galois coverings of weakly shod algebras

Communications in Algebra®, 38: 1291–1318, 2010Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870902897939

GALOIS COVERINGS OF WEAKLY SHOD ALGEBRAS

Patrick Le MeurCMLA, ENS Cachan, CNRS, UniverSud, Cachan, France

We investigate the Galois coverings of weakly shod algebras. For a weakly shod algebranot quasi-tilted of canonical type, we establish a correspondence between its Galoiscoverings and the Galois coverings of its connecting component. As a consequence weshow that a weakly shod algebra which is not quasi-tilted of canonical type is simplyconnected if and only if its first Hochschild cohomology group vanishes.

Key Words: Galois covering; Hochschild cohomology; Orbit-graph; Simple connectedness; Weaklyshod algebra.

2000 Mathematics Subject Classification: Primary 16G10; Secondary 16E40, 16E65.

INTRODUCTION

Let A be a finite dimensional k-algebra where k is an algebraically closed field.In order to study the category modA of finite dimensional (right) A-modules weassume that A is basic and connected. The study of modA has risen importantclasses of algebras. For example: The representation-finite algebras, that is, withonly finitely many isomorphism classes of indecomposable modules; the hereditaryalgebras, that is, path algebras kQ of finite quivers Q with no oriented cycles;the tilted algebras of type Q, that is, endomorphism algebras EndkQ�T� of tiltingkQ-modules (see [19]); and the quasi-tilted algebras, that is, endomorphism algebrasEnd��T� of tilting objects T in a hereditary abelian category � (see [18], a quasi-tilted algebra which is not tilted is called of canonical type). For the last threeclasses, each class is a generalisation of the previous one. More recently, a newclass of algebras has arisen (see [2, 27, 31]): That of laura algebras. The algebraA is called laura if there is an upper bound in the number of isomorphism classesof indecomposable modules which can appear in an oriented path of nonzeromorphisms between indecomposable A-modules starting from an injective andending at a projective. It appears that this class contains the four classes citedabove. A laura algebra which not quasi-tilted is characterised by the existence ofa unique nonsemiregular component (that is, containing both a projective and aninjective) in its Auslander–Reiten quiver. It is called the connecting component asa generalisation of the connecting components of tilted algebras. Hence a laura

Received October 7, 2008; Revised February 26, 2009. Communicated by D. Zacharia.Address correspondence to Patrick Le Meur, CMLA, ENS Cachan, CNRS, UniverSud, 61

Avenue du Président Wilson, Cachan F-94235, France; E-mail: [email protected]

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algebra which is not quasi-tilted of canonical type has at least one, and at most two,connecting components (actually, it has two if and only if A is concealed). Recallthat quasi-tilted algebras of finite representation type are tilted [18, Cor. 2.3.6] andthose of infinite representation type are characterised by the existence of a sincereseparating family of semiregular standard tubes [24]. Laura algebras comprise theweakly shod algebras defined by the existence of an upper bound for the length ofa path of nonzero nonisomorphisms from an injective to a projective. Actually alaura algebra which is not quasi-tilted is weakly shod if and only if the connectingcomponent contains no oriented cycles. Weakly shod algebras were introducedin [14] as a generalisation of shod algebras which were defined in [15, 26] as theclass of algebras for which any indecomposable module has injective dimension orprojective dimension at most 1. For example, quasi-tilted algebras are shod andtherefore weakly shod.

On the other hand, the covering techniques [11, 28] have permitted importantprogress in the study of representation-finite algebras (see [9, 12, 17]). Thesetechniques need to consider algebras as k-categories. If � → A is a Galois covering,then modA and mod� are related by the so-called push-down functor F� �mod� → modA. When A has no proper Galois covering by a connected andlocally bounded k-category (or, equivalently, when the fundamental group ofany presentation of A in the sense of [25] is trivial), we say that A is simplyconnected (see [8]). Simply connected algebras are of special interest because of thereduction allowed by the push-down functors. Also they have been object of manyinvestigations (see [8, 10] for instance). For example, Bongartz and Gabriel [11]have classified the simply connected representation-finite standard algebras usinggraded trees. Therefore a nice characterisation of simply connected algebras wouldbe very useful. In [29, Pb. 1], Skowronski asked the following question for a tameand triangular algebra A:

Is A simply connected if and only if HH1�A� = 0? (�)

Up to now, there have been partial answers to � (regardless the tame assumption):For algebras derived equivalent to a hereditary algebra in [22] (and therefore fortilted algebras), for tame quasi-tilted algebras in [3] and for tame weakly shodalgebras in [5]. Therefore, it is natural to try to answer � for laura algebras. Thisshall be done in a forthcoming text [1]. In the present text we study the case ofweakly shod algebras not quasi-tilted of canonical type, which will serve for thestudy made in [1]. For this purpose we prove the following main result.

Theorem A. Let A be connected, weakly shod and not quasi-tilted of canonical type.Let �A be a connecting component of ��modA�. Let G be a group. Then A admits aGalois covering with group G by a connected and locally bounded k-category if andonly if �A admits a Galois covering with group G of translation quivers. In particular,A admits a Galois covering with group �1��A� by a connected and locally boundedk-category.

By [11, 4.2], the fundamental group �1��A� of a connecting component �Ais free and isomorphic to the fundamental group of its orbit-graph. If A isconcealed, then its two connecting components are the unique postprojective and

GALOIS COVERINGS OF WEAKLY SHOD ALGEBRAS 1293

the unique preinjective components, so they have isomorphic fundamental groups.As a consequence of our main result we prove that � has a positive answer forweakly shod algebras.

Corollary B. Let A be connected, weakly shod and not quasi-tilted of canonical type.Let �A be a connecting component of ��modA�. The following conditions are equivalent:

(a) A is simply connected;(b) HH1�A� = 0;(c) The orbit-graph ��A� of �A is a tree;(d) �A is simply connected.

Our proof of Corollary B is independent of the one given in [5] for the tamecase. Actually we make no distinction between the different representation types(finite, tame, or wild). The proof of Theorem A decomposes in two main steps:

1. If F � � → A is a Galois covering with group G, then every module X ∈ �A isisomorphic to F�X for some X ∈ mod�. The modules X, for X in �A, form anAuslander–Reiten component of �. This component is a Galois covering withgroup G of �A.

2. A admits a Galois covering with group �1��A� associated to the universal coverof the orbit-graph ��A�.

As an application of the methods we use, we prove the last main result ofthe text.

Theorem C. Let A′ → A be a Galois covering with finite group G where A′ is a basicand connected finite dimensional k-algebra. Then:

(a) A is tilted if and only if A′ is tilted;(b) A is quasi-tilted if and only if A′ is quasi-tilted;(c) A is weakly shod if and only if A′ is weakly shod.

The text is organised as follows. In Section 1 we fix some notations and recallsome useful definitions. In Section 2 we give some preliminary results: First, weprove some useful facts on covering techniques; second, we compare the Auslander–Reiten quiver of A and the one of B when A = B�M�. Section 3 is the very core ofthe text and is devoted to the first step of Theorem A. In Section 4 we proceed thesecond step. In Section 5, we prove Theorem A and Corollary B. Finally, we proveTheorem C in Section 6.

1. DEFINITIONS AND NOTATIONS

Notations on k-Categories

We refer the reader to [11, 2.1] for notions on k-categories and locally boundedk-categories. All locally bounded k-categories are small and all functors betweenk-categories are k-linear (of course, our module categories will be skeletally small).For a locally bounded k-category �, its objects set is denoted by �o and the space of

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morphisms from an object x to an object y is denoted by ��x y�. If A is a basic finitedimensional k-algebra, it is equivalently a locally bounded k-category as follows.Fix a complete set e1 � � � en� of pairwise orthogonal primitive idempotents. ThenAo = e1 � � � en� and A�ei ej� = ejAei for every i j. In the sequel, A will alwaysdenote a basic finite dimensional k-algebra.

Notations on Modules

Let � be a k-category. Following [11, 2.2], a (right) �-module is a k-linearfunctor M � �op → MOD k where MOD k is the category of k-vector spaces. If�′ is another k-category, a � −�′-bimodule is a k-linear functor � ×�′op →MOD k. We write MOD� for the category of �-modules and mod� for the fullsubcategory of finite dimensional �-modules, that is, those modules M such that∑

x∈�odimk M�x� < �. The standard duality Homk�− k� is denoted by D. We write

ind� for a full subcategory of mod� consisting of exactly one representative ofeach isomorphism class of indecomposable modules. A set � of modules is calledfaithful if

⋂X∈� Ann�X� = 0 where Ann�X� is the annihilator of X, that is, the � −

�-subbimodule of � such that Ann�X��x y� = u ∈ ��x y� �mu = 0 for every m ∈X�y��. If S is a set of finite dimensional �-modules, then add�S� denotes the smallestfull subcategory of mod� containing S and stable under direct sums and directsummands.

Assume that � is locally bounded. We write ��mod�� for the Auslander–Reiten quiver and � = DTr for the Auslander–Reiten translation. Let � be acomponent of ��mod��. Then � is called generalised standard if rad��X Y� = 0 forevery X Y ∈ � (see [30]). Here rad denotes the radical of mod�, that is, the idealgenerated by the nonisomorphisms between indecomposable modules, radn denotesthe nth power of the radical and rad� = ⋂

n≥1 radn. The component � is called

nonsemiregular if it contains both an injective module and a projective module. Werecall the definition of the orbit-graph �� of � in the case � has no periodic module(see [11, 4.2] for the general case). First, fix a polarisation in � , that is, for everyarrow � � x → y in � with y non-projective we fix an arrow �� y → x in sucha way that � induces a bijection from the set of arrows x → y to the set of arrows �y → x (see [11, 1.1]). Then �� is the graph whose vertices are the �-orbits �X�

�

of the vertices X in � and such that there is an edge �X� � − �Y� � for every �-orbitof arrows between a module in �X� � and a module in �Y� � .

We refer the reader to [7, Chap. VIII, IX] for a background on tilting theory.

Weakly Shod Algebras ([14])

Let � be a locally bounded k-category and X Y ∈ ind�. A path X � Y inind� (or in ��mod��) is a sequence of nonzero morphisms (or of irreducible

morphisms, respectively) between indecomposable �-modules X = X0

f1−→ X1 →· · · → Xn−1

fn−→ Xn = Y (with n ≥ 0). We then say that X is a predecessor of Y andthat Y is a successor of X in ind� (or in ��mod��, respectively). Hence X is asuccessor and a predecessor of itself.

The algebra A is called weakly shod if and only if the length of paths inindA from an injective to a projective is bounded. We write �f

A for the set


of indecomposable projectives which are successors of indecomposable injectives.When A is weakly shod this set is partially ordered [5, 4.3] by the relation: P ≤ Q ifand only if P is a predecessor of Q in indA. We need the following properties whenA is weakly shod and connected:

(a) If �fA = ∅, then A is quasi-tilted [18, Thm. II.1.14];

(b) If �fA = ∅, then ��modA� has a unique nonsemiregular component [14, 1.6, 5.4].

This component is generalised standard, faithful, has no oriented cycle andcontains all the modules lying on a path in indA form an injective to aprojective. In particular, every module lying on it is a brick [7, IV.1.4]. Thiscomponent is called the connecting component of ��modA� (or of A).

Assume that A is connected, weakly shod and that �fA = ∅. Let Pm ∈ �f

A be maximaland e the idempotent such that Pm = eA. Then A = B�M� where M = rad�Pm� andB = �1− e�A�1− e�. Moreover:

(a) Any component B′ of B is weakly shod. It is moreover tilted if �fB′ = ∅ [14, 4.8];

(b) Let M ′ ∈ indB be a summand of M and B′ the component of B such thatM ′ ∈ indB′. Then B′ is weakly shod and not quasi-tilted of canonical type andM ′ lies on a connecting component of ��modB′� [5, 5.3].

Recall [27, Thm. 3.1] that if a connected algebra A admits a nonsemiregularcomponent which is faithful, generalised standard and has no oriented cycle, then Ais weakly shod.

Galois Coverings of Translation Quivers ([11, 28])

Let � and � ′ be translation quivers and assume that � is connected. A coveringof translation quivers p � � ′ → � is a morphism of quivers such that: (a) p is acovering of unoriented graphs; (b) p�x� is projective (or injective, respectively) if andonly if so is x; (c) p commutes with the translations. It is called a Galois coveringwith group G if, moreover, the group G acts on � ′ in such a way that: (d) G actsfreely on vertices; (e) p g = p for every g ∈ G; (f) the translation quiver morphism� ′/G → � induced by p is an isomorphism; (g) � ′ is connected. Given a connectedtranslation quiver � , there exists a group �1�� (called the fundamental group of�) and a Galois covering � → � with group �1�� called the universal cover of � ,which factors through any covering � ′ → � . If p � � ′ → � is a covering (or a Galoiscovering with group G), then it naturally induces a covering (or a Galois coveringwith group G, respectively) �� ′� → �� between the associated orbit-graphs. It isproved in [11, 4.2] that if � has only finitely many -orbits and if p � � ′ → � is theuniversal cover of translation quiver, then �� ′� → �� is the universal cover ofgraphs, that is, �1�� is isomorphic to �1�� (and therefore is free).

Group Actions on Module Categories ([17])

Let G be a group. A G-category is a k-category � together with a groupmorphism G → Aut��. This defines an action of G on MOD�: If M ∈ MOD�and g ∈ G, then gM = M g−1. We write GM �= g ∈ G � gM � M� for the stabiliserof M . We say that G acts freely on � if the induced action on �o is free. Assume

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that � is locally bounded. Then this G-action preserves Auslander–Reiten sequencesand commutes with �. Also it induces an action on ��mod�� and on �� for anyG-stable component � of ��mod��.

Galois Coverings of Categories ([17])

Let G be a group and F � � → � a functor between k-categories. We setAut�F� = g ∈ Aut�� F g = F�. We say that F is a Galois covering with groupG if there is a group morphism G → Aut�F� such that the induced action of Gon � is free and the induced functor F � �/G → � is an isomorphism. We needthe following characterisation for a functor F � � → � to be a Galois covering [17,Sect. 3]. The group morphism G → Aut�F� is such that F is Galois with group Gif and only if: (a) the fibres F−1�x� (x ∈ �o) are nonempty and G acts on thesefreely and transitively and (b) F is a covering functor in the sense of [11, 3.1], that is,for every x y ∈ �o the two maps

⊕g∈G ��x gy� → ��Fx Fy� and

⊕g∈G ��gy x� →

��Fy Fx� induced by F are isomorphisms. A Galois covering F � � → � with � and� locally bounded and connected is called connected. In such a case, the morphismG → Aut�F� is an isomorphism [20, Prop. 6.1.37]. A connected and locally boundedk-category � is called simply connected if and only if there is no connected Galoiscovering � → � with nontrivial group. This definition is equivalent [21, Cor. 4.5] tothe original one of [8], and it is more convenient for our purposes.

Covering Techniques ([11, 17])

Let F � � → � be a Galois covering between locally bounded k-categories.We write F� � MOD� → MOD� and F• � MOD� → MOD� for the push-downfunctor and the pull-up functor, respectively. Recall [11, 3.2] that F• = X F forevery X ∈ MOD� and that for M ∈ MOD�, the �-module F�M is such thatF�M�x� = ⊕

Fx′=x M�x′� for every x ∈ �o. We list some needed properties on thesefunctors. Both F� and F• are exact; �F� F•� is adjoint; F�M is projective (or injective)if and only if M is projective (or injective, respectively); F��mod�� ⊆ mod�; thefunctor F� is G-invariant, that is, F� g = F� for every g ∈ G; for every M ∈ mod�we have F•F�M � ⊕

g∈GgM [17, 3.2]; and F� commutes with the duality, that is,

D F� � Fop� D on mod�.

Finally, it satisfies a property which will be refered to as the coveringproperty of F�: For MN ∈ mod�, the two maps

⊕g∈G Hom��

gMN� →Hom��F�M F�N� and

⊕g∈G Hom��M gN� → Hom��F�M F�N� induced by F� are

k-linear isomorphisms. A module X ∈ ind� is called of the first kind (with respectto F ) if and only if there exists X ∈ mod� (necessarily indecomposable) such thatF�X � X in modB. Note that if X exists, then X = F�X for some X ∈ ind�; and, ifX X ∈ ind� are such that F�X � F�X � X, then X � gX for some g ∈ G (see [17,3.5]).

2. PRELIMINARIES

Some Results on Covering Techniques

Let F � � → A be a Galois covering with group G where � is locally bounded.We prove some useful comparisons between of ��modA� and ��mod��. First,


we give a necessary condition on a morphism in mod� to be mapped by F� to asection or a to retraction.

Lemma 2.1. Let X Y ∈ mod� and f ∈ Hom��X Y�.

(a) F��f� is a section (or a retraction) if and only if so is f ;(b) If F��f� is irreducible, then so is f ;(c) Let u � E → X (or v � X → E) be a right (or left) minimal almost split morphism

in mod�. Assume that GX = 1. Then so is F��u� (or F��v�, respectively);(d) F� �X � AF�X.

Proof. (a) Obviously, if f is a section (or a retraction), then so is F��f�. Assumethat F��f� is a section. So IdF�X

= r F��f� with r ∈ HomA�F�X F�Y�. Moreover,r = ∑

g F��rg� with �rg�g∈G ∈ ⊕g∈G Hom��Y

gX�, using the covering propery of F�.Therefore, IdF�X

= ∑g F��rg f�. The covering property of F� then implies that

IdX = r1 f , that is, f is a section. Dually, if F��f� is a retraction, then so is f .

(b) is a direct consequence of (a).

(c) is due to the proof of [17, 3.6, (b)].

(d) follows from the fact that F� is exact, maps projective modules to projectivemodules (in particular, F� maps a minimal projective resolution in mod� to aminimal projective resolution in modA) and commutes with the duality. �

Lemma 2.2. Let � be a component of ��modA� made of modules of the first kindand � the full subquiver of ��mod�� generated by X ∈ ��mod�� F�X ∈ ��. Then:

(a) Let u � M → P be a right minimal almost split morphism in mod� with Pindecomposable projective. Then F��u� is right minimal almost split.

(b) Let X ∈ � be non projective. Then F� transforms any almost split sequence endingat X into an almost split sequence ending at F�X.

(c) Let u ∈ Hom��X Y� with X Y ∈ � . Then u is irreducible if and only if so is F��u�.(d) � is stable under predecessors and under successors in ��mod�� and under the

action of G.

Proof. (a) follows from [11, 3.2].

(b) Fix an almost split sequence 0 → �X�−→E

�−→X → 0 in mod�.

By 2.1(d), we have an exact sequence 0 → AF�XF��−−→ F�E

F��−−→ F�X → 0 in modA.By 2.1(a), it does not split. Moreover, F�X is indecomposable and non-projective.Let v � Z → F�X be right minimal almost split. We only need to prove that v factorsthrough F��. Write v � Z → F�X as v = �v1 · · · vn� � Z1 ⊕ · · · ⊕ Zn → F�X whereZ1 � � � Zn ∈ indA. We prove that each vi factors through F��. We have Zi ∈ �because vi is irreducible. Therefore, Zi = F�Zi for some Zi ∈ mod� indecomposable.So vi =

∑g F��vig� where �vig�g∈G ∈ ⊕

g∈G Hom��gZi X�. Note that gZi � X for

every g ∈ G because Zi � F�X. Thus vig = � wig for some wig ∈ Hom��gZi E� for

every g. We may assume that wig = 0 if vig = 0. Then vi = F�� ∑

g∈G F��wig��where

∑g∈G F��wig� ∈ HomA�Zi F�X� for every i. Thus v1 � � � vn factor through

F��. Therefore, so does v. This proves (b).

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(c) is a direct consequence of (a), (b) and 2.1.

(d) Clearly, � is stable under the action of G. We prove the stability underpredecessors (the proof for successors is dual). Let u ∈ Hom��X Y� be irreduciblewith X ∈ ind� and Y ∈ � . We claim that F�X ∈ add��. If Y is projective, thenX is a direct summand of rad�Y� and u � X → Y is the inclusion. So F�Y isindecomposable projective, F�X is a direct summand of F��rad�Y�� = rad�F�Y� [11,3.2] and F��u� � F�X → F�Y is injective. Since F�Y ∈ � we have rad�F�Y� ∈ add��and therefore F�X ∈ add��. Assume that Y is not projective. So there is an almostsplit sequence in mod�

0 → �Y → E ⊕ X

[?u

]−−→ Y → 0�

By (a), there is an almost split sequence in modA

0 → AF�Y → F�E ⊕ F�X

[?

F�u

]−−−→ F�Y → 0�

Since F�Y ∈ � , we have F�X ∈ add��. This proves the claim. Now we prove thatF�X is indecomposable. Since F�X ∈ add��, there exist E1 � � � En ∈ � and anisomorphism � � F�X

∼→ E1 ⊕ · · · ⊕ En. From the covering property of F�, we have� = ∑

g∈G F��g� where ��g�g∈G ∈ ⊕g∈G Hom��

gX E1 ⊕ · · · ⊕ En�. Since � is anisomorphism, there exists g ∈ G such that F��g� ∈ rad�F�X F�E1 ⊕ · · · ⊕ F�En�. Sothere exists i such that the restriction F�

gX → F�Ei of F��g� is an isomorphism sothat gX � Ei ∈ � . �

The following proposition describes the action of F� on almost split sequencesin mod� under suitable conditions. Note that if we assume that G acts freely onindecomposable �-modules (that is, GX = 1 for any X ∈ ind�), then the last threepoints follow at once from [17, 3.6].

Proposition 2.3. Keep the hypotheses and notations of 2.2.

(a) � is faithful if and only if � is.(b) � is generalised standard if and only if rad��X Y� = 0 for every X Y ∈ � .(c) � is a (disjoint) union of components of ��mod��. In particular, � is a translation

subquiver of ��mod��.(d) The map X �→ F�X extends to a covering of translation quivers � → � . If �

is connected and GX = 1 for every X ∈ � , then this is a Galois covering withgroup G.

(e) � has an oriented cycle if and only if � has a nontrivial path of the form X � gX.

Proof. (a) Assume that � is faithful. Let u ∈ Ann��x y�, that is, u ∈ ��x y�

and mu = 0 for every m ∈ X�y�, X ∈ � . We claim that F�u� ∈ Ann��Fx Fy�.Let X ∈ � and m ∈ X�Fy�. We may assume that X = F�X with X ∈ � . So m =�mg�g∈G ∈ ⊕

g∈G X�gy� and, therefore, mF�u� = �mgg�u��g∈G. On the other hand,g�u� ∈ Ann��gx gy� because � is G-stable. So mgg�u� = 0 for every g ∈ G so thatmF�u� = 0. Thus F�u� ∈ Ann��Fx Fy� = 0 and, therefore, u = 0. So � is faithful.


Conversely, assume that � is faithful, and let u ∈ Ann��Fx Fy�. So u =∑g F�ug� where �ug�g∈G ∈ ⊕

g∈G ��gx y�. We claim that ug ∈ Ann��gx y� forevery g ∈ G. Indeed, let X ∈ � and m ∈ X�y�. Then m ∈ F�X�Fy� and 0 = mu =�mug�g∈G ∈ ⊕

g∈G F�X�gx�. So mug = 0 for every g. Thus ug ∈ Ann��gx y� forevery g ∈ G and, therefore, u = 0 because � is faithful. So � is faithful.

(b) Assume that rad��X Y� = 0 for every X Y ∈ � . Let X Y ∈ � . Weprove that rad��F�X F�Y� = 0. Since HomA�F�X F�Y� is finite dimensional andisomorphic to

⊕g∈G Hom��X

gY�, there exists n ≥ 1 such that radn�X gY� = 0for every g ∈ G. Let f ∈ radl�F�X F�Y� with l ≥ 1. Let �u1 � � � ut�

t � X → E1 ⊕· · · ⊕ Et be left minimal almost split in mod�. By 2.2(a) and (b), there existfi ∈ HomA�F�Ei F�Y� for every i, such that f = ∑

i fi F��ui�. More generally,an induction on l shows that there exist morphisms �1 � X → X1 � � � �s � X →Xs in mod� all lying in radI and there exist hi ∈ HomA�F�Xi F�Y� for every i,such that f = ∑

i hi F��i�. On the other hand, hi =∑

g F��hig� with �hig�g∈G ∈⊕g∈G Hom��Xi

gY� by the covering property of F�. Therefore,

f = ∑g

F�

(∑i

hig �i)

where∑

i hig �i ∈ radl�X gY� for every g. In the particular case where l = n,we have f = 0. Thus radn�F�X F�Y� = 0. This proves that � is generalised standard.

Conversely, assume that � is generalised standard. Let f ∈ radl�X Y� withX Y ∈ � and l ≥ 1. The arguments used above show that there exist morphisms�1 � X → X1 � � � �s � X → Xs in mod� all lying in radI and there exist morphismsh1 � X1 → Y � � � hs � Xs → Y such that f = ∑

i hi �i. By 2.2(c), we therefore haveF��f� ∈ radl�F�X F�Y�. Hence F��rad

��X Y�� ⊆ rad��F�X F�Y� = 0. Since F� isfaithful, we have rad��X Y� = 0 for every X Y ∈ � .

(c) This is a direct consequence of 2.2(d).

(d) By assumption and 2.2, F� preserves indecomposability, irreducibility,and almost split sequences in � . Consequently, for each X ∈ � there is a bijectionbetween the set of arrows in � which leave (or arrive at) X and the set of arrows in� which leave (or arrive at, respectively) F�X. Whence the covering � → � extendingthe map X �→ F�X. The rest of the assertion is a consequence of the argumentspresented in the proof of [17, 3.6].

(e) follows from (d). �

Remark 2.4. Assume, in 2.2, that � is connected and GX = 1 for every X ∈ � . By2.3(d), there is a Galois covering with group G of graphs p � �� → �� suchthat p ��X� � � = �F�X�

A for every vertex X ∈ � . The G-action on �� is given byg ��X� � � = �gX� � for every g ∈ G, X ∈ � . In particular, if g � �� → �� is anautomorphism of graphs such that p g = p, then there exists g′ ∈ G such that g isinduced by g′.

Remark 2.5. In view of the proof of 2.3(a), if X ∈ mod� is faithful, then so isF�X. However, one can easily find examples where F�X is faithful and X is not.

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Comparisons between the Auslander–Rieten Quiversof A and B When A = B�M�

In this paragraph we assume that A is connected and weakly shod and �fA = ∅.

Let Pm ∈ �fA be maximal and A = B�M� the associated one-point extension. We give

a useful relationship between the connecting component �A of ��modA� and theconnecting components associated to the connected components of B. It followsfrom the work made in [14] (see also [5, Lem. 4.1] who treated the case where theextension point is separating). For convenience, we give the details below. Note thefollowing:

(a) Any strict predecessor of Pm in indA is a B-module;(b) If P ∈ indB is projective, then any predecessor of P in indA is a B-module.

We begin with the following lemma.

Lemma 2.6. Let � be the full subcategory of indA generated by

X ∈ indA �X � Pm and X is a predecessor in indA of an indecomposable

projective A-module��

Then:

(a) � is made of B-modules;(b) � is stable under predecessors in indA and contains no successor of Pm in indA;(c) A and B coincide on �;(d) The full subquivers of ��modA� and ��modB� generated by � coincide.

Proof. (a) and (b) follow from the definition of � . For L ∈ modA let L be theB-module obtained by restriction of scalars, that is, L = L · �1− e� if e ∈ A is theidempotent such that Pm = eA. Assume that 0 → AX

u−→Ev−→X → 0 is an almost

split sequence in modA with X ∈ indB. Then it is easily verified that AX = BX and

0 → BXu−→E

v−→X → 0 is almost split in modB. Also, if X is not a successor ofPm, then the two exact sequences coincide. Then (c) and (d) follow from these facts.

�

The category � of the preceding lemma serves to compare connectingcomponents as follows.

Lemma 2.7. Let � be as in the preceding lemma, M ′ ∈ indB a direct summand of Mand B′ the component of B such that M ′ ∈ indB′. If � ′ is the component of ��modB′�containing M ′, then:

(a) The connecting component �A of ��modA� contains every module lying on both � ′

and �;(b) The full subquivers of �A and � ′ generated by the modules lying on both � ′ and �

coincide;(c) Every B′ -orbit of � ′ contains a module lying on � .


Proof. (a) Let X lie on both � ′ and � . By [5, 1.1], mB′X is a predecessor in��modB′� (and therefore in ��modA�, by 2.6) of M ′ or of a projective P ∈ indB′

for some m ≥ 0. By [14, Lem. 5.3], P ∈ �A. So mB′X ∈ �A. On the other hand, 2.6(c),implies that mB′X = mAX. So X ∈ �A.

(b) Let �1 and �2 be the full subquivers of �A and � ′, respectively, generatedby the modules lying on both � and � ′. By (a), �1 and �2 have the same vertices.Then 2.6(d), implies that �1 = �2.

(c) is obtained using similar arguments as those used to prove (a). �

Remark 2.8. Using 2.7 we get the following description of the orbit-graph ��A�.For simplicity, we write ��A�\�Pm�

A� for the full subgraph of ��A� generated bythe vertices different from �Pm�

A .

(a) Let B′ be a component of B and �B′ the (unique) connecting componentof B′ containing a direct summand of M . Then ��B′� is a component of��A�\�Pm�

A�, and all the components of ��A�\�Pm� A� have this form.

(b) If X is an indecomposable direct summand of M with multiplicity d, then �X� B

lies on exactly one of the connected components of ��A�\�Pm�� A and ��A�

contains exactly d edges �X� A − �Pm� A . Moreover, all the arrows connected to

�Pm� A have this form.

3. COMPONENTS OF THE FIRST KIND FOR WEAKLY SHOD ALGEBRAS

Let A be weakly shod. We examine when a component of ��modA� is madeof modules of the first kind with respect to any Galois covering of A. We studytwo cases: When the component is connecting and when it is semiregular and notregular.

Connecting Components of the First Kind

The aim of this paragraph is to prove the following proposition.

Proposition 3.1. Let A be connected, weakly shod and not quasi-tilted of canonicaltype, �A a connecting component of A, and F � � → A a connected Galois covering withgroup G. Then �A is made of modules of the first kind. Moreover, the full subquiver�� of ��mod�� generated by the modules X ∈ ind� such that F�X ∈ �A is a G-stablefaithful and generalised standard component of ��mod�� with no nontrivial path ofthe form X � gX. Finally, the map X �→ F�X on the vertices of �� extends to a Galoiscovering of translation quivers �� → �A with group G.

In order to prove this result, we proceed along the following steps:

(a) Any X ∈ �A satisfies X � F�X for some X ∈ ind� such that GX = 1;(b) rad��X Y� = 0 for every X Y ∈ ��;(c) �� is a component of ��mod��.

We prove each step in a separate lemma.

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Lemma 3.2. Let A be connected, weakly shod and not quasi-tilted of canonical type,�A a connecting component of A, and F � � → A a Galois covering with group G where� is locally bounded. Then for every X ∈ �A there exists X ∈ ind� such that F�X � Xand GX = 1.

Proof. Note that if Y = mAX for some m ∈ , then the conclusion holds true for Xif and only if it holds true for Y . We prove the lemma by induction on rk�K0�A��and begin with the case where A is tilted. If A is tilted then �A has a completeslice T1 � � � Tn�. By [22, Cor. 4.5, Prop. 4.6] and the above remark, the lemmaholds true for A. Now assume that A is not tilted and that the lemma holds truefor algebras whose rank of the Grothendieck group is smaller than rk�K0�A��. So�f

A = ∅. Let Pm ∈ �fA be maximal and A = B�M� the associated one-point extension.

Recall [20, Prop. 6.1.40, Prop. 6.1.41] that for any component B′ of B the Galoiscovering F � � → A restricts to a Galois covering F−1�B′� → B′ with group G. Theconclusion of the lemma clearly holds true for X = Pm. Let B′ be a componentof B and X lie in a connecting component of B′. By the induction hypothesis, wehave X � F ′

�X where X ∈ ind F−1�B′� is such that GX = 1 and F ′ � F−1�B′� → B′

is the restriction of F . In particular X � F�X. By the above remark and 2.8, theproposition therefore holds true for A. �

Lemma 3.3. Keep the notations and hypotheses of 3.2. Let �� be the full subquiver of��mod�� generated by the modules X ∈ ind� such that F�X ∈ �A. Then:

(a) �� is a (disjoint) union of components of ��mod��;(b) �� is faithful, has no nontrivial path of the form X � gX and rad��X Y� = 0 for

every X Y ∈ ��.

Proof. This follows from 2.3 and the fact that �A is faithful, generalised standard,and has no oriented cycle. �

Lemma 3.4. Keep the notations and hypotheses of 3.1. Then �� is a component of��mod��.

Proof. Following [18], we define the left part A of modA as the full subcategoryof indA generated by:

M ∈ indA � pdA L ≤ 1 for every predecessor L of M in indA�

where pdA is the projective dimension. Let T be the direct sum of theindecomposable A-modules which are either Ext-injective in A or not in A andprojective. Then T is a basic tilting A-module [3, 4.2,4.4] and for every X ∈ �Athere exists m ∈ such that mAX is a direct summand of T . Fix an indecomposabledecomposition T = T1 ⊕ · · · ⊕ Tn in modA. By 3.2, there exist T1 � � � Tn ∈ �� suchthat F�Ti � Ti and GTi

= 1, for every i. Let � be the full subcategory of ind�generated by gTi � i ∈ 1 � � � n� and g ∈ G�. So � and � have equivalent derivedcategories (see the proof of [22, Lem. 4.8]). In particular � is connected. So,by 3.3(b), there is a component � of �� which contains gTi�ig. We claim that� = ��. If X ∈ ��, then F�X ∈ �A so that mAF�X � Ti for some i ∈ 1 � � � n� and


m ∈ . Consequently, m�X � gTi for some g and therefore X ∈ � . Thus �� = �is connected. �

Now we prove 3.1.

Proof of 3.1. The proposition is a direct consequence of 3.2–3.4. �

Remark 3.5. Assume that �fA = ∅ and A admits two connecting components:

Its unique postprojective component and its unique preinjective one. With thehypotheses and notations of 3.1, assume that �A is the postprojective component(or the preinjective component) of A. Then it is not difficult to check that ��is the unique postprojective component (or the unique preinjective component,respectively) of ��mod��.

SEMIREGULAR COMPONENTS OF THE FIRST KIND

Now we treat the case of semiregular components containing a projective oran injective. Most of the work in this paragraph is based on the following lemmawhich does not assume A to be weakly shod.

Lemma 3.6. Let F � � → A be a Galois covering with group G where � is locallybounded. Let � be a component of ��modA� such that:

(a) � has no multiple arrows, and every vertex in � is the source of at most two arrowsand the target of at most two arrows;

(b) There exists M0 ∈ � which is either the source of exactly one arrow or the target ofexactly one arrow, and which is isomorphic to F�M0 where M0 ∈ ind� is such thatGM0

= 1.

Then every X ∈ � is isomorphic to F�X for some X ∈ ind� such that GX = 1.

Proof. Let � be the set of those modules X ∈ � for which the conclusion of thelemma holds. Therefore, � contains M0 and � is stable under A and −1

A becauseof 2.1(d). Assume by absurd that � � � . Then by considering an unoriented pathin � starting from a module X ∈ �\� , ending at M0 and of minimal length, wehave the following (or its dual treated dually): There exists an irreducible morphismu � Y → X with X ∈ � , Y ∈ �\� and such that if E → X is right minimal almostsplit, then either E = Y , or E = Y ⊕ Y ′ for some Y ′ ∈ � . We prove that Y � F�Y forsome Y ∈ ind�. For this purpose, we distinguish two cases according to whether Eis indecomposable or not. We fix X ∈ ind� such that F�X � X and GX = 1. Assumefirst that E = Y is indecomposable. Let u � Y → X be a right minimal almost splitmorphism in mod�. Thus 2.1(c), implies that so is F��u� � F�Y → F�X. Therefore,F�Y � Y . Now assume that E = Y ⊕ Y ′ with Y ′ ∈ � . In particular, Y ′ � F�Y

′ forsome Y ′ ∈ ind�. We thus have a right minimal almost split morphism �u u′� � Y ⊕Y ′ → X in modA. Let f � E → X be a right minimal almost split morphism inmod�. As above, we deduce that so is F��f� � F�E → F�X in modA. Therefore,F�E � Y ⊕ F�Y

′. Applying F• yields⊕

g∈GgE � F•Y ⊕⊕

g∈GgY ′. Since Y ∈ ind�,

we deduce that gE = Y ′ ⊕ Y for some g ∈ G and some Y ∈ mod�. Consequently

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F�E � F�Y′ ⊕ F�Y and finally Y � F�Y . Hence, in any case, we have Y � F�Y and

an irreducible morphism Y → X for some Y ∈ ind�. Since Y ∈ � , there exists g ∈G\1� such that gY � Y . Therefore, the morphism Y → X defines two irreduciblemorphisms � � Y → X and � � Y → gX. Since GX = 1, and by 2.1(c), both F�� F�Y → F�X and F�� F�Y → F�

gX = F�X are irreducible. On the other hand, �has no multiple arrows so there exists an isomorphism � � F�X

∼→F�X such thatF�� = � F��. By the covering property of F� we have � = ∑

h∈G F��h� with��h�h ∈

⊕h∈G Hom��Y

hX�. So F�� =∑

h∈G F��h ��, and therefore, � = �g �because of the covering property of F�. This implies that �g � X → gX is a retractionand therefore an isomorphism. We get a contradiction because GX = 1. �

We apply this lemma to our situation where A is weakly shod and not quasi-tilted of canonical type.

Proposition 3.7. Let A be connected, weakly shod and not quasi-tilted of canonicaltype, F � � → A a Galois covering with group G where � is locally bounded and � asemiregular component of ��modA� containing a projective or an injective. Then forevery X ∈ � there exists X ∈ ind� such that F�X � X and GX = 1.

Proof. It follows from [14, 6.2] that at least one of the following cases is satisfied:

(a) � is a postprojective or a preinjective component;(b) � is obtained from a tube or from A� by ray or coray insertions.

In case (a), the proposition follows from: 2.1(d); the fact that the G-action on mod�commutes with �; and, the fact that the conclusion of the proposition holds true forindecomposable projective or injective modules. In case (b), there exists a projectiveor an injective M0 ∈ � such that � and M0 satisfy the conditions of 3.6. Whence theproposition. �

Remark 3.8. Keep the notations and hypotheses of the 3.7. Let � be the fullsubquiver of ��mod�� generated by the vertices X ∈ ind� such that F�X ∈ � .Then � is a union of semiregular components and contains a projective or aninjective.

The following example shows that 3.7 does not necessarily hold for regularcomponents, even for tilted algebras.

Example 3.9. Let A be the path algebra of the Kronecker quiver 1a⇒b2. It admits

a Galois covering F � A′ → A with group /2 = �� where A′ is the path algebraof the following quiver of type �3:


with F�x� = F��x� = x for every x ∈ 1 2 a b�. Then the indecomposable A-module

kId⇒idk lying on a homogeneous tube is not of the first kind with respect to F and, in

general, with respect to any nontrivial connected Galois covering of A.

4. THE GALOIS COVERING OF A ASSOCIATED TO THE UNIVERSALCOVER OF THE CONNECTING COMPONENT

Let A be weakly shod and not quasi-tilted of canonical type and �A aconnecting component. Recall that given a connected Galois covering F � A′ → Awith group G there is a component �A′ of ��modA′� and a Galois covering of graphs��A′� → ��A� with group G (see 3.1 and 2.4). This Galois covering of graphsis called associated to F . In this section, we prove the following result which is acounter-part of the work made in 3.1.

Proposition 4.1. Let A be connected, weakly shod and not quasi-tilted of canonicaltype, and �A a connecting component. Then there exists a connected Galois coveringF � A → A with group the fundamental group �1��A� such that the associated Galoiscovering of graphs ��A� → ��A� is the universal cover.

Remark 4.2. Recall that if A has more than one connecting component, then ithas two of them: The unique preinjective component and the unique postprojectivecomponent. In particular the isomorphism class of �1��A� does not depend on theconnecting component.

Until the end of the section we adopt the hypotheses and notations of theabove proposition. Here is the strategy of its proof. We use an induction onrk�K0�A��. If A is tilted of type Q, then ��A� is the underlying graph of Q. So 4.1follows from [22, Thm. 1] in that case. If A is not tilted, there exists Pm ∈ �f

A

maximal and defining the one-point extension A = B�M�. Then we use 2.8 and theGalois covering of B given by the inductive step to construct the desired Galoiscovering of A.

From now on we assume that A is not tilted, Pm ∈ �fA is maximal and

A=B�M� is the associated one-point extension. The extending object is denoted byx0 ∈ Ao. Also we assume that 4.1 holds true for the components B1 � � � Bt of B (B =B1 × · · · × Bt). Thus for every i ∈ 1 � � � t� there is a connected Galois coveringF�i� � Bi → Bi with group �1��i� equal to the fundamental group of the (unique)connecting component �i of Bi containing a direct summand of M . We write �i → �ifor the universal cover of translation quivers. The construction of a connectedGalois covering F � � → A with group �1��A� is decomposed into the followingsteps:

(a) A reminder on the universal cover of ��A�;(b) The construction of a Galois covering F � B → B with group �1��A� using

F�1� � � � F �t�;(c) The construction of the locally bounded k-category A and the Galois covering

F � A → A;(d) The proof that A is connected.

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Reminder: The Universal Cover ��A�

For simplicity, we still denote by x0 the vertex �Pm� A of ��A� and use it as

the base-point for the computation of the universal cover of ��A�. Recall that theuniversal cover p � � → ��A� is such that:

(a) � is the graph with vertices the homotopy classes �� of paths � � x0 � x in ��A�(where x is any vertex) and such that for every edge � � x − y in ��A� and everyvertex �� in � with end-point x, there is an edge � � ��− �� in �;

(b) With the notations of (a), p maps the vertex �� to x and the edge � � ��− ��to � � x − y.

The Galois Covering of F � B → B with Group �1��A�

We construct a Galois covering F � B → B with group �1��A� usingF�1� � � � F �t�. We define B as a disjoint union

∐ti=1

∐? Bi of (infinitely many)

copies of Bi (i ∈ 1 � � � t�). More precisely, let i ∈ 1 � � � t�. Every component �of p−1��i�� is simply connected so the restriction � → ��i� of p fits into acommutative diagram of graphs

where the horizontal arrow is an isomorphism and the oblique arrow on the right isinduced by �i → �i. We then attach to B one copy of Bi for each component � ofp−1��i��. The Galois coverings F�1� � � � F �t� then clearly define a functor F � B →B such that F and F�i� coincide on each copy of Bi.

Now we endow B with a �1��A�-action such that F g = g for every g ∈�1��A�. Let g ∈ �1��A� and Bi be a copy of Bi in B. We define the action of g onBi. Let � be the component of p−1��i�� associated to Bi. Then g�� is also acomponent of p−1��i�� to which corresponds a copy Bi of Bi in B. Moreover,the graph morphism g � � → g�� and the diagrams (D�) and (Dg��) determine an

automorphism ��i�∼→��i� making the following diagram commute:

Therefore, the automorphism ��i�∼→��i� extends the map �X�

Bi �→ �gX� Bi

associated to some g ∈ �1��i� (see 2.4). The action of g on Bi is therefore definedas follows: g maps the component Bi of B to the component Bi and, as a functor,it acts like g � Bi = Bi

∼→ Bi = Bi. This way, we get a �1��A�-action on B such thatF g = F for every g ∈ G.

Lemma 4.3. The �1��A�-action on B is free, B is locally bounded and F � B → B isa Galois covering with group �1��A�.


Proof. Let x ∈ Bo and g ∈ �1��A� be such that gx = x. We write Bi for the copyof Bi in B containing x and � for the corresponding component of p−1��i��. Inparticular, g�� = �, and there exists g′ ∈ �1��i� such that the action of g on Bi isgiven by g′ � Bi = Bi

∼→ Bi = Bi. Since gx = x, this means that g′x = x. So g′ = IdBi

and g is the identity map on �. Thus, g is the identity on the universal cover �, andtherefore, on B. This proves that the �1��A�-action on B is free.

By construction, B is locally bounded.Now we prove that �1��A� acts transitively on F−1�x� for every x ∈ Bo. Let

x y ∈ Bo be such that Fx = Fy. By construction of F , there exists i such that xand y lie on copies Bi and Bi of Bi in B, respectively. We write � and for thecomponents of p−1��i�� corresponding to Bi and Bi, respectively. So there existsg ∈ �1��A� such that g�� = . Therefore, gx lies on Bi and F�gx� = Fy. So we mayassume that Bi = Bi. Using (��), we identify the map � → ��i� induced by p withthe universal cover ��i� → ��i�. Since F coincides with F�i� � Bi → Bi on Bi, thereexists g′ ∈ �1��i� such that g′�x� = y. Moreover, there exists g′′ ∈ �1��A� such thatg′′ and g′ coincide on some vertex of � (because p � � → ��A� is a Galois coveringwith group �1��A�) and therefore on � (because � → ��i� is a Galois covering).We thus have g′′x = y with g′′ ∈ �1��A�. This shows the transitivity of �1��A� on thefibres of F � Bo → Bo.

Therefore, F � B → B is, by construction, a covering functor, �1��A� is agroup acting freely on B such that F g = g for every g ∈ �1��A� and �1��A� actstransitively on the fibres of F � Bo → Bo. So F is a Galois covering with group�1��A�. �

The Galois Covering F � A → A with Group �1��A�

Now we extend F � B → B to a Galois covering F � A → A with group �1��A�.Recall that A = B�M�. Accordingly, let A be the category:

A =[S M

0 B

] (�)

where S is the category with objects set So = �1��A�× x0� and no nonzeromorphism except the scalar multiples of the identity morphisms and M is anS − B-bimodule defined as follows. Fix an indecomposable decomposition M =⊕t

i=1

⊕nij=1 Mij such that Mij ∈ indBi for every i j. Let i j be such indices. Then

the homotopy class of the edge x0 − �Mij� A associated to the inclusion morphism

Mij ↪→ Pm is a vertex in � (see 2.8). Also it lies on some component � of p−1��i��to which corresponds a copy Bi of Bi in B. By 3.2, there exists Mij ∈ ind Bi suchthat F�i�

� Mij = Mij . We thus consider Mij as an indecomposable B-module such thatMij ∈ ind Bi. In particular, we have F�Mij = Mij . The S − B-bimodule M is thendefined as follows:

M � S × Bop → mod k

��g x0� x� �→t⊕

i=1

ni⊕j=1

gMij�x��

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The k-category A is thus completely defined. Now we extend the �1��A�-action onB to an action on A. We let �1��A� act on �1��A�× x0� in the obvious way. Letg ∈ �1��A� and u ∈ Mij�h

−1x� ⊆ M��h x0� x�. We define g · u to be the morphism u

viewed as an element of Mij�h−1x� ⊆ M�g · �h x0� g · x�.

Lemma 4.4. The above construction defines a locally bounded k-category A endowedwith a free �1��A�-action.

Proof. We clearly have defined a k-category and the �1��A�-action is well-definedand free because �1��A� acts freely on �1��A�× x0� and on B. We prove that A islocally bounded. Recall that B is locally bounded. Moreover, for every g ∈ �1��A� wehave

⊕x∈Bo

A��g x0� x� =⊕

x∈BoijMij�g

−1x� = ⊕x∈Bo

M�x� because F�Mij = Mij

for every i j. Thus⊕

x∈BoA��g x0� x� is finite dimensional for every g ∈ �1��A�.

Finally, for every x ∈ Bo we have⊕

g∈�1��A� A��g x0� x� =⊕

g∈�1��A�ij Mij�g−1x� =

M�F�x��. So⊕

g∈�1��A� A��g x0� x� is finite dimensional for every x ∈ Bo. This provesthat A is locally bounded. �

We extend the Galois covering F � B → B to a functor F � A → A as follows:

(a) F��g x0�� = x0 for every g ∈ �1��A�;(b) Let u ∈ Mij�g

−1x� ⊆ M��g x0� x�. Then Mij�g−1x� ⊆ ⊕

h∈�1��A� Mij�h−1x� =

Mij�F�x�� ⊆ M�F�x�� (recall that F�Mij = Mij). So we set F�u� = u ∈ M�F�x��.

Lemma 4.5. The above construction defines a Galois covering F � A → A with group�1��A�.

Proof. F � A → A is a k-linear functor such that F g = g for every g ∈ �1��A�.Moreover, it is a covering functor because so is F � B → B and F�Mij = Mij forevery i j. Finally, the group �1��A� acts transitively on F−1�x� for every x ∈ Ao.Indeed, this is the case if x ∈ Bo because F � B → B is a Galois covering with group�1��A� and it is clearly the case if x = x0. So F is a Galois covering with group�1��A�. �

The Category A is Connected

We denote by Pm the indecomposable projective A-module associated to theobject �1 x0� of A. Therefore, rad�Pm�. We need the following lemma whose prooffollows from the definitions and where x0 = �x0 1�.

Lemma 4.6. Let g ∈ �1��A� and gx0 − x1 be an edge in �. Then there exist i j suchthat x1 is the homotopy class of the edge � � x0 − �Mij�

A in ��A� associated to theinclusion Mij ↪→ Pm. Let � be the component of p−1��i�� containing x1 and Bi

the associated copy of Bi in B. Then gMij ∈ ind Bi (and Mij is a direct summand ofrad�gPm�).

We use 4.6 to prove that A is connected.


Lemma 4.7. A is connected.

Proof. It suffices to prove that two indecomposable projective A-modules lie on thesame component of mod A. Let g ∈ �1��A�. Since � is connected, there is a sequenceof edges in �

where g = gn and, for every j, the vertices xj and x′j lie on the same component ofp−1��ij �� for some ij . By 4.6 and because B1 � � � Bt are connected, the modules

Pm and gPm lie on the same connected component of mod A.Now let P be an indecomposable projective A-module associated to an object

x ∈ Bo. So F�P is the indecomposable projective B-module associated to Fx. Let i ∈1 � � � t� be such that Fx is an object of Bi. So x is an object of some copy Bi

of Bi in B and we let � be the associated component of p−1��i��. On the otherhand, we let Bi be the copy of Bi in B such that Mi1 ∈ indBi and the associatedcomponent of p−1��i��. In particular there exists g ∈ �1��A� such that g� � = �so that gMi1 ∈ ind Bi. Therefore, P and gMi1 lie on the same component of mod Abecause they are indecomposable Bi-modules and Bi is connected; gMi1 and gPm

lie on the same component of mod A because of the inclusion Mij ↪→ Pm; and wealready proved that so do Pm and gPm. This shows that P and Pm lie on the samecomponent of mod A. So A is connected. �

Now we are in position to prove the main result of the section.

Proof of 4.1. We use an induction on rk�K0�A��. If A is tilted, then the resultfollows from [22, Thm. 1]. Assume that A is not tilted and that the conclusion ofthe proposition holds true for algebras B such that rk�K0�B�� < rk�K0�A��. Hencethere exists a maximal element Pm ∈ �f

A. Let A = B�M� be the associated one-point extension. Let B = B1 × · · · × Bt be an indecomposable decomposition. ThenB1 � � � Bt are connected, weakly shod and not quasi-tilted of canonical type. Let�1 � � � �t be the connecting components of B1 � � � Bt, respectively, containing asummand of M . The induction hypothesis implies that, for every i, there exists aconnected Galois covering F�i� � Bi → Bi with group �1��i� whose associated Galoiscovering of ��i� is the universal cover of graph. By 4.5 and 4.7, there exists aconnected Galois covering F � A → A with group �1��A�. Let ��A� → ��A� be theassociated Galois covering with group �1��A� (see 3.1). Since �1��A� is free, thisGalois covering is necessarily the universal covering of graphs. �

We give some examples to illustrate 4.1. In these examples, we write Px, Ixor Sx for the corresponding indecomposable projective, indecomposable injective orsimple, respectively.

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Example 4.8. Let A be the radical square zero algebra with ordinary quiver Q asfollows:

Let M = rad�P6�. Then A = B�M� where B is the radical square zero algebra withordinary quiver

Note that B is of finite representation type and ��modB� is equal to

The algebra A is wild and weakly shod, it has a unique connecting component ofthe following shape:

Note that A is not quasi-tilted because the projective dimension of S5 is equal to 4.The orbit-graph of the connecting component of A is equal to

The fundamental group of this graph is free of rank 2. So 4.1 implies that A admitsa connected Galois covering with group a free group with rank 2. Actually thisGalois covering is given by the fundamental group of the monomial presentation ofA (see [25]).

Recall that weakly shod algebras are particular cases of Laura algebras. Thefollowing example from [14] shows that 4.1 holds for some Laura algebras whichare not weakly shod.


Example 4.9 (see [14, 2.6]). Let A be the radical square zero algebra with ordinaryquiver Q as follows:

Then A is a Laura algebra. The component of ��modA� consist of:

1. The postprojective components and the homogeneous tubes of the Kroneckeralgebra with quiver 1 ⇒ 2;

2. The preinjective component and the homogeneous tubes of the Kroneckeralgebra with quiver 4 ⇒ 5;

3. A unique nonsemiregular component of the following shape:

where the two copies of the S3 are identified.

In this example, the orbit-graph of the unique nonsemiregular component is thefollowing:

The fundamental group of this graph is the free group of rank 3. On the other hand,if one denotes by �kQ+� for the ideal of kQ generated by the set of arrows, then thefundamental group of the natural presentation kQ/�kQ+�2 � A (in the sense of [25])is also isomorphic to the free group of rank 3. Hence A admits a connected Galoiscovering with group isomorphic to the orbit-graph of the connecting component.

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5. PROOF OF THEOREM A AND OF COROLLARY B

Throughout the section, we assume that A is connected and weakly shod.We prove the first two main results of the text presented in the introduction.

Proof of Theorem A. We assume that A is not quasi-tilted of canonical type. LetG be a group and �A a connecting component of ��modA�. If F � � → A is aconnected Galois covering then 3.1 yields a Galois covering of translation quiverswith group G of �A. Conversely, let � ′ → �A be a Galois covering of translationquivers with group G. Therefore G � �1��A�/N for some normal subgroup N �

�1��A� [11, 1.4]. On the other hand, 4.1 yields a connected Galois covering A → Awith group �1��A�. Factoring out by N yields a connected Galois covering A/N → Awith group G. �

Now we turn to the proof of Corollary B. We need the three following lemmas.The first one follows directly from Theorem A so we omit the proof.

Lemma 5.1. Assume that A is not quasi-tilted of canonical type. Let �A be aconnecting component of A. Then the following conditions are equivalent:

(a) A is simply connected;(b) The orbit-graph ��A� is a tree;(c) �A is simply connected.

The following lemma expresses the simple connectedness of A = B�M� interms of the simple connectedness of the components of B. In the case where Ais tame weakly shod, the necessity was proved in [5, Lem. 5.1]. We recall that ifA is connected and x0 ∈ Ao is the extension object in A = B�M�, then x0 is calledseparated if M has exactly as many indecomposable summands as the numberof components of B (that is, M restricts to an indecomposable module on eachcomponent of B).

Lemma 5.2. Assume that �fA = ∅. Let Pm ∈ �f

A be maximal, A = B�M� the associatedone-point extension and x0 ∈ Ao the extending object. Then A is simply connected if andonly if the two following conditions are satisfied:

(a) B is a product of simply connected algebras;(b) x0 is separating (that is, M is multiplicity-free).

Proof. By [14, 4.5, 4.8], B is a product of connected, weakly shod and not quasi-tilted of canonical type algebras. Assume that A is simply connected. By [4, 2.6], theobject x0 is separating. Let B

′ be a connected component of B. Since A is connected,M admits an indecomposable summand lying on indB′. By 2.8 and because theorbit-graph of any connecting component of A is simply connected, the orbit-graphof any connecting component of B′ is simply connected. So B′ is simply connectedby Theorem A. Conversely assume that x0 is separating and B is a product of simplyconnected algebras. By Theorem A, for every component B′ of B, the orbit-graphof any connecting component of B′ is a tree. By 2.8 and because x0 is separating,we deduce that the orbit-graph of any connecting component of A is a tree. ByTheorem A, this implies that A is simply connected. �


Finally, we recall the following lemma which was proved in [5, 2.5].

Lemma 5.3. Under the hypothesis and notations of 5.2, the following conditions areequivalent:

(a) HH1�A� = 0;(b) HH1�B� = 0 and x0 is separating.

Now we can prove Corollary B.

Proof of Corollary B. We use an induction on rk�K0�A��. By [22, Thm. 1], thecorollary holds true if A is tilted. So we assume that A is not quasi-tilted and thecorollary holds true for algebras B such that rk�K0�B�� < rk�K0�A��. Since �f

A = ∅,there exists Pm ∈ �f

A maximal. Let A = B�M� be the associated one-point extension.Using the induction hypothesis applied to the components of B and using 5.1–5.3,we deduce that A is simply connected if and only if HH1�A� = 0. On the other hand,Theorem A shows that A is simply connected if and only if ��A� is a tree. �

We finish this section with an example to illustrate Corollary B.

Example 5.4. Let A be as in 4.8. Then A is not simply connected and neither isthe orbit-graph of its connecting component. On the other hand, a straightforwardcomputation shows that dimHH0�A� = 1, dimHH1�A� = 3 and dimHHi�A� = 0 ifi ≥ 2.

6. THE CLASS OF WEAKLY SHOD ALGEBRAS IS STABLE UNDER FINITEGALOIS COVERINGS AND UNDER QUOTIENTS

In this section we prove Theorem C. At first we study the implications ofTheorem C in the more general setting of Galois coverings with non-necessarilyfinite groups.

Lemma 6.1. Let F � � → A be a connected Galois covering with group G. If Ais weakly shod and not quasi-tilted, then ��mod�� has a unique nonsemiregularcomponent ��. Moreover it is faithful, generalised standard and has no non trivial pathof the form X � gX with X ∈ �� and g ∈ G.

Proof. Let �A be the connecting component of A. Let �� be as in 3.1. We only needto prove that �� is the unique nonsemiregular component of ��mod��. Note that�� contains both a projective and an injective because so does �A. Let P ∈ ind�\��be projective. Then F�P ∈ indA\�A is projective and, therefore, lies on a semiregularcomponent of ��modA�. By 3.8, so does P. Whence the lemma. �

The preceding lemma has a converse under the additional assumption that thegroup G acts freely on the indecomposable modules lying on ��. This last conditionis always verified when G is torsion-free.

Lemma 6.2. Let F � � → A be a connected Galois covering with group G. Assumethat ��mod�� has a unique nonsemiregular component �� and that the following

1314 LE MEUR

conditions are satisfied:

(a) �� is faithful and generalised standard;(b) �� has no nontrivial path of the form X � gX;(c) GX = 1 for every X ∈ ��.

Then A is weakly shod.

Proof. Note that �� is G-stable because of its uniqueness. If follows from thearguments presented in the proof of [17, 3.6] that there is a component � of��modA� such that � = F�X �X ∈ ��. Also the map X �→ F�X extends to aGalois covering of translation quivers �� → � with group G. In particular, � isnonsemiregular. Moreover 2.3 implies that � is faithful, generalised standard, andhas no oriented cycles. Therefore, A is weakly shod. �

Now we prove the equivalences of Theorem C. Part of the tilted case wastreated in [22, Rem. 4.10]. We recall it for convenience.

Proposition 6.3. Let F � A′ → A be a connected Galois covering with finite group G.Then A′ is tilted if A is tilted.

Now we prove the equivalence of Theorem C in the quasi-tilted case.

Proposition 6.4. Let F � A′ → A be a connected Galois covering with finite group G.Then A′ is quasi-tilted if and only if A is quasi-tilted.

Proof. Recall that A denotes the left part of A. We use the following descriptionof A [6, Thm. 1.1]:

A = M ∈ indA � pdA�L� ≤ 1 for every L ∈ indA such that HomA�LM� = 0��

Also, by [18, II Thm. 1.14, II Thm. 2.3], the following conditions are equivalent forany algebra A:

(a) A is quasi-tilted;(b) A has global dimension at most 2 and idA�X� ≤ 1 or pdA�X� ≤ 1 for every X ∈

indA;(c) A contains all the indecomposable projective A-modules.

Assume that A is quasi-tilted. Let u � X → P be a nonzero morphism of A′-moduleswith XP ∈ indA′ and P projective. So F��u� � F�X → F�P is non zero and F�Pis indecomposable projective. Fix an indecomposable decomposition F�X = X1 ⊕· · · ⊕ Xr in modA. So the restriction Xi → F�P of F��u� is nonzero for some i.Since A is quasi-tilted, we have F�P ∈ A and, therefore, pdA�Xi� ≤ 1. On the otherhand, F•F�X = ⊕

g∈GgX, F•F�X = F•X1 ⊕ · · · ⊕ F•Xr and the projective dimension

is unchanged under F•, F� and under the action of G. Consequently pdA′�X� =pdA′�F•Xi� = pdA�Xi� ≤ 1. So P ∈ A′ . Thus, A′ is quasi-tilted.

Conversely, assume that A′ is quasi-tilted. In particular, A and A′ have thesame global dimension, that is, at most 2. Let X ∈ indA. Since G is finite, F•X ∈


modA′. Fix an indecomposable decomposition F•X = X1 ⊕ · · · ⊕ Xr in modA′. Weclaim that X1 � � � Xr have the same projective dimension. Indeed, let d = pdA′�X1�and I = i ∈ 1 � � � r� � pdA′�Xi�� = d�. Then F•X = L⊕M where L = ⊕

i∈I Xi andM = ⊕

i∈Ic Xi. Since the G-action on modA′ preserves the projective dimension,we have gL = L and gM = M for every g ∈ G. By [16, 1.2], we deduce thatthere exist Y Z ∈ modA such that X = Y ⊕ Z, L = F•Y and M = F•Z. Since Xis indecomposable and I = ∅, we have Z = 0 and, therefore, I = 1 � � � r�. ThuspdA′�Xi� = pdA′�Xj� = pdA�X� (and, dually, idA′�Xi� = idA′�Xj� = idA�X�) for everyi j. Since A′ is quasi-tilted, we infer that pdA�X� ≤ 1 of idA�X� ≤ 1. This proves thatA is quasi-tilted. �

Now we end the proof of Theorem C.

Proof of Theorem C. The necessity in (a) follows from 6.3 and (b) was provedin 6.4. We prove (c) and may assume that neither A nor A′ is quasi-tilted. Assumethat A is weakly shod and not quasi-tilted. Then 6.1 implies that ��modA′� has aunique nonsemiregular component which is moreover faithful, generalised standard,and has no oriented cycle. Therefore A′ is weakly shod. This proves the necessityin (c). From now on, we assume that A′ is weakly shod and not quasi-tilted ofcanonical type. We prove that A is weakly shod. In view of 6.2, we need thefollowing result.

Lemma 6.5. Assume that A′ is weakly shod and not quasi-tilted of canonical type. Wehave GX = 1 for every indecomposable A′-module X lying on a connecting componentof ��modA′�.

Proof of 6.5. The conclusion of the lemma holds true for any indecomposableprojective or injective A′-module. So does it for nonstable modules because A′commutes with the G-action. Let �A′ be a connecting component of A′ and X ∈ �A′be stable. We still write A′ for the left part of modA′, and we write �A′ for the rightpart of modA′, defined dually. Since A′ is weakly shod, the set indA′\ �A′ ∪�A′�is finite, contained in �A′ , and has no periodic module. Therefore, there existsn∈ such that nA′X ∈ �A′ ∩ �A′ ∪�A′�. Assume for example that X′ = nA′X ∈ �A′ ∩A′ (the remaining case is dealt with dually). Let e be the sum of the primitiveidempotents e′ of A′ such that e′A′ ∈ A′ , and let B′ = eA′e. Therefore, B′ is a fullconvex subcategory of A′, it is a product of tilted algebras, X′ ∈ indB′ (see [3]) andB′ is stable under G because so is A′ . In particular, F restricts to a Galois coveringF ′ � B′ → B with group G, where B �= F�B′�. In order to prove that GX = 1, weprove that GX′ = 1. By absurd assume that there is g ∈ G\1� such that gX′ � X′.After replacing g by some adequate power, we assume that g is of prime order p.The quotient � � B → B/�g� is a Galois covering with group �g� � /p. Therefore,Ext1B/�g��X

′ ��X′� � ⊕p−1

j=0 Ext1B�X

′ gjX′� = 0 because of [22, 2.1], the isomorphismgX′ � X′ and the equality Ext1B�X

′ X′� = 0. In order to get a contradiction we firstprove that ��X

′ is indecomposable. Fix an indecomposable decomposition ��X′ =

M1 ⊕ · · · ⊕Ml in mod�B/�g��. Hence Ext1B/�g��MiMi� = 0 for all i. We claim thatMi lies in the image of �� for all i. Indeed, we distinguish two cases accordingto whether car�k� = p or car�k� = p. If car�k� = p, then the claim follows from[23, Lem. 6.1]. If car�k� = p, then B/�g� is Morita equivalent to the skew-group

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algebra B��g�� [13, Thm. 2.8] and B��g�� is tilted [6, Thm. 1.2(g)]. Therefore,B/�g� is tilted and the claim follows from [22, Prop. 4.6]. Thus, in all cases, Mi ��M

′i for some M ′

i ∈ modB (necessarily indecomposable). So⊕p−1

j=0gjM ′

i � �•Mi isa summand of �•��X

′ � ⊕p−1j=0

gjX′ � ⊕p−1j=0 X

′. We thus have M ′i � X′ for all i,

whereas ��X′ = ��M

′1 ⊕ · · · ⊕ ��M

′l . This proves that ��X

′ is indecomposable. Thecontradiction is therefore the following. On the one hand, Ext1B/�g��X

′ ��X′� = 0,

��X′ ∈ ind�B/�g�� and B/�g� is a product of quasi-tilted algebras (because B is a

product of tilted algebras and by 6.4), which imply that EndB/�g��X′�� k. On the

other hand, EndB/�g��X′� � ⊕p−1

j=0 HomB�X′ gjX′� � ⊕p−1

j=0 EndB�X′� as k-vector

spaces. This is absurd. So GX′ = 1 and, therefore, GX = 1. �

Now we can prove that A is weakly shod by applying 6.2. As remarked inthe proof of 6.5, a nontrivial path in indA of the form X � gX with X ∈ �A′ givesrise to a nontrivial path X � X in indA′ which is impossible because A′ is weaklyshod. Therefore all the hypotheses of 6.2 are satisfied and A is weakly shod. Thisproves (c).

It only remains to prove the necessity in (a). We assume that A′ is tilted andprove that so is A. Let �A′ be a connecting component of ��modA′�. It admitsa complete slice �′. Clearly, �A′ is G-stable whatever the number of connectingcomponents of A′ is (one or two). By 6.2, 6.5 and [17, 3.6], there exists a component� of ��modA� such that � = F�X �X ∈ �A′�. Moreover, there is a Galois coveringof translation quivers �A′ → � with group G extending the map X �→ F�X. We provethat � has a complete slice. For this purpose we use the following lemma.

Lemma 6.6. gX ∈ �′ for every g ∈ GX ∈