ORBIT ALGEBRAS AND PERIODICITY

PETTER ANDREAS BERGH

Abstract. Given an object in a category, we study its orbit algebra with respect to an endofunctor. We show that if the object is periodic, then its orbit algebra modulo nilpotence is a polynomial ring in one variable. This specializes to a result on Ext-algebras of periodic modules over Gorenstein algebras. We also obtain a criterion for an algebra to be of wild representation type.

1. Introduction

Given a suitably nice category and an endofunctor, we may construct the orbit algebra of a given object (cf. [Len]). These algebras are well suited for studying various types of periodicity. For example, when the endofunctor is an equivalence, then we show that, modulo nilpotence, the orbit algebra of an indecomposable periodic object is the polynomial ring in one variable. Using this, we generalize [GSS, Proposition 1.3] and obtain a periodicity result for modules over Gorenstein algebras. Moreover, we use this theory to study modules over selfinjective algebras, in particular modules which are periodic with respect to the Auslander-Reiten translate. Namely, we show that, modulo nilpotence, the Ext-orbit algebra of such a module with respect to the Nakayama automorphism is the polynomial ring in one variable. As an application of the latter, we provide a criterion for an algebra to be of wild representation type.

2. Orbit algebras

Throughout this paper, we let k be an algebraically closed field and Λ a finite dimensional k-algebra. We denote by mod Λ the category of finitely generated left Λ-modules. Whenever we deal with Λ-modules, we assume they belong to mod Λ.

Recall that a category C is k-linear if for all objects X, Y, Z ∈ C the set Hom_{C}(X, Y) is a k-vector space, and the composition

Hom_{C}(X, Y) × Hom_{C}(Y, Z) → Hom_{C}(X, Z)

is k-bilinear. Furthermore, such a category is Hom-finite if all the morphism spaces are finite dimensional over the ground field. Now let C ⟶^{F} C be an endofunctor. Then we may define a graded algebra

Hom_{C}(F^{∗}(X), X) =^{def} ⨁_{i=0}^{∞} Hom_{C}(F^{i}(X), X),

with ring structure given as follows: for f ∈ Hom_{C}(F^{i}(X), X) and g ∈ Hom_{C}(F^{j}(X), X), the product fg is the composition f ∘ F^{i}(g), which is an element in Hom_{C}(F^{i+j}(X), X).

The algebra just defined is the orbit algebra of M with respect to F (cf. [Len]). As an example, suppose Λ is selfinjective, and denote its enveloping algebra Λ⊗_{k}Λ^{op} by Λ^{e}. Furthermore, let \underline{mod}Λ^{e} be the stable module category of bimodules, and denote by τ_{Λ^{e}} the Auslander-Reiten translate of Λ^{e}. The orbit algebra

2000 Mathematics Subject Classification. 16B50, 16E05, 16E30, 16G60.

Key words and phrases. Orbit algebras, periodicity, Auslander-Reiten translate.

The author was supported by NFR Storforsk grant no. 167130.

𝔸(Λ, τ_{Λ^{e}}) =^{def} ⨁_{i=0}^{∞} \underline{Hom}_{Λ^{e}}(τ_{Λ^{e}}^{i}(Λ), Λ)

is called the Auslander-Reiten orbit algebra of Λ. In [Po1] it was shown that these algebras are invariant under stable equivalences of Morita type between symmetric algebras, and this was generalized in [Po2] to arbitrary finite dimensional selfinjective algebras. In [Po4] the Auslander-Reiten orbit algebras of a class of finite dimensional basic connected selfinjective Nakayama algebras were computed. In particular, it was shown that if Γ is such an algebra over a field K, and τ_{Γ^{e}}(Γ) ≃ Γ, then there are two possibilities. Namely, if Γ is a radical square zero algebra then 𝔸(Γ, τ_{Γ^{e}}) ≃ K[x], and if not then there exists a natural number t such that 𝔸(Γ, τ_{Γ^{e}}) ≃ K[x, y]/(y^{t}). In [Po3] τ-periodicity was investigated using similar techniques as was used in [Sc1] to study syzygy-periodicity.

