Laura skew group algebras

Ibrahim Assem, Marcelo Lanzilotta and María Julia Redondo ^{∗}

1 Introduction

In the representation theory of artin algebras, an important line of research consists in studying those processes which allow to modify, in a predictable way, certain features of the module category of an artin algebra. In this paper, the features we are interested in are the left and the right parts of the module category, introduced by Happel, Reiten and Smalø in [15]. Let A be an artin algebra, mod A denote the category of finitely generated left A-modules and ind A denote a full subcategory of mod A having as objects exactly one representative from each isomorphism class of indecomposable A-modules. The left part L_{A} of mod A is the full subcategory of ind A having as objects those indecomposable A-modules whose predecessors have projective dimension at most one. The right part R_{A} is defined dually. These classes were used successfully in [15] to study the representation theory of quasi-tilted algebras, then, later, to study the many generalisations of this class such as the shod, weakly shod, laura and left (or right) supported algebras (see the survey [5]). However, the definition of L_{A} and R_{A} is not very practical: it is difficult to find all predecessors (or successors) of a given indecomposable module and thus to say whether it lies in L_{A} (or R_{A}, respectively) or not. Our first theorem, which generalises [15, II.1.5] and [11, 1.2], simplifies this task: it says that instead of considering all predecessors (or successors) of an indecomposable module, it suffices to look at the “immediate” ones.

THEOREM 1.1 Let A be an artin algebra, and M be an indecomposable A-module. Then:

(a) M belongs to L_{A} if and only if, for every object L in ind A with projective dimension at least two, we have Hom_{A}(L, M) = 0.

(b) M belongs to R_{A} if and only if, for every object N in ind A with injective dimension at least two, we have Hom_{A}(M, N) = 0.

As a first application of this theorem we consider the indecomposable Ext-injective modules in the additive subcategory of mod A generated by L_{A}, studied and characterised in [4, 3.1]. We give here handier characterisations.

∗The third author is a researcher from CONICET, Argentina.

Our main interest in this paper, however, lies in a construction which turns out to behave well when it comes to the left and the right parts: that of the skew group algebra. Let G be a finite group acting on an artin algebra A, the skew group algebra A[G] is the free left A-module with basis all the elements in G endowed with the multiplication given by (aσ)(bζ) = aσ(b)σζ for all a, b ∈ A and σ, ζ ∈ G. The study of the representation theory of skew group algebras was started in [19, 17, 18, 14]. We are partly motivated by the fact that finite coverings, and the algebra of invariants, as well as the smash algebras of [10], are particular cases of skew group algebras [17, 7]. The algebra A[G] retains many features from A, such as being representation-finite, being an Auslander algebra, or being a Nakayama algebra (see [7, 19]). However, many properties are not preserved by this construction, like being a basic algebra, or being connected, so we are dealing with essentially different algebras, making it worthwhile to compare their representation theories.

It is shown in [19] that an algebra A and the skew group algebra A[G] share most homological information. Thus, it has been shown that if A is a tilted (or a quasi-tilted) algebra, then so is A[G], see [19] (or [15], respectively). However, for studying generalisations of these classes of algebras (such as, for instance, laura algebras) homological information, by itself, is not sufficient. We also need a nice correspondence between paths in ind A and in ind A[G]. After establishing this correspondence, we are able to prove our second main theorem.

THEOREM 1.2 Let A be an artin algebra, G be a finite group acting on A such that |G| is invertible in A, and R = A[G]^{b} be the basic algebra associated to the skew group algebra. Then:

(a) A is left (or right) supported if and only if so is R.

(b) A is laura if and only if so is R.

(c) A is left (or right) glued if and only if so is R.

(d) A is weakly shod if and only if so is R.

(e) A is shod if and only if so is R.

(f) A is quasi-tilted if and only if so is R.

(g) A is tilted if and only if so is R.

For the definitions of the above classes, we refer to [5] or to section 5 below. Here, statements (f) and (g) are included for completeness.

Finally, we apply this result to the toupie algebras of [9]. We define a new class, which we call skew toupie algebras, and exhibit a familiy of laura (actually, weakly shod) skew toupie algebras.

2 A characterisation of the left and right parts.

2.1 Notation.

For an artin algebra A, we denote by mod A the category of finitely generated left A-modules, and by ind A a full subcategory of mod A having as objects a full set of representatives of the isomorphism classes of the indecomposable A-modules. For a subcategory C of mod A, we write M ∈ C to express that M is an object in C. We denote by add C the full subcategory of mod A having as objects the direct sums of indecomposable summands of objects in C and, if M is an A-module, we abbreviate add{M} as add M. We say that a full subcategory C of ind A is finite if it has only finitely many objects. Given two A-modules L and M, we write L ∈∣ M to express that L is a direct summand of M. We denote the projective (or injective) dimension of an A-module M by pd_{A} M (or id_{A} M, respectively) and the global dimension of A by gl.dim. A. Finally, we denote by Γ(mod A) the Auslander-Reiten quiver of A, and by τ_{A} its Auslander-Reiten translation DTr. For further definitions or facts on mod A, Γ(mod A) and τ_{A} we refer the reader to [7, 20].

Given M, N ∈ ind A, we write M ⇝ N in case there exists a path

M = M_{0} →^{f_{1}} M_{1} → ⋯ → M_{t−1} →^{f_{t}} M_{t} = N  (1)

(t ≥ 0) from M to N, that is, the f_{i} are non-zero morphisms, and the M_{i} are indecomposable A-modules. We then say that M is a predecessor of N, and N is a successor of M. If each f_{i} in (1) is an irreducible morphism, we say that (1) is a path of irreducible morphisms. A path (1) of irreducible morphisms is sectional if τ_{A}M_{j+1} ≠ M_{j−1} for all j such that 1 ≤ j ≤ t.

Let A be a basic and connected artin algebra. Following [15], we define the left part L_{A} of mod A to be the full subcategory of ind A consisting of all the modules M such that, if L ⇝ M, then pd_{A} L ≤ 1. Dually, the right part R_{A} is the full subcategory of ind A consisting of all the modules M such that, if M ⇝ N, then id_{A} N ≤ 1. Clearly, L_{A} is closed under predecessors, while R_{A} is closed under successors.

For the sake of brevity, we refrain from now on from stating the dual of each statement and leave the primal-dual translation to the reader.

The following result generalises [15, II.1.5] and [11, 1.2] and its proof is heavily inspired from the proofs of these statements.

LEMMA 2.1 Let A be an artin algebra, and M be an indecomposable A-module such that there exists a path M_{0} →^{f_{0}} M_{1} →^{f_{1}} M in ind A with pd_{A} M_{0} ≥ 2. Then there exists an indecomposable A-module L with pd_{A} L ≥ 2 and Hom_{A}(L, M) ≠ 0.

