2) f induces a bijection Zsp Λ \ Add X → Zsp Γ \ Add Y, M ↦ M_{f}, such that fM = M_{f} ∐ Y_{fM} and Y_{fM} is a maximal direct summand of fM in Add Y.

(3) There exists c ∈ ℕ such that ℓ_{end}(fM) ≤ c · ℓ_{end}(M) for all M in Mod Λ.

(4) If M is a generic Λ-module, then M_{f} is generic.

Proof. (1) The functor f induces the following commutative diagram of functors:

mod Λ ⟶^{f} mod Γ
↓  ↓
\underline{mod}_{X} Λ ⟶^{∼} \underline{mod}_{Y} Γ

The canonical functor mod Λ → \underline{mod}_{X} Λ extends uniquely to a functor p_{Λ} : Mod Λ → \underrightarrow{lim} \underline{mod}_{X} Λ which commutes with direct limits. Here, \underrightarrow{lim} \underline{mod}_{X} Λ denotes the category Flat((\underline{mod}_{X} Λ)^{op}, Ab) of flat functors (\underline{mod}_{X} Λ)^{op} → Ab which is characterized by the following three properties [6]:

(i) \underrightarrow{lim} \underline{mod}_{X} Λ contains, up to equivalence, \underline{mod}_{X} Λ as a full subcategory.

(ii) \underrightarrow{lim} \underline{mod}_{X} Λ is an additive category with direct limits.

(iii) Every object in \underrightarrow{lim} \underline{mod}_{X} Λ is a direct limit of objects in \underline{mod}_{X} Λ. We obtain therefore the following commutative diagram of functors which commute with direct limits:

Mod Λ  ⟶^{f}  Mod Γ
↓p_{Λ}  ↓p_{Γ}
\underrightarrow{lim} \underline{mod}_{X} Λ ⟶^{∼} \underrightarrow{lim} \underline{mod}_{Y} Γ

It follows from Proposition 2.2 and Proposition 3.2 in [6] that p_{Λ} induces an equivalence \underline{Pinj}_{X} Λ → Pinj(\underrightarrow{lim} \underline{mod}_{X} Λ) where the notion of purity in \underrightarrow{lim} \underline{mod}_{X} Λ is defined as for Mod Λ. The assertion in (1) is now a consequence since \underrightarrow{lim} \underline{mod}_{X} Λ → \underrightarrow{lim} \underline{mod}_{Y} Γ is an equivalence.

(2) Every indecomposable pure-injective module has a local endomorphism ring. Therefore the canonical functor Mod Λ → \underline{Mod}_{X} Λ induces an injective map from Zsp Λ \ Add X into the set of isomorphism classes of indecomposable objects in \underline{Pinj}_{X} Λ. The map is surjective by the preceding lemma. Keeping the notation from this lemma, it follows from (1) that M ↦ M_{f} = (fM)_{Y} induces a bijection between Zsp Λ \ Add X and Zsp Γ \ Add Y.

(3) This has been shown in [4]. However, we sketch the argument for the convenience of the reader. The composition F = Hom_{Γ}(Γ, −)∘f: Mod Λ → Ab has a presentation Hom_{Λ}(B, −) → Hom_{Λ}(A, −) → F → 0 with A and B in mod Λ, and there is a homomorphism φ: Γ → End(F) such that the Γ-action on fM coincides with the Γ-action on F M via φ [4, Corollary 12.2]. Choose an epimorphism Λ^{c} → A in mod Λ. Then c · ℓ_{end}(M) bounds the length of the End_{Λ}(M)^{op}-module F M and therefore also the endolength of fM since the End_{Λ}(M)^{op}-module structure on F M is induced by that of End_{Γ}(fM)^{op} via the canonical homomorphism End_{Λ}(M) → End_{Γ}(fM).

(4) It follows from the decomposition fM = M_{f} ∐ Y_{fM} and part (3) that M_{f} is endofinite. We claim that M_{f} is not finitely presented. Otherwise we find a finitely presented Λ-module L such that fL and M_{f} are isomorphic in \underline{mod}_{Y} Γ since f induces an equivalence \underline{mod}_{X} Λ → \underline{mod}_{Y} Γ. Therefore fM and fL are isomorphic in \underline{Pinj}_{Y} Γ since we assume that every finitely presented module is pure-injective. Using (1) it follows that M and L are isomorphic in \underline{Pinj}_{X} Λ. Thus there are morphisms φ: M → L and ψ: L → M in Mod Λ such that id_{M} − ψ∘φ factors through an object in Add X. Therefore id_{M} − ψ∘φ ∈ rad End_{Λ}(M) and φ is a split monomorphism, a contradiction. We conclude that M_{f} is generic.