We end this section with the following result. It shows that if the endofunctor F is an equivalence and M is “indecomposable” and F -periodic, then its orbit algebra modulo nilpotence is a polynomial ring. The result and its proof are inspired by [GSS, Proposition 1.3].

Theorem 2.1. Let C be a k-linear Hom-finite category, and let C ⟶^{F} C be an equivalence. Furthermore, let M ∈ C be an object whose endomorphism ring is local, and suppose F^{n}(M) ≃ M for some n ≥ 1, where n is minimal with this property. Then

Hom_{C}(F^{∗}(M), M)/I ≃ k[x],

where I is the ideal in Hom_{C}(F^{∗}(M), M) generated by the homogeneous nilpotent elements, and x is a homogeneous element in degree n.

Proof. Let f ∈ Hom_{C}(F^{u}(M), M) be a homogeneous nilpotent element. We first show that for any homogeneous element g ∈ Hom_{C}(F^{v}(M), M), the products fg and gf are also nilpotent in Hom_{C}(F^{∗}(M), M). Choose a positive number p with the property that p(u + v) is a multiple of n. If the element (fg)^{p} is is an isomorphism, then there is a morphism h ∈ Hom_{C}(M, F^{u}(M)) such that f ∘ h is the identity on M. This implies that the map h∘f is not nilpotent in End_{C}(F^{u}(M)), and therefore an isomorphism since this endomorphism ring is local. But then f is an isomorphism, contradicting the assumption that it is nilpotent in Hom_{C}(F^{∗}(M), M). Thus the element (fg)^{p} cannot be an isomorphism in Hom_{C}(F^{p(u+v)}(M), M). Since Hom_{C}(F^{p(u+v)}(M), M) is isomorphic to End_{C}(M), we see that (fg)^{p} corresponds to an element in the radical of End_{C}(M), and the same holds for F^{ni}((fg)^{p}) for any i ≥ 0. However, the radical is nilpotent, hence (fg)^{p}, and therefore also fg, is nilpotent. A similar argument shows that gf is also nilpotent in Hom_{C}(F^{∗}(M), M).

Next, let u be a multiple of n, and let f and g be homogeneous nilpotent elements in Hom_{C}(F^{∗}(M), M) of degree u. We show that f + g is also nilpotent. Both f and g correspond to radical elements of End_{C}(M), and the same hold for F^{ni}(f) and F^{ni}(g), for every i ≥ 0. Since the radical of End_{C}(M) is nilpotent, we see that f + g must be nilpotent in Hom_{C}(F^{∗}(M), M).

We now show that if u is not a multiple of n, then any element of Hom_{C}(F^{u}(M), M) is nilpotent in Hom_{C}(F^{∗}(M), M). Let f be such an element, and choose a number p with the property that pu is a multiple of n. Using the same arguments as above, we see that the element f^{p} cannot be an isomorphism, hence it corresponds to a radical element of End_{C}(M). Again using arguments from above, we see that f^{p}, and therefore also f, must be nilpotent in Hom_{C}(F^{∗}(M), M).

Finally, let F^{n}(M) ⟶^{f} M be an isomorphism, and suppose f^{u} belongs to I for some u. Then by the first part of the proof, we can write f^{u} = f_{1} + ⋯ + f_{t}, where each f_{i} is an element of Hom_{C}(F^{un}(M), M) nilpotent in Hom_{C}(F^{∗}(M), M). By the second part of the proof, the element f^{u}, and therefore also f, is nilpotent, an obvious contradiction. Therefore f^{u} does not belong to I for any u ≥ 0. Since k is algebraically closed, the ring End_{C}(M) modulo its radical is just k itself. Therefore, up to scalars, for any i ≥ 0 there is only one non-nilpotent element in Hom_{C}(F^{in}(M), M), namely the element f^{i}. This shows that Hom_{C}(F^{∗}(M), M)/I is isomorphic to the polynomial ring k[f]. □