Proof. Assume that this is not the case, that is, there exist M ∈ ind A and a path M_{0} →^{f_{0}} M_{1} →^{f_{1}} M with pd_{A} M_{0} ≥ 2 and moreover Hom_{A}(L, M) = 0 for all L ∈ ind A with pd_{A} L ≥ 2. We may clearly choose the path M_{0} →^{f_{0}} M_{1} →^{f_{1}} M with pd_{A} M_{0} ≥ 2 so that the length ℓ(M_{1}) of M_{1} is minimal.

It follows from our hypothesis that f_{1}f_{0} = 0 (in particular, f_{0} is not an epimorphism) and also that pd_{A} M_{1} ≤ 1.

We claim that C = Coker f_{0} is indecomposable. Since f_{1}f_{0} = 0, we have Hom_{A}(C, M) ≠ 0. So, if C were decomposable, there would exist an indecomposable C′ ∈∣ C such that Hom_{A}(C′, M) ≠ 0. Let C ↠^{p} C′ ↪^{i} C denote respectively the canonical projection and injection. We have a fiber product diagram

(δi)  0 → Im f_{0} →^{f′} M′_{1} →^{g′} C′ → 0
∥ ↓ h ↓ i
(δ)  0 → Im f_{0} →^{f} M_{1} →^{g} C → 0

where f is the canonical inclusion. We claim that the upper sequence is not zero. Indeed, assume that it is, and let g′′ : C′ → M′_{1} be such that g′g′′ = 1_{C′}. We have (pg)(hg′′) = p(gh)g′′ = p(ig′)g′′ = 1_{C′}, hence C′ ∈∣ M_{1}. Since M_{1} is indecomposable, C′ ≃ M_{1} and then Im f_{0} = 0, a contradiction. This shows that the upper sequence is not split.

Since, by hypothesis, Hom_{A}(C′, M) ≠ 0 and g′ is an epimorphism, Hom_{A}(M′_{1}, M) ≠ 0. Thus, there exists an indecomposable summand M′′_{1} of M′_{1} such that Hom_{A}(M′′_{1}, M) ≠ 0. On the other hand, Im f_{0} maps non-trivially on every indecomposable summand of M′_{1} (because the upper sequence is not split). In particular, Hom_{A}(Im f_{0}, M′′_{1}) ≠ 0. Composing with the canonical projection f′_{0} : M_{0} ↠ Im f_{0} yields a non-zero morphism M_{0} → M′′_{1}. Then we have a path M_{0} → M′′_{1} → M in ind A. Our minimality assumption yields ℓ(M_{1}) ≤ ℓ(M′′_{1}) ≤ ℓ(M′_{1}) ≤ ℓ(M_{1}) so that M_{1} ≃ M′′_{1} = M′_{1}. Therefore, h is an isomorphism. Hence so is i. This establishes our claim that C is indecomposable.

The indecomposability of C implies that pd_{A} C ≤ 1, because Hom_{A}(C, M) ≠ 0. This, and the short exact sequence

0 → Im f_{0} →^{f} M_{1} →^{g} C → 0

imply that pd_{A} Im f_{0} ≤ 1. Therefore, f_{0} is not a monomorphism.

On the other hand, pd_{A} C ≤ 1 implies Ext_{A}^{2}(C, Ker f_{0}) = 0. In particular, the class in Ext_{A}^{2}(C, Ker f_{0}) of the exact sequence

(ϵ) 0 → Ker f_{0} → M_{0} →^{f_{0}} M_{1} →^{g} C → 0

vanishes. Letting ϵ_{1} and ϵ_{2} denote respectively the short exact sequences

(ϵ_{1}) 0 → Ker f_{0} → M_{0} →^{f′_{0}} Im f_{0} → 0

(ϵ_{2}) 0 → Im f_{0} →^{f} M_{1} →^{g} C → 0

we have 0 = ϵ = ϵ_{1}ϵ_{2}. Applying Hom_{A}(−, Ker f_{0}) to ϵ_{2} yields an exact sequence

⋯ → Ext_{A}^{1}(M_{1}, Ker f_{0}) → Ext_{A}^{1}(Im f_{0}, Ker f_{0}) → Ext_{A}^{2}(C, Ker f_{0}) = 0.

Then, there exists ζ ∈ Ext_{A}^{1}(M_{1}, Ker f_{0}) such that ζf = ϵ_{1}. That is, there exists an A-module N and a commutative diagram with exact rows and columns

0  0
↓  ↓
0 → Ker f_{0} → M_{0} → Im f_{0} → 0
∥ ↓ ↓ f
0 → Ker f_{0} → N → M_{1} → 0
↓  ↓
C = C
↓  ↓
0  0

from which we deduce a short exact sequence

0 → M_{0} → Im f_{0} ⊕ N → M_{1} → 0.

Since pd_{A} M_{0} ≥ 2 while pd_{A} Im f_{0} ≤ 1 and pd_{A} M_{1} ≤ 1, then N has an indecomposable summand N′ with pd_{A} N′ ≥ 2. On the other hand, the middle column of the above diagram is not split (otherwise, the right column would split too, a contradiction). Hence, every indecomposable summand of N maps non-trivially to C. We consider the resulting path N′ → C → M in ind A. Since pd_{A} N′ ≥ 2 while ℓ(C) < ℓ(M_{1}), we get a contradiction to our minimality assumption. □

Proof of Theorem 1.1: It suffices to prove (a), since (b) is dual. Since the necessity of the condition is obvious, we prove the sufficiency. Assume that M is such that, for every L ∈ ind A with pd_{A} L ≥ 2, we have Hom_{A}(L, M) = 0. We must show that every predecessor L′ of M has projective dimension at most one. Assume to the contrary that pd_{A} L′ ≥ 2, and use induction on the length t of a shortest path from L′ to M in ind A:

L′ = M_{0} →^{f_{1}} M_{1} →^{f_{2}} M_{2} … →^{f_{t}} M_{t} = M.

If t ∈ {0, 1}, we have a contradiction to the hypothesis. If t = 2, the contradiction follows from Lemma 2.1. Assume that t ≥ 3. By Lemma 2.1, there exists an indecomposable N and a non-zero morphism f : N → M_{2} such that pd_{A} N ≥ 2. But then the path

N →^{f} M_{2} →^{f_{3}} M_{3} … →^{f_{t}} M_{t} = M

of length t − 1 yields a contradiction to the induction hypothesis. □

3 Ext-injectives in add L_{A}.

Let A be a basic and connected artin algebra. We recall from [8] that an indecomposable module M ∈ L_{A} is called Ext-injective in add L_{A} whenever Ext_{A}^{1}(L, M) = 0 for all L ∈ L_{A}. It is shown in [8, 3.4] that M ∈ L_{A} is Ext-injective in add L_{A} if and only if τ_{A}^{−1}M ∉ L_{A}. Further, it is shown in [4, 3.1] that an indecomposable A-module M is Ext-injective in add L_{A} if and only if it belongs to one of the following subsets of ind A:

(a) E_{1} = E_{1}(A) = {L ∈ L_{A} : there exists an injective I ∈ ind A, and a path I ⇝ L in ind A}, and

(b) E_{2} = E_{2}(A) = {L ∈ L_{A} \ E_{1} : there exists a projective P ∈ ind A \ L_{A}, and a sectional path P ⇝ τ_{A}^{−1}L}.

