STABLE EQUIVALENCE AND GENERIC MODULES

HENNING KRAUSE AND GRZEGORZ ZWARA

Abstract. Let Λ and Γ be finite dimensional algebras. It is shown that any stable equivalence f: \overline{mod} Λ → \overline{mod} Γ between the categories of finitely generated modules induces a bijection M ↦ M_{f} between the sets of isomorphism classes of generic modules over Λ and Γ such that the endolength of M_{f} is bounded by the endolength of M up to a scalar which only depends on f. Using Crawley-Boevey’s characterization of tame representation type in terms of generic modules, one obtains as a consequence a new proof for the fact that a stable equivalence preserves tameness. This proof also shows that polynomial growth is preserved.

Generic modules were introduced by Crawley-Boevey in order to give a conceptual description of the representation type of a finite dimensional algebra [1]. In this note we study the relation between the generic modules over two finite dimensional algebras Λ and Γ under the assumption that there exists an equivalence \overline{mod} Λ → \overline{mod} Γ between the stable categories of finitely generated modules.

Recall that a Λ-module is generic if it is indecomposable and endofinite but not of finite length over Λ. A Λ-module M is endofinite if the endolength endol(M) which is the length of M over its endomorphism ring End_{Λ}(M) is finite.

The endolength of a module is not a categorical invariant. In other words, a Morita equivalence F: Mod Λ → Mod Γ between two module categories does not necessarily preserve the endolength. However, it is not hard to see that there exists a constant c_{F} such that endol(F M) ≤ c_{F} . endol(M) for every Λ-module M. The purpose of this paper is to show that this fact remains true for a stable equivalence.

Using Crawley-Boevey’s characterization of tame representation type in terms of generic modules, we obtain in this way a simple proof for the fact that a stable equivalence preserves tame representation type [2]. In fact, we improve the results obtained in [2] and show that a stable equivalence preserves polynomial growth.

In [2], it was shown that an equivalence f: \overline{mod} Λ → \overline{mod} Γ extends to an equivalence F: \overline{Pinj} Λ → \overline{Pinj} Γ between the full subcategories of (not necessarily finitely generated) pure-injective modules in \overline{Mod} Λ and \overline{Mod} Γ. Any endofinite module is pure-injective, and our main result states that the functor F controls the endolength as follows.

Theorem. Let Λ and Γ be artin algebras and suppose that f: \overline{mod} Λ → \overline{mod} Γ is an equivalence. Denote by F: \overline{Pinj} Λ → \overline{Pinj} Γ the equivalence between the full subcategories of pure-injective modules in \overline{Mod} Λ and \overline{Mod} Γ which extends f (see [2]). Then there exists a constant c_{f} ∈ ℕ such that

endol(F M) ≤ c_{f}. endol(M)

for every endofinite non-zero Λ-module M.

Corollary 1. An equivalence f: \overline{mod} Λ → \overline{mod} Γ induces a bijection M ↦ M_{f} between the sets of isomorphism classes of indecomposable endofinite and non-injective modules over Λ and Γ such that endol(M_{f}) ≤ c_{f}. endol(M) for all M.

Before we can give the proof of the main theorem we need some preparations. Let us start with some notation. We fix an artin algebra Λ and denote by Mod Λ the category of all Λ-modules. The full subcategory of finitely generated modules is denoted by mod Λ. The stable category \overline{Mod} Λ has the same objects as Mod Λ but \overline{Hom}_{Λ}(M, N) is the group Hom_{Λ}(M, N) of Λ-maps modulo the subgroup of maps which factor through some injective module. Analogously, the stable category \overline{mod} Λ of finitely generated modules, and the stable category \overline{Pinj} Λ of pure-injective modules are defined.

We denote by S_{1}, …, S_{p} a representative set of simple Λ-modules, and T_{1}, …, T_{q} denotes a representative set of simple Γ-modules. Keeping the equivalence f: \overline{mod} Λ → \overline{mod} Γ fixed, we choose Λ-modules U_{1}, …, U_{q} sucht that f(U_{j}) ≅ T_{j} in \overline{mod} Γ for all j. Now let M and X be Λ-modules and suppose that X has finite length. We denote by [X, M] the length of Hom_{Λ}(X, M) as End_{Λ}(M)-module, and ⟦X, M⟧ denotes the length of \overline{Hom}_{Λ}(X, M) as \overline{End}_{Λ}(M)-module. The multiplicity of S_{i} in a composition series of X is denoted by m_{i}(X). We note the following elementary fact.

Lemma 1. Let M and X be Λ-modules and suppose that X has finite length. Then

⟦X, M⟧ ≤ [X, M] ≤ ∑_{i=1}^{p} m_{i}(X)[S_{i}, M].

Proof. The first inequality follows from the fact that the length of \overline{Hom}_{Λ}(X, M) as an \overline{End}_{Λ}(M)-module equals the one as an End_{Λ}(M)-module. In order to prove the second inequality, observe that any exact sequence 0 → X′ → X → X′′ → 0 of Λ-modules gives an exact sequence

0 ⟶ Hom_{Λ}(X′′, M) ⟶ Hom_{Λ}(X, M) ⟶ Hom_{Λ}(X′, M)

and so [X, M] ≤ [X′, M] + [X′′, M]. By induction on the length of X we get the second inequality.

The next lemma is simple but extremely useful.

Lemma 2. Let M be a Λ-module without any non-zero injective submodule, and let S be a simple Λ-module. Then Hom_{Λ}(S, M) = \overline{Hom}_{Λ}(S, M) and therefore [S, M] = ⟦S, M⟧.