Given a ring Λ and n ∈ ℕ, we denote by g_{Λ}(n) the number of isomorphism classes of generic Λ-modules of endolength n, and γ_{Λ}(n) = ∑_{i=1}^{n} g_{Λ}(i).

Corollary. Suppose there is an equivalence \underline{mod}_{X} Λ → \underline{mod}_{Y} Γ which is induced by a functor Mod Λ → Mod Γ commuting with direct limits and products. Then there exists c ∈ ℕ such that γ_{Λ}(n) ≤ γ_{Γ}(cn) for all n. In particular, the following holds:

(1) If Γ is generically tame, i.e. γ_{Γ}(n) < ∞ for all n, then Λ is generically tame.

(2) If Γ is generically of polynomial growth, i.e. there exists a polynomial p such that γ_{Γ}(n) ≤ p(n) for all n, then Λ is generically of polynomial growth.

(3) If Γ is generically domestic, i.e. there exists N ∈ ℕ such that γ_{Γ}(n) ≤ N for all n, then Λ is generically domestic.

Proof. The formula γ_{Λ}(n) ≤ γ_{Γ}(cn) is an immediate consequence of the proposition, and (1) – (3) follow from this formula.

Suppose that Λ is a finite dimensional algebra over an algebraically closed field and denote by μ_{Λ}(n) the minimal number of continuous one-parameter families which is needed to parametrize all but finitely many indecomposable Λ-modules of dimension n. In [3, Theorem 4.4 and 5.6], Crawley-Boevey has shown that μ_{Λ}(n) is finite for all n iff Λ is generically tame; moreover in this case μ_{Λ}(n) = ∑_{i|n} g_{Λ}(i). Therefore the ‘generical’ definition of the representation type coincides with the ‘classical’ one. More precisely:

(i) Λ is of tame representation type (i.e. μ_{Λ}(n) < ∞ for all n) iff Λ is generically tame.

(ii) Λ is of polynomial growth (i.e. there exists a polynomial p such that μ_{Λ}(n) ≤ p(n) for all n) iff Λ is generically of polynomial growth.

(iii) Λ is of domestic representation type (i.e. there exists N ∈ ℕ such that μ_{Λ}(n) ≤ N for all n) iff Λ is generically domestic.

We now obtain our promised result. To this end we call two finite dimensional algebras over a field k stably equivalent of Morita type if there are bimodules B in mod Λ^{op} ⊗_{k} Γ and C in mod Γ^{op} ⊗_{k} Λ such that the functors

− ⊗_{Λ} B: mod Λ ⟶ mod Γ  and  − ⊗_{Γ}C: mod Γ ⟶ mod Λ

induce mutually inverse equivalences between \underline{mod} Λ and \underline{mod} Γ.

Corollary. Let Λ and Γ be finite dimensional algebras over an algebraically closed field. Suppose that Λ and Γ are stably equivalent of Morita type. Then there exists c ∈ ℕ such that γ_{Λ}(n) ≤ γ_{Γ}(cn) and γ_{Γ}(n) ≤ γ_{Λ}(cn) for all n. In particular, the following holds:

(1) Λ is of tame representation type if and only Γ is of tame representation type.

(2) Λ is of polynomial growth if and only Γ is of polynomial growth.

(3) Λ is of domestic representation type if and only Γ is of domestic representation type.

Proof. Denote by _{Λ}B_{Γ} and _{Γ}C_{Λ} the bimodules which induce the stable equivalence of Morita type. The corresponding tensor functors − ⊗_{Λ} B: Mod Λ → Mod Γ and − ⊗_{Γ} C: Mod Γ → Mod Λ commute with direct limits and products since B is finitely presented over Λ and C is finitely presented over Γ. The assertion therefore follows from the preceding corollary and Crawley-Boevey’s result.

References

[1] I. Assem and J.A. de la Peña, On the tameness of trivial extension algebras, Fund. Math. 149 (1996) 171–181.

[2] M. Broué, Equivalences of blocks of group algebras, in: V. Dlab and L.L. Scott (eds.), Finite dimensional algebras and related topics, Kluwer (1994) 1–26.

[3] W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc. 63 (1991) 241–264.

[4] H. Krause, Functors on locally finitely presented categories, Colloq. Math., to appear.

[5] H. Krause, Exactly definable categories, J. Algebra, to appear.

[6] H. Krause, Stable equivalence preserves representation type, Comment. Math. Helv., to appear.

[7] M. Linckelmann, Stable equivalences of Morita type for self-injective algebras and p-groups, Math. Z. 223 (1996) 87–100.

[8] J.A. de la Peña, Constructible functors and the notion of tameness, Comm. Algebra 24 (1996) 1939-1955.

[9] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra, 61 (1989) 303–317.

[10] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991) 37–48.

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

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