Clearly, E_{2} is contained in the (apparently) larger set:

E′_{2} = E′_{2}(A) = {L ∈ L_{A}\E_{1} : there exists a projective P ∈ ind A\L_{A}, and a path P ⇝ τ_{A}^{−1}L}.

Our objective in this section is to prove that E_{2} = E′_{2}. This yields an easier characterisation of the Ext-injectives in add L_{A} which, apart from its theoretical interest, will be used essentially in section 5.

PROPOSITION 3.1 Let A be an artin algebra. An indecomposable A-module M is Extinjective in add L_{A} if and only if M ∈ E_{1} ∪ E′_{2}.

Proof. Assume M to be indecomposable and Ext-injective in add L_{A}. Then τ_{A}^{−1}M ∉ L_{A}. By Theorem 1.1, there exists an indecomposable A-module L such that pd_{A} L ≥ 2 and Hom_{A}(L, τ_{A}^{−1}M) ≠ 0. Hence, there exists an indecomposable injective I such that Hom_{A}(I, τ_{A}L) ≠ 0. So, either Hom_{A}(τ_{A}L, M) ≠ 0 and the path I → τ_{A}L → M yields M ∈ E_{1}, or else Hom_{A}(τ_{A}L, M) = 0 in which case the Auslander-Reiten formula gives \underline{Hom}_{A}(L, τ_{A}^{−1}M) ≃ \overline{Hom}_{A}(τ_{A}L, M) = 0. Since Hom_{A}(L, τ_{A}^{−1}M) ≠ 0, there exists an indecomposable projective P and a path L → P → τ_{A}^{−1}M. Since pd_{A} L ≥ 2, we have L ∉ L_{A}, so P ∉ L_{A} and consequently, M ∈ E′_{2}.

Conversely, let M ∈ E_{1} ∪ E′_{2}. Clearly, if M ∈ E′_{2}, then M ∈ L_{A} but τ_{A}^{−1}M ∉ L_{A} (because τ_{A}^{−1}M succedes a projective not in L_{A}) so that M is Ext-injective in add L_{A}. If M ∈ E_{1}, then either M is injective, or else there exists an indecomposable injective I and a path I ⇝ M → ∗ → τ_{A}^{−1}M. By [3, 1.6], we infer that τ_{A}^{−1}M ∉ L_{A}. Thus, M is Ext-injective in add L_{A}.

This establishes the first assertion. The second follows immediately. □

The following corollary gives equivalent characterisations of the sets E_{1} and E_{2}.

COROLLARY 3.2 Let M ∈ L_{A}. Then:

(a) The following conditions are equivalent: i) There exist an injective I ∈ ind A and a path I ⇝ M; ii) There exist an injective I ∈ ind A and a path of irreducible morphisms I ⇝ M; iii) There exist an injective I ∈ ind A and a sectional path I ⇝ M; iv) There exists an injective I ∈ ind A such that Hom_{A}(I, M) ≠ 0.

(b) The following conditions are equivalent:

i) There exist a projective P ∈ ind A \ L_{A} and a path P ⇝ τ_{A}^{−1}M; ii) There exist a projective P ∈ ind A \ L_{A} and a path of irreducible morphisms

P ⇝ τ_{A}^{−1}M; iii) There exist a projective P ∈ ind A \ L_{A} and a sectional path P ⇝ τ_{A}^{−1}M; iv) There exists a projective P ∈ ind A \ L_{A} such that Hom_{A}(P, τ_{A}^{−1}M) ≠ 0.

Proof.

(a) i) implies ii) and ii) implies iii) follow from [3, 1.6], the other implications are trivial.

(b) i) implies iii) by the above proposition. The other implications are trivial. □

4 Preliminaries on skew group algebras.

We recall the relevant definitions, see [7, 19]. Let A be an artin k-algebra, and G be a finite group with identity 1. We say that G acts on A if there is a function G × A → A, (σ, a) ↦ σ(a) such that:

(a) For each σ ∈ G, the map a ↦ σ(a) is a k-linear automorphism of A.

(b) (σ_{1}σ_{2})(a) = σ_{1}(σ_{2}(a)) for all σ_{1}, σ_{2} ∈ G and a ∈ A.

(c) 1(a) = a for all a ∈ A.

Such an action induces an action of G on mod A as follows. Let M be an A-module, and σ ∈ G. We define ^{σ}M to be the A-module with the additive structure of M but where the multiplication is given by a.x = σ^{−1}(a)x, for a ∈ A and x ∈ M. LEMMA 4.1 Let σ ∈ G. The mapping M ↦ ^{σ}M (where M is an A-module) induces an homomorphism of G into the group of automorphisms of the category mod A. Proof. We define a functor ^{σ}(−) : mod A → mod A on objects by M ↦ ^{σ}M. Let now f : L → M be a morphism of A-modules and define ^{σ}f : ^{σ}L → ^{σ}M by x ↦ f(x) for x ∈ L. This is an A-linear map because

σf(a.x) = ^{σ}f(σ^{−1}(a)x) = f(σ^{−1}(a)x) = σ^{−1}(a)f(x) = a. ^{σ}f(x) for all a ∈ A and x ∈ L. This clearly defines an endofunctor of mod A. The lemma then follows from the observation that, for any σ_{1}, σ_{2} ∈ G, we have ^{σ_{1}}(^{σ_{2}}(−)) = ^{σ_{1}σ_{2}}(−) and, in particular, ^{σ}(−). ^{σ^{−1}}(−) = id_{mod A} = ^{σ^{−1}}(−). ^{σ}(−) for any σ ∈ G. □

Assume that G acts on A. The skew group algebra A[G] has as underlying A-module the free left A-module having as basis all the elements in G, with the multiplication defined by

(aσ)(bζ) = aσ(b)σζ

for all a, b ∈ A and σ, ζ ∈ G.

Throughout this paper, we assume that the order |G| of G is invertible in A. Since the skew group algebra A[G] is not basic in general, even if A is so (see [7]), we agree to always consider the basic form R = A[G]^{b} of A[G]. Also, in order to avoid confusion, we denote the A-modules by the letters L, M, N, … and the R-modules by the letters X, Y, Z, . . ..