Proof. Any non-zero map S → M which factors through some injective module, also factors through the injective envelope E(S) of S, and induces therefore a monomorphism E(S) → M. This is impossible by our assumption on M. Thus Hom_{Λ}(S, M) = \overline{Hom}_{Λ}(S, M).

Proof of the Theorem. Let M be an endofinite non-zero Λ-module and let N = F M. We denote by I_{N} a maximal injective submodule of N which exists by Zorn’s lemma since direct limits of injective modules are injective. This gives a direct decomposition N = N_{I} ∐ I_{N}. Using the preceding lemmas we obtain the following bound for the endolengh of N_{I}:

endol(N_{I}) = [Γ, N_{I}] ≤ ∑_{j=1}^{q} m_{j}(Γ)[T_{j}, N_{I}] = ∑_{j=1}^{q} m_{j}(Γ)⟦T_{j}, N_{I}⟧

= ∑_{j=1}^{q} m_{j}(Γ)⟦U_{j}, M⟧

≤ ∑_{j=1}^{q} m_{j}(Γ)[U_{j}, M]

≤ ∑_{j=1}^{q} m_{j}(Γ)( ∑_{i=1}^{p} m_{i}(U_{j})[S_{i}, M])

= ∑_{i=1}^{p} ( ∑_{j=1}^{q} m_{j}(Γ)m_{i}(U_{j}))[S_{i}, M]

≤ n_{f} ∑_{i=1}^{p}[S_{i}, M]
≤ n_{f}[Λ/rad Λ, M]
≤ n_{f}[Λ, M] = n_{f}. endol(M)

where n_{f} = max{∑_{j=1}^{q} m_{j}(Γ)m_{i}(U_{j}) | 1 ≤ i ≤ p}. The endolength of any injective Γ-module I is bounded by the composition length ℓ(Γ) of Γ since I is isomorphic to a direct factor of a product of copies of the injective cogenerator DΓ (D denotes the usual duality between mod Γ and mod Γ^{op}), and

endol(∏_{κ} DΓ) = endol(DΓ) = endol(Γ) = ℓ(Γ)

for any cardinal κ > 0. Defining c_{f} = n_{f} + ℓ(Γ), we obtain

endol(F M) = endol(N_{I}) + endol(I_{N}) ≤ n_{f}. endol(M) + ℓ(Γ) ≤ c_{f}. endol(M).

This finishes the proof of the Theorem.

We are now in a position to prove that a stable equivalence preserves the representation type. To this end denote for an algebra Λ by g_{Λ}(n) the number of isomorphism classes of generic Λ-modules of endolength n, and put γ_{Λ}(n) = ∑_{i=1}^{n} g_{Λ}(i).

Corollary 2. Let \overline{mod} Λ → \overline{mod} Γ be an equivalence. Then there exists a constant c ∈ ℕ such that γ_{Λ}(n) ≤ γ_{Γ}(cn) for all n ≥ 1. In particular, the following holds:

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

(2) If Γ is generically of polynomial growth, i.e., there exist k ≥ 1 and d ≥ 1 such that g_{Γ}(n) ≤ dn^{k} for all n ≥ 1, then Λ is generically of polynomial growth.

(3) If Γ is generically domestic, i.e., there exist only finitely many isomorphism classes of generic modules, then Λ is generically domestic.

Proof. The formula γ_{Λ}(n) ≤ γ_{Γ}(cn) is an immediate consequence of Corollary 1. Observe that in the definition of generically tame algebras one can replace g_{Λ}(n) by γ_{Λ}(n). Hence, (1) follows from the formula γ_{Λ}(n) ≤ γ_{Γ}(cn). Now assume that g_{Γ}(n) ≤ dn^{k} for some k ≥ 1, d ≥ 1, and all n ≥ 1. Then

g_{Λ}(n) ≤ γ_{Λ}(n) ≤ γ_{Γ}(cn) = ∑_{i=1}^{cn} g_{Γ}(i) ≤ ∑_{i=1}^{cn} di^{k} ≤ (cn)d(cn)^{k} = (c^{k+1}d)n^{k+1},

for all n ≥ 1. This implies (2). An algebra Γ is generically domestic if and only if there exists N ∈ ℕ such that γ_{Γ}(n) ≤ N for all n ≥ 1. Hence, also (3) is a consequence of the formula γ_{Λ}(n) ≤ γ_{Γ}(cn).

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 [1, Theorem 4.4 and 5.6], Crawley-Boevey has shown that μ_{Λ}(n) is finite for all n ≥ 1 if and only if 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:

• Λ is of tame representation type (i.e., μ_{Λ}(n) < ∞ for all n ≥ 1) if and only if Λ is generically tame.

• Λ is of polynomial growth (i.e., there exist k ≥ 1 and d ≥ 1 such that μ_{Λ}(n) ≤ dn^{k} for all n ≥ 1) if and only if Λ is generically of polynomial growth. Indeed, the inequality g_{Λ}(n) ≤ μ_{Λ}(n) shows that polynomial growth implies ‘generical’ polynomial growth. On the other hand, if g_{Λ}(n) ≤ dn^{k} for some k ≥ 1, d ≥ 1, and all n ≥ 1, then

μ_{Λ}(n) ≤ ∑_{i=1}^{n} g_{Λ}(i) ≤ ∑_{i=1}^{n} di^{k} ≤ ndn^{k} = dn^{k+1}.

• Λ is of domestic representation type (i.e., there exists N ∈ ℕ such that μ_{Λ}(n) ≤ N for all n ≥ 1) if and only if Λ is generically domestic.

Acknowledgement. The results in this paper evolved during a visit of the first named author to U. Toruń.

References

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

[2] H. Krause, Stable equivalence preserves representation type, Comment. Math. Helv. 72 (1997), 266–284.

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

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

Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

   E-mail address: gzwara@mat.uni.torun.pl