3. Applications

As a first application of Theorem 2.1, we generalize [GSS, Proposition 1.3] on extension algebras of periodic modules. Suppose the algebra Λ is Gorenstein, that is, the injective dimension of Λ as a module over itself is finite. Denote by MCM(Λ) the category of maximal Cohen-Macaulay Λ-modules, i.e.

MCM(Λ) = {M ∈ mod Λ | Ext_{Λ}^{i}(M, Λ) = 0 for all i > 0}.

It follows from general cotilting theory that this is a Frobenius exact category, in which the projective-injective objects are the projective Λ-modules, and the injective envelopes are the left (add Λ)-approximations. Therefore the stable category \underline{MCM}(Λ), which is obtained by factoring out all morphisms which factor through projective Λ-modules, is a triangulated category. Its shift functor is given by cokernels of left (add Λ)-approximations, the inverse shift is the usual syzygy functor.

Theorem 3.1. Suppose that Λ is Gorenstein, and let M be an indecomposable Λ-module such that Ω_{Λ}^{n}(M) ≃ M for some n ≥ 1, where n is minimal with this property. Then

Ext_{Λ}^{∗}(M, M)/I ≃ k[x],

where I is the ideal in Ext_{Λ}^{∗}(M, M) generated by the homogeneous nilpotent elements, and x is a homogeneous element in degree n.

Proof. Since M is periodic, it must be a maximal Cohen-Macaulay module. Consider the equivalence Ω_{Λ} on \underline{MCM}(Λ). It follows from Theorem 2.1 that

Hom_{\underline{MCM}(Λ)}(Ω_{Λ}^{∗}(M), M)/J ≃ k[x],

where J is the ideal in Hom_{\underline{MCM}(Λ)}(Ω_{Λ}^{∗}(M), M) generated by the homogeneous nilpotent elements. Now when i is positive, we may identify Hom_{\underline{MCM}(Λ)}(Ω_{Λ}^{i}(M), M) with Ext_{Λ}^{i}(M, M). Furthermore, modulo nilpotence the rings End_{\underline{MCM}(Λ)}(M) and End_{Λ}(M) are isomorphic. □

Now let Λ be arbitrary. Let mod Λ ⟶^{F} mod Λ be an exact functor such that F (Λ) is a projective left module (as happens for example when F is an equivalence). Then F preserves projective modules (cf. [Ber, Section 2]). For a positive integer t, we then define a graded algebra

Ext_{Λ}^{t∗}(F^{∗}(M), M) =^{def} ⨁_{i=0}^{∞} Ext_{Λ}^{ti}(F^{i}(M), M),

with ring structure given as follows: for η ∈ Ext_{Λ}^{tu}(F^{u}(M), M) and θ ∈ Ext_{Λ}^{tv}(F^{v}(M), M), the product ηθ is the Yoneda product η ∘ F^{u}(θ), which is an element in Ext_{Λ}^{t(u+v)}(F^{u+v}(M), M). This ring structure is well defined by [Ber, Lemma 2.1], and in the following result we apply Theorem 2.1 to these algebras.

Theorem 3.2. Suppose that Λ is selfinjective, and let mod Λ ⟶^{F} mod Λ be an equivalence. Let M be an indecomposable Λ-module, and suppose Ω_{Λ}^{tn}(F^{n}(M)) ≃ M for some n ≥ 1, where n is minimal with this property. Then

Ext_{Λ}^{t∗}(F^{∗}(M), M)/I ≃ k[x],

where I is the ideal in Ext_{Λ}^{t∗}(F^{∗}(M), M) generated by the homogeneous nilpotent elements, and x is a homogeneous element in degree tn.

Proof. The proof is analogous to that of Theorem 3.1. □

Recall that when Λ is selfinjective, then the Nakayama automorphism N is the equivalence D∘Hom_{Λ}(−, Λ), where D is the usual k-dual. The composition Ω_{Λ}^{2}∘N is isomorphic to the Auslander-Reiten translate τ of Λ (cf. [ARS, Proposition IV.3.7]). An immediate application of Theorem 3.2 is the following result on τ-periodic modules.

Corollary 3.3. Suppose that Λ is selfinjective, and let M be an indecomposable Λ-module. Suppose τ^{n}(M) ≃ M for some n ≥ 1, where n is minimal with this property. Then

Ext_{Λ}^{2∗}(N^{∗}(M), M)/I ≃ k[x],

where I is the ideal in Ext_{Λ}^{2∗}(N^{∗}(M), M) generated by the homogeneous nilpotent elements, and x is a homogeneous element in degree 2n. □

We now look at orbit algebras of modules whose minimal projective resolutions are not “too big”. Let M be a Λ-module with minimal projective resolution

⋯ → P_{2} → P_{1} → P_{0} → M → 0,

say. Recall that the complexity of M, denoted cx M, is defined as

cx M =^{def} inf{t ∈ ℕ ∪ {0} | ∃a ∈ ℝ such that dim_{k} P_{n} ≤ an^{t−1} for n ≫ 0}.

In general, the complexity of a module may be infinite. From the definition, we see that cx M = 0 if and only if M has finite projective dimension, and that cx M = 1 precisely when {dim_{k} P_{n}}_{n=0}^{∞} is bounded. The following lemma shows that any equivalence of mod Λ preserves the complexity of a module.

Lemma 3.4. If mod Λ ⟶^{F} mod Λ is an equivalence, then the following hold:

(i) cx M = cx F (M) for every M ∈ mod Λ, (ii) if {M_{i}}_{i=1}^{∞} is a set of modules such that {dim_{k} M_{i}}_{i=1}^{∞} is bounded, then {dim_{k} F^{t_{i}}(M_{i})}_{i=1}^{∞} is also bounded, where each t_{i} is an arbitrary number.

Proof. Since F is an equivalence, it maps a projective resolution of M to a projective resolution of F (M). Moreover, it maps the minimal projective resolution ℙ_{M} of M to that of F (M). Namely, if F (ℙ_{M}) contains a direct summand of the form Q ⟶^{id} Q, where Q is a projective module, then F^{−1}(F (ℙ_{M})) contains F^{−1}(Q) ⟶^{id} F^{−1}(Q) as a summand. However, since F^{−1}(F (ℙ_{M})) is isomorphic to ℙ_{M}, the projective module F^{−1}(Q) must be the zero module. Then Q is the zero module, hence F (ℙ_{M}) contains no nontrivial direct summands of the form Q ⟶^{id} Q. This shows that F (ℙ_{M}) is the minimal projective resolution of F (M), and so cx M = cx F (M). The second part follows from the fact that F preserves the length of a module. □

With the help of this lemma, we obtain the following converse to Theorem 3.2, for modules of complexity one.

Theorem 3.5. Suppose that Λ is selfinjective, and let mod Λ ⟶^{F} mod Λ be an equivalence. Let M be an indecomposable Λ-module of complexity one. If the ring

Ext_{Λ}^{t∗}(F^{∗}(M), M)/I

is a polynomial ring in one variable, where I is the ideal in Ext_{Λ}^{t∗}(F^{∗}(M), M) generated by the homogeneous nilpotent elements, then M is (Ω_{Λ}^{t} ∘ F)-periodic.

Proof. Suppose that Ext_{Λ}^{t∗}(F^{∗}(M), M)/I is a polynomial ring k[x]. Let η be a homogeneous element in Ext_{Λ}^{t∗}(F^{∗}(M), M) corresponding to x in the factor algebra,

and choose a map Ω_{Λ}^{tn}(F^{n}(M)) ⟶^{f_{η}} M representing this element. Note that by Dickson’s lemma, the module Ω_{Λ}^{ti}(M), and therefore also Ω_{Λ}^{ti}(F^{i}(M)), is indecomposable for all i. Moreover, since M is of complexity one, it follows from Lemma 3.4 that {dim_{k} Ω_{Λ}^{ti}(F^{i}(M))}_{i=1}^{∞} is bounded. Therefore, if f_{η} is not an isomorphism, then the Harada-Sai Lemma and a proof analogous to that of [Sc2, Theorem 2] shows that η is nilpotent. This clearly cannot be the case, hence f_{η} must be an isomorphism. □

Using Lemma 3.4, we now show that, under certain conditions, given a number t and an equivalence F on mod Λ, there exists an infinite set {M_{i}}_{i=1}^{∞} of modules satisfying the following: the modules are of the same dimension and pairwise nonisomorphic, and each of them is indecomposable and not periodic with respect to Ω_{Λ}^{t} ∘ F.

Theorem 3.6. Suppose that Λ is selfinjective, and let mod Λ ⟶^{F} mod Λ be an equivalence. Furthermore, suppose there exists an indecomposable Λ-module M such that the following hold:

(i) cx M = 1, (ii) the ring Ext_{Λ}^{t∗}(F^{∗}(M), M)/I is not a polynomial ring, where I is the ideal

generated by the homogeneous nilpotent elements. Then M is not periodic with respect to Ω_{Λ}^{t} ∘ F. Moreover, there exists a positive integer d and infinitely many non-isomorphic indecomposable Λ-modules of dimension d, none of which are (Ω_{Λ}^{t} ∘ F)-periodic.

Proof. That M is not (Ω_{Λ}^{t} ∘ F)-periodic is an immediate consequence of Theorem 3.2. Now consider the modules {Ω_{Λ}^{tn}(F^{n}((M))}_{n=0}^{∞}. These are all pairwise nonisomorphic, and not (Ω_{Λ}^{t} ∘ F)-periodic. Furthermore, from Lemma 3.4 we see that {dim_{k} Ω_{Λ}^{tn}(F^{n}((M))}_{n=0}^{∞} is bounded. □

Applying this theorem to the Nakayama automorphism of a selfinjective algebra, we obtain the following result, which provides infinitely many modules of the same dimension, none of which are τ-periodic.

Corollary 3.7. Let Λ be selfinjective, and suppose there exists an indecomposable Λ-module M such that the following hold:

(i) cx M = 1, (ii) the ring Ext_{Λ}^{2∗}(N^{∗}(M), M)/I is not a polynomial ring, where I is the ideal

generated by the homogeneous nilpotent elements. Then M is not τ-periodic, and there exists a positive integer d and infinitely many non-isomorphic indecomposable Λ-modules of dimension d, none of which are τperiodic.

Recall that an algebra is of wild representation type if its representation theory is at least as complicated as the classification of finite dimensional vector spaces together with two non-commuting endomorphisms (see [Ben], [C-B] or [SiS] for a precise definition). We end this paper with the following application of Corollary 3.7, providing a criterion for an algebra to be of wild representation type. See also [BGL] for a related result on hereditary algebras.

Corollary 3.8. Let Λ be selfinjective, and suppose there exists an indecomposable Λ-module M such that the following hold:

(i) cx M = 1, (ii) the ring Ext_{Λ}^{2∗}(N^{∗}(M), M)/I is not a polynomial ring, where I is the ideal

generated by the homogeneous nilpotent elements. Then Λ has wild representation type.

Proof. By Corollary 3.7 and [C-B, Theorem D], the algebra is not of tame representation type, hence it is wild. □

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Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

   E-mail address: bergh@math.ntnu.no