The natural ring inclusion A ↪ R given by a ↦ a.1 for a ∈ A, induces the change of rings functors _{R}R_{A} ⊗ − : mod A → mod R and Hom_{R}(_{R}R_{A}, −) : mod R → mod A. The following proposition summarises the properties of these functors, as in [19, 1.1 and 1.8 (a)(b)(c)].

PROPOSITION 4.2 Let G be a finite group acting on an artin algebra A, and R = A[G]^{b}. Assume that |G| is invertible in A. Then:

(a) (R ⊗_{A} −, Hom_{R}(R, −)) and (Hom_{R}(R, −), R ⊗_{A} −) are two adjoint pairs of functors.

(b) (1) The natural morphism id_{mod A} → Hom_{R}(R, R ⊗_{A} −) is a section of functors.

(2) The natural morphism R ⊗_{A} Hom_{R}(R, −) → id_{mod R} is a retraction of functors.

(c) If M, N ∈ ind A, then

(1) Hom_{R}(R, R ⊗_{A} M) ≃ ⨁_{σ∈G} ^{σ}M.

(2) R ⊗_{A} M ≃ R ⊗_{A} N if and only if there exists σ ∈ G such that M ≃ ^{σ}N.

(3) If R ⊗_{A} M ≃ ⨁_{i=1}^{n} X_{i} is an indecomposable decomposition, then, for each i, the A-module Hom_{R}(R, X_{i}) has an indecomposable summand from each isomorphism class of the ^{σ}M, with σ ∈ G.

It follows from (a) above that both functors R ⊗_{A} − and Hom_{R}(R, −) are exact and preserve projectives and injectives. On the other hand, as seen in (c), they do not preserve indecomposability.

COROLLARY 4.3 Let X be an indecomposable R-module. Then there exists M ∈ ind A such that M ∈∣ Hom_{R}(R, X) and X ∈∣ R ⊗_{A} M.

Proof. By 4.2(b)(2), there is a retraction R ⊗_{A} Hom_{R}(R, X) → X. The statement follows from the indecomposability of X. □

LEMMA 4.4 Let M, N be indecomposable A-modules such that Hom_{A}(M, N) ≠ 0.

(a) For any indecomposable X ∈∣ R ⊗_{A} M, we have Hom_{R}(X, R ⊗_{A} N) ≠ 0.

(b) For any indecomposable Y ∈∣ R ⊗_{A} N, we have Hom_{R}(R ⊗_{A} M, Y) ≠ 0.

Proof. We only prove (a) since (b) is similar. By 4.2(c)(3), we have an indecomposable decomposition in mod R

R ⊗_{A} M = ⊕_{i=1}^{m}X_{i}

such that Hom_{R}(R, X_{i}) = ⨁_{σ∈H_{i}} ^{σ}M for some H_{i} ⊆ G. Moreover, for any i, and any ζ ∈ G, there exists σ ∈ H_{i} such that

σM ≃ ^{ζ}M.

We need to show that, for any i, Hom_{R}(X_{i}, R ⊗_{A} N) ≃ Hom_{A}(Hom_{R}(R, X_{i}), N) ≠ 0. It follows from Hom_{A}(M, N) ≠ 0 and 4.2(b)(1) that

Hom_{A}(Hom_{R}(R, R ⊗_{A} M), Hom_{R}(R, R ⊗_{A} N)) ≠ 0.

Therefore there exist σ, ζ ∈ G such that Hom_{A}( ^{σ}M, ^{ζ}N) ≠ 0. By 4.1, Hom_{A}( ^{ζ^{−1}σ}M, N) ≠ 0. Thus Hom_{A}(Hom_{R}(R, X_{i}), N) ≠ 0 for any i. □

COROLLARY 4.5 Let M_{1} → M_{2} → ⋯ → M_{t} be a path in ind A. Then:

(a) For any indecomposable X_{1} ∈∣ R ⊗_{A} M_{1}, there exists a path X_{1} → X_{2} → ⋯ → X_{t} in ind R such that, for any i, X_{i} ∈∣ R ⊗_{A} M_{i} and M_{i} ∈∣ Hom_{R}(R, X_{i}).

(b) For any indecomposable Y_{t} ∈∣ R⊗_{A} M_{t}, there exists a path Y_{1} → Y_{2} → ⋯ → Y_{t} in ind R such that, for any i, Y_{i} ∈∣ R ⊗_{A} M_{i} and M_{i} ∈∣ Hom_{R}(R, Y_{i}).

Proof. We only prove (a) since (b) is similar. By 4.2(c)(3), X_{i} ∈∣ R ⊗_{A} M_{i} implies M_{i} ∈∣ Hom_{R}(R, X_{i}) for any i. Since X_{1} ∈∣ R ⊗_{A} M_{1} and Hom_{A}(M_{1}, M_{2}) ≠ 0, we have Hom_{A}(X_{1}, R ⊗_{A} M_{2}) ≠ 0 by 4.4(a). Hence there exists an indecomposable R-module X_{2} ∈∣ R ⊗_{A} M_{2} such that Hom_{R}(X_{1}, X_{2}) ≠ 0. The statement follows from an obvious induction. □

LEMMA 4.6 Let X, Y be indecomposable R-modules such that Hom_{R}(X, Y) ≠ 0. Let M be an indecomposable summand of Hom_{R}(R, X) such that X ∈∣ R ⊗_{A} M. Then there exist σ ∈ G and an indecomposable summand N ∈∣ Hom_{R}(R, Y) such that Y ∈∣ R ⊗_{A} ^{σ}N and Hom_{A}(M, ^{σ}N) ≠ 0.

Proof. By 4.2(c)(2), R⊗_{A}N ≃ R⊗_{A} ^{σ}N for all σ ∈ G and all N ∈ ind A. Since X ∈∣ R⊗_{A}M and Hom_{R}(X, Y) ≠ 0 then, for each N ∈ ind A such that N ∈∣ Hom_{R}(R, Y) and Y ∈∣ R ⊗_{A} N, we have Hom_{R}(R ⊗_{A} M, R ⊗_{A} N) ≠ 0. Adjunction gives Hom_{A}(M, Hom_{R}(R, R ⊗_{A} N)) ≠ 0. By 4.2(c)(1), Hom_{R}(R, R ⊗_{A} N) ≃ ⨁_{σ∈G} ^{σ}N. Hence there exists σ ∈ G such that Hom_{A}(M, ^{σ}N) ≠ 0 and Y ∈∣ R ⊗_{A} N ≃ R ⊗_{A} ^{σ}N. □

COROLLARY 4.7 Let X_{1} → X_{2} → ⋯ → X_{t} be a path in ind R. Then:

(a) For any indecomposable M_{1} ∈∣ Hom_{R}(R, X_{1}) such that X_{1} ∈∣ R ⊗_{A} M_{1}, there exist σ_{2}, ⋯, σ_{t} ∈ G and a path

M_{1} → ^{σ_{2}}M_{2} → ⋯ → ^{σ_{t−1}}M_{t−1} → ^{σ_{t}}M_{t}

in ind A, with M_{i} ∈∣ Hom_{R}(R, X_{i}) and X_{i} ∈∣ R ⊗_{A} ^{σ_{i}}M_{i} for any i.

(b) For any indecomposable N_{t} ∈∣ Hom_{R}(R, X_{t}) such that X_{t} ∈∣ R ⊗_{A} N_{t}, there exist ζ_{1}, ⋯, ζ_{t−1} ∈ G and a path

ζ_{1}N_{1} →^{ζ_{2}} N_{2} → ⋯ →^{ζ_{t−1}} N_{t−1} → N_{t}

in ind A, with N_{i} ∈∣ Hom_{R}(R, X_{i}) and X_{i} ∈∣ R ⊗_{A}^{ζ_{i}} N_{i} for any i.

Proof. The proof is easy and left to the reader. □

5 Laura skew group algebras.

Throughout this section, we assume that A is a basic and connected artin algebra, that G is a group acting on A, with |G| invertible in A, and we let R = A[G]^{b}.

LEMMA 5.1 For any σ ∈ G, we have ^{σ}L_{A} = L_{A}.

Proof. We first show that ^{σ}L_{A} ⊆ L_{A}. Let L ∈ ^{σ}L_{A}, then ^{σ^{−1}}L ∈ L_{A}. Let M be an indecomposable A-module such that Hom_{A}(M, L) ≠ 0. Then Hom_{A}( ^{σ^{−1}}M, ^{σ^{−1}}L) ≠ 0. Since ^{σ^{−1}}L ∈ L_{A}, we have pd_{A} ^{σ^{−1}}M ≤ 1. Hence pd_{A} M ≤ 1. By Theorem 1.1, L ∈ L_{A}, as required. Conversely, if N ∈ L_{A}, then ^{σ^{−1}}N ∈ ^{σ^{−1}}L_{A} so that ^{σ^{−1}}N ∈ L_{A} and N ∈ ^{σ}L_{A}. □

LEMMA 5.2

(a) add(R ⊗_{A} L_{A}) = add L_{R}.

(b) add Hom_{R}(R, L_{R}) = add L_{A}.

Proof. We first show that add(R ⊗_{A} L_{A}) ⊆ add L_{R}. Let M ∈ L_{A}, and R ⊗_{A} M = ⨁_{i=1}^{m} X_{i} be a decomposition into indecomposable modules in mod R. We claim that X_{i} ∈ L_{R} for any i. By Theorem 1.1, we must prove that, for each Y ∈ ind R such that Hom_{R}(Y, X_{i}) ≠ 0, we have pd_{R} Y ≤ 1. By 4.6, there exist L, N ∈ ind A such that L ∈∣ Hom_{R}(R, Y), Y ∈∣ R ⊗_{A} L, N ∈∣ Hom_{R}(R, X_{i}), X_{i} ∈∣ R ⊗_{A} ^{σ}N for some σ ∈ G, and Hom_{A}(L, ^{σ}N) ≠ 0. On the other hand, by 4.2(c), we have an indecomposable decomposition in mod A

Hom_{R}(R, X_{i}) = ⊕_{ζ∈H_{i}} ^{ζ}M

for some H_{i} ⊆ G. Hence, there exists ζ ∈ H_{i} such that Hom_{A}(L, ^{σζ}M) ≠ 0. By 5.1, σζM ∈ L_{A} hence pd_{A} L ≤ 1. Therefore pd_{R} R ⊗_{A} L ≤ 1. Since Y ∈∣ R ⊗_{A} L, we infer that pd_{R} Y ≤ 1, as required.

We next prove that add Hom_{R}(R, L_{R}) ⊆ add L_{A}. Let X ∈ L_{R} and Hom_{R}(R, X) = ⊕_{i=1}^{m}M_{i} be a decomposition into indecomposable modules in mod A. We claim that M_{i} ∈ L_{A} for any i. By Theorem 1.1, we must prove that for each L ∈ ind A such that Hom_{A}(L, M_{i}) ≠ 0, we have pd_{A} L ≤ 1. Let R⊗_{A}L = ⨁_{j=1}^{n} Y_{j} be an indecomposable decomposition in mod R. Since Hom_{A}(L, M_{i}) ≠ 0, we have Hom_{R}(R ⊗_{A} L, X) ≃ Hom_{A}(L, Hom_{R}(R, X)) ≠ 0, so there exists j_{0} ∈ {1, ⋯, n} such that Hom_{R}(Y_{j_{0}}, X) ≠ 0. Then pd_{R} Y_{j_{0}} ≤ 1. On the other hand, Hom_{R}(R, Y_{j_{0}}) ≃ ⨁_{σ∈H_{j_{0}}} ^{σ}L for some H_{j_{0}} ⊆ G. Hence pd_{R} Y_{j_{0}} = pd_{A} Hom_{R}(R, Y_{j_{0}}) = pd_{A} ^{σ}L = pd_{A} L for each σ ∈ H_{j_{0}}. Therefore, pd_{A} L ≤ 1.

There remains to prove that equality holds in each case. Assume X ∈ L_{R}. Then Hom_{R}(R, X) ∈ add L_{A}. By 4.2(b), X ∈∣ R ⊗_{A} Hom_{R}(R, X). Hence X ∈ add(R ⊗_{A} L_{A}). Similarly, if M ∈ L_{A}, then M ∈∣ Hom_{R}(R, R ⊗_{A} M) by 4.2(b) so M ∈ add Hom_{R}(R, L_{R}). □

COROLLARY 5.3

(a) add(R ⊗_{A} ind A \ L_{A}) = add(ind R \ L_{R}).

(b) add Hom_{R}(R, ind R \ L_{R}) = add(ind A \ L_{A}).

Proof. We only prove (a) since (b) is similar. Let M ∈ ind A \ L_{A} and R ⊗_{A} M ≃ ⨁_{i=1}^{m} X_{i} be an indecomposable decomposition in mod R. By 4.2(c), for any i, we have Hom_{R}(R, X_{i}) = ⨁_{σ∈H_{i}} ^{σ}M for some H_{i} ⊆ G. By 5.1, ^{σ}M ∉ L_{A} for any σ. Hence, by 5.2, X_{i} ∉ L_{R} for any i. Conversely, assume X ∈ ind R \ L_{R}. By 4.3, there exists M ∈ ind A such that X ∈∣ R ⊗_{A} M. By 5.2, M ∉ L_{A}. □

We now show that the Ext-injectives in add L_{A} correspond to those in add L_{R}. For this purpose, we denote by E_{1}(A), E′_{2}(A) and by E_{1}(R), E′_{2}(R), respectively, the sets described in section 3 for the algebras A and R.

LEMMA 5.4

(a) add(R ⊗_{A} E_{1}(A)) = add E_{1}(R), add(R ⊗_{A} E′_{2}(A)) = add E′_{2}(R).

(b) add Hom_{R}(R, E_{1}(R)) = add E_{1}(A), add Hom_{R}(R, E′_{2}(R)) = add E′_{2}(A).

Proof. Let E_{1} ∈ E_{1}(A) and X ∈∣ R ⊗_{A} E_{1} be indecomposable. By 5.2, X ∈ L_{R}. Now, there exists a path I ⇝ E_{1} in ind A with I injective. By 4.5, this path induces a path I′ ⇝ X in ind R, with I′ ∈∣ R ⊗_{A} I so that I′ is an injective R-module. Thus X ∈ E_{1}(R). Let now E_{2} ∈ E′_{2}(A) and X ∈∣ R ⊗_{A} E_{2} be indecomposable. By 5.2, X ∈ L_{R}. Moreover, there exists a path P ⇝ τ^{−1}E_{2} in ind A, with P ∉ L_{A} projective. By [19, 3.8], we have τ_{R}^{−1}X ∈∣ R ⊗_{A} (τ_{A}^{−1}E_{2}). Applying 4.5 yields a path P′ ⇝ τ_{R}^{−1}X in ind R, with P′ ∈∣ R ⊗_{A} P, that is, P′ is a projective R-module. By 5.3, P′ ∉ L_{R}. Then, X ∈ E_{1}(R) ∪ E′_{2}(R).

One proves in exactly the same way (using 4.7 instead of 4.5) that add Hom_{R}(R, E_{1}(R)) ⊆ add E_{1}(A) and add Hom_{R}(R, E′_{2}(R)) ⊆ add(E_{1}(A) ∪ E′_{2}(A)).

Since add(R ⊗_{A} E_{1}(A)) ⊆ add E_{1}(R) and add Hom_{R}(R, E_{1}(R)) ⊆ add E_{1}(A), a simple application of 4.2(b) implies that equality holds in each of these cases.

Now, let E_{2} ∈ E′_{2}(A) and assume that an indecomposable summand X ∈∣ R ⊗_{A} E_{2} lies in E_{1}(R). Then Hom_{R}(R, X) ∈ add E_{1}(A). However, Hom_{R}(R, X) = ⨁_{σ∈H} ^{σ}E_{2} for some H ⊆ G so ^{σ}E_{2} ∈ E_{1}(A) for any σ, hence E_{2} ∈ E_{1}(A) a contradiction. This shows that add(R ⊗_{A} add E′_{2}(A)) ⊆ add E′_{2}(R). Similarly, add Hom_{R}(R, E′_{2}(R)) ⊆ add E′_{2}(A). Finally, applying 4.2(b) yields equality in each of these cases. □

We recall some definitions. An artin algebra A is left supported if add L_{A} is contravariantly finite [4]. We know that an algebra A is left supported if and only if add L_{A} is cogenerated by the direct sum of a complete set of representatives of the isomorphism classes of indecomposable Ext-injectives in add L_{A} (see [4]). Right supported algebras are defined dually. An artin algebra A is a laura algebra if L_{A} ∪ R_{A} is cofinite in ind A, that is, if ind A \ (L_{A} ∪ R_{A}) is finite [3]. It is called right glued if the class of all M ∈ ind A such that pd_{A} M ≤ 1 is cofinite in ind A, see [1], or, equivalently, if L_{A} is cofinite in ind A, see [3, 2.2]. Left glued algebras are defined dually. Clearly, left and right glued algebras are laura.

An artin algebra is called weakly shod if the length of any path in ind A from an injective to a projective is bounded [12] or, equivalently, if there exists l ≥ 0 such that any path in ind A from an indecomposable not in L_{A} to one not in R_{A} has length at most l, see [2, 1.4]. The algebra A is called shod if, for each M ∈ ind A, we have pd_{A} M ≤ 1 or id_{A} M ≤ 1, or, equivalently, if L_{A} ∪ R_{A} = ind A, see [11]. Shod algebras are weakly shod algebras, and weakly shod algebras are laura algebras. Finally, A is quasi-tilted if it is shod and gl.dim. A ≤ 2, see [15]. Those laura algebras which are not quasi-tilted are left and right supported, see [4, 4.4].

Proof of Theorem 1.2: The proof of (f) is in [15, II.1.6]. Actually the same proof establishes (e). The proof of (g) in the representation-finite case is in [19, 4.6]. It carries over to the general case using, for instance, the Liu-Skowroński criterion. We just show statements (a) to (d).

(a) Assume A to be left supported and denote by E (or Q) the direct sum of a complete set of representatives of the isomorphism classes of indecomposable Ext-injectives in add L_{A} (or add L_{R}, respectively). We know that add L_{A} is the class of A-modules cogenerated by E. Let X ∈ L_{R}. Then Hom_{R}(R, X) ∈ add L_{A} by 5.2, hence there exist m > 0 and a monomorphism Hom_{R}(R, X) ↪ E^{(m)}. Since X ∈∣ R ⊗_{A} Hom_{R}(R, X) by 4.2 (b), and R ⊗_{A} − is exact, we deduce a monomorphism

X ↪ R ⊗_{A} Hom_{R}(R, X) ↪ (R ⊗_{A} E)^{(m)}.

Since R ⊗_{A} E ∈ add Q by 5.4, X is cogenerated by Q. So R is left supported. The converse is proven in the same way.

(b) Assume A to be laura. Then ind A \ (L_{A} ∪ R_{A}) is finite. Hence the set X_{R} of all indecomposable R-modules in add(R⊗_{A}ind A\(L_{A}∪R_{A})) is finite. Let X ∉ L_{R}∪R_{R} be an indecomposable R-module. By 4.3, there exists M ∈ ind A such that M ∈∣ Hom_{R}(R, X) and X ∈∣ R ⊗_{A} M. By 5.2, M ∉ L_{A} ∪ R_{A}. Hence X ∈ X_{R} and ind R \ (L_{R} ∪ R_{R}) ⊆ X_{R}. In particular, it is finite so R is laura.

Conversely, assume R to be laura. Hence the set M_{A} of all indecomposable A-modules in add Hom_{R}(R, ind R\(L_{R}∪R_{R})) is finite. Let M ∉ L_{A}∪R_{A} be an indecomposable Amodule. If R⊗_{A}M = ⨁_{i=1}^{m} X_{i} is an indecomposable decomposition in mod R, then, by 5.3, X_{i} ∉ L_{R} ∪R_{R} for any i. By 4.2(b), M ∈∣ Hom_{R}(R, R⊗_{A} M) ≃ ⨁_{i=1}^{m} Hom_{R}(R, X_{i}). Consequently, M ∈ M_{A} and ind A \ (L_{A} ∪ R_{A}) ⊆ M_{A}. In particular, it is finite, so A is laura.

(c) Assume A to be right glued. Then ind A \ L_{A} is finite. By 5.3 and 4.3, ind R \ L_{R} is finite. Hence R is right glued. The converse is proven similarly.

(d) Assume A to be weakly shod, and let l ≥ 0 be such that any path from an indecomposable not in L_{A} to one not in R_{A} has length at most l. Let

X = X_{0} → X_{1} → ⋯ → X_{s} = Y  (2)

be a path in ind R with X ∉ L_{R} and Y ∉ R_{R}. By 4.7, it induces a path in ind A of the form

M_{0} → ^{σ_{1}}M_{1} → ^{σ_{2}}M_{2} → ⋯ → ^{σ_{s}}M_{s}

with M_{j} ∈∣ Hom_{R}(R, X_{j}) and X_{j} ∈∣ R ⊗_{A} ^{σ_{j}}M_{j} ≃ R ⊗_{A} M_{j}. By 5.3 and 5.1, M_{0} ∉ L_{A} and ^{σ_{s}}M_{s} ∉ R_{A}. Then s ≤ l and R is weakly shod.

The converse is proven in the same way, using 4.5 instead of 4.7.

□

We now assume that the group G acts freely on the idempotents. In this case, R = A[G]^{b} and the algebra of invariants A^{G} are Morita equivalent. Indeed, recall, for instance from [13], that this is the case if and only if there exist an element x ∈ A such that Σ_{g∈G}g(x) = 1 and two families of elements {x_{1}, x_{2}, ⋯, x_{r}}, {y_{1}, y_{2}, ⋯, y_{r}} of A such that Σ_{i=1}^{r}x_{i}.y_{i} = 1 and Σ_{i=1}^{r}x_{i}.g(y_{i}) = 0 for all g ≠ 1. Now, since |G| is invertible in A, the element x = |G|^{−1} verifies the first condition. The second follows upon taking x_{i} = y_{i} = e_{i}, where {e_{1}, e_{2}, ⋯, e_{r}} is a complete set of primitive orthogonal idempotents. This shows our assertion. Observe also, that, in this case, A is a finite Galois covering of A[G] (or equivalently A^{G}) with group G (see [6] or [13]). We may now state:

COROLLARY 5.5 Let A be an artin algebra, G be a finite group acting on A such that |G| is invertible in A and G acts freely on the idempotents of A. Then:

(a) A is left (or right) supported if and only if so is A^{G}; (b) A is laura if and only if so is A^{G};

(c) A is left (or right) glued if and only if so is A^{G};

(d) A is weakly shod if and only if so is A^{G};

(e) A is shod if and only if so is A^{G};

(f) A is quasitilted if and only if so is A^{G};

(e) A is tilted if and only if so is A^{G}.

The Auslander-Reiten components of a laura or a supported algebra have been described in [3, 4]. We notice that, if A, R are as above, and Γ is a component of the Auslander-Reiten quiver Γ(mod A) of A (or Γ(mod R) of R), then, in general, the images of the indecomposables of Γ lie in several components of Γ(mod R) (or Γ(mod A), respectively).

LEMMA 5.6 Let A be an artin algebra, Γ be a component of Γ(mod A) and Γ′ be the unique component of Γ(mod R) containing an indecomposable X ∈∣ R ⊗_{A} M, with M ∈ Γ. Then:

(a) Γ is postprojective if and only if so is Γ′;

(b) Γ is preinjective if and only if so is Γ′;

(c) Γ is regular if and only if so is Γ′;

(d) Γ is non-semiregular if and only if so is Γ′;

(e) Γ is semiregular if and only if so is Γ′.

Proof. For the proofs of (a), (b) and (c), we refer to [19, 4.3]. We now prove (d). Suppose that Γ is non-semiregular, then there exist an indecomposable injective A-module I ∈ Γ, an indecomposable projective A-module P ∈ Γ, and a walk of irreducible morphisms between indecomposables

I = L_{0} − L_{1} − ⋯ − L_{s} = M = N_{0} − N_{1} − ⋯ − N_{t} = P.

Applying [19, 4.1] and induction yields a walk of irreducible morphisms between indecomposable R-modules in Γ′

I′ = Y_{0} − Y_{1} − ⋯ − Y_{s} = X = Z_{0} − Z_{1} − ⋯ − Z_{t} = P′

with Y_{i} ∈∣ R ⊗_{A} L_{i} for any i (thus, I′ is injective) and Z_{j} ∈∣ R ⊗_{A} N_{j} for any j (thus, P′ is projective). This shows the sufficiency. The necessity is shown in the same way taking into account that, under the stated hypothesis, M ∈∣ Hom_{R}(R, X). Finally, the proof of (e) is also similar. □

Here, we are interested mainly in the case where A is a strict (that is, not quasi-tilted) laura algebra.

COROLLARY 5.7 Let A be a strict laura algebra and Γ, Γ′ be as in the hypothesis of the lemma. Then Γ is the unique faithful non-semiregular component of Γ(mod A) if and only if Γ′ is the unique faithful non-semiregular component of Γ(mod R).

EXAMPLE 5.8

Let k be a field, and A be the radical square zero k-algebra given by the quiver

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Hence, A is a strict laura algebra. We let the group ℤ/2ℤ act on A, where the only nontrivial element of ℤ/2ℤ fixes the point 3, permutes the points x and x′ (for x ∈ {1, 2, 4, 5}) and the arrows ξ and ξ′ (for ξ ∈ {α, β, γ, λ, σ, μ}). Then, by [19, 2.3], R is the radical square zero k-algebra given by the quiver

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According to our Theorem 1.2, R is also a strict laura algebra.

We now draw the unique non-semiregular faithful component of Γ(mod A).

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(where indecomposable modules are represented by their Loewy series and horizontal dotted lines describe the Auslander-Reiten translation). The unique non-semiregular faithful component of Γ(mod R) is

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Notice that Γ(mod A) has two postprojective (or preinjective) components, while Γ(mod R) has only one.

6 Skew toupie algebras.

Throughout this section, all algebras are finite dimensional over an algebraically closed field, thus are bound quiver algebras.

Let n, l be two positive integers. We define the complete bipartite quiver Q_{l}^{n} to have as its only points n sources a_{1}, ⋯, a_{n} and l sinks b_{1}, ⋯, b_{l} and, for each pair (i, j) with 1 ≤ i ≤ n, 1 ≤ j ≤ l, there is an arrow a_{i} → b_{j} and these are the only arrows of Q_{l}^{n}. A skew toupie quiver Q is defined as follows: it consists of a complete bipartite quiver Q_{l}^{n}, its opposite quiver (Q_{l}^{n})^{op} as well as l disjoint paths w_{1}, ⋯, w_{l} from the sinks of Q_{l}^{n} to the sources of (Q_{l}^{n})^{op}. Thus, a skew toupie is a quiver of the form:

•  _{•}  _{•}^{•}  ^{•}  ^{•}  ^{•}^{•}  ^{•}  ^{•}  ^{•}^{•}  ^{•}  ^{•} a_{1}  a_{2}  a_{n} b_{1}  b_{2}  b_{3}  b_{l} w_{1}  w_{2}  w_{3}  w_{l} b′_{1}  b′_{2}  b′_{3}  b′_{l} a′_{1}  a′_{2}  a′_{n}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  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An algebra R = kQ/I is called a skew toupie algebra if its quiver Q is a skew toupie quiver. Our objective in this section in to exhibit a family of skew toupie algebras which are laura, and even weakly shod.

If, above, Q has exactly one source and one sink (that is, n = 1), then Q is a toupie quiver, and R = kQ/I is a toupie algebra, as defined and studied in [9].

PROPOSITION 6.1 Let R = kQ/I be a skew toupie algebra. Then R is a weakly shod algebra provided:

(1) The ideal I is generated by all possible commutativity relations. In this case, R is tilted.

(2) The ideal I is monomial, and generated by at least one subpath of each of the w_{i}. In this case, R is tilted if and only if each path w_{i} is bound by exactly one relation.

(3) The ideal I is generated by the sums of all paths from each source to each sink, and moreover nl ∈ {2, 3} or the length of each w_{i} does not exceed one. In this case, R is canonical if l = 3, and tilted otherwise.

Proof. We consider the toupie quiver Q′ with m = nl branches

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For each i with 1 ≤ i ≤ m, we set i′ = i + l if i + l ≤ m and i′ = i + l − m if i + l > m. We suppose that, for each i with 1 ≤ i ≤ m, we have p_{i} = p_{i′} and, if i ≤ l, we set p_{i} = ℓ(w_{i}) + 1, where ℓ(w_{i}) denotes the length of the path w_{i}. We finally denote by γ_{i} the path c_{i_{1}} → c_{i_{2}} → ⋯ → c_{i_{p_{i}}}, for 1 ≤ i ≤ m.

We define on Q′ an action of the cyclic group ℤ/nℤ =< σ > as follows: we set σ(c) = c, σ(c′) = c′ and, for each pair (i, j), with 1 ≤ i ≤ m and 1 ≤ j ≤ p_{i}, σ(c_{ij}) = c_{i′j} where i′ is as above. We let σ have the induced action on the arrows. This defines indeed an action on Q′ (and hence on the path algebra kQ′) because of our assumption on the p_{i}.

Let A = kQ′/I′, where I′ is an admissible ideal of one of the following forms:

(1) I′ is generated by all possible commutativity relations.

(2) I′ is a monomial ideal generated by at least one subpath of each of the γ_{i}. Moreover, for each i such that 1 ≤ i ≤ m, the path γ_{i} is isomorphic to γ_{i′}, as full convex subcategories of A (again, i′ is as above).

(3) I′ is generated by the sum of all paths from c to c′ and moreover m ∈ {2, 3} or p_{i} ≤ 2 for all i.

Clearly, the action of ℤ/nℤ on kQ′ leaves invariant the ideal I′. Thus, by [19, 2.1], ℤ/nℤ acts on A.

Now, it follows from the main result of [9] that, in each of these cases, A is a weakly shod algebra. Furthermore, in the case (1) it is always tilted, while, in the case (2), it is tilted if and only if each path γ_{i} is bound by exactly one relation, and, finally, in the case (3), it is canonical if and only if m = 3 and l > 1 (thus l = 3) and tilted in all the other cases.

By [19, 2.3], we get A[ℤ/nℤ]^{b} = R, as given in the statement of our proposition. The assertion now follows from Theorem 1.2. □

6.1 Remarks and examples.

(a) If l = 1, then R has as quiver the following tree

•^{•}_{•}  ^{•}  ^{•}  ^{•}  ^{•}  ^{•}^{•}_{•} b  b′ a_{1} a_{2} a_{n}  a′_{1} a′_{2} a′_{n}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} 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(b) The tubular canonical algebra given by the quiver

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bound by the ideal generated by the sum of all paths from the source to the sink provides an example of a (skew) toupie algebra which is quasi-tilted, but not tilted.

(c) Let A be given by the toupie quiver

•^{•}^{•}  ^{•}  ^{•}  ^{•}  ^{•}  ^{•}^{•}  ^{•}  ^{•}  ^{•}  ^{•}  ^{•}^{•}^{•}  ^{•}^{•}  ^{•}^{•}^{•}^{•}^{•}  ^{•}^{•}^{•}  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c c′ c_{11} c_{12} c_{13} c_{14}  c_{21} c_{22} c_{23} c_{24} c_{25}  c_{31} c_{32} c_{33} c_{34}  c_{41} c_{42} c_{43} c_{44} c_{45}  c_{51} c_{52} c_{53} c_{54}  c_{61} c_{62} c_{63} c_{64} c_{65}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  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bound by the relations denoted by the shown dotted lines. We define an action of ℤ/3ℤ =< σ > as follows: σ(c) = c, σ(c′) = c′, σ(c_{1i}) = c_{3i}, σ(c_{3i}) = c_{5i} for 1 ≤ i ≤ 4, σ(c_{2j}) = c_{4j}, σ(c_{4j}) = c_{6j} for 1 ≤ j ≤ 5. In this case, R is given by the quiver

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bound by the relations denoted by the shown dotted lines. According to 6.1 or, directly, by the main result of [16], R is tilted.

ACKNOWLEDGEMENTS. The first author gratefully acknowledges partial support from the NSERC of Canada. This paper was written while the second author was a CRM-ISM postdoctoral fellow at the universities of Sherbrooke and Bishop’s. This paper was started while the third author was visiting the University of Sherbrooke in Québec. She acknowledges support from the NSERC of Canada, and would like to express her gratitude to Ibrahim and Marcelo for their warm hospitality.

References

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I. Assem; Département de Mathématiques, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1.

E-mail address: ibrahim.assem@usherbrooke.ca

M. Lanzilotta; Centro de Matemática (CMAT), Iguá 4225, Universidad de la República, CP 11400, Montevideo, Uruguay.

E-mail address: marclan@cmat.edu.uy

M. J. Redondo; Instituto de Matemática, Universidad Nacional del Sur, Av. Alem 1253, (8000) Bahía Blanca, Argentina.

E-mail address: mredondo@criba.edu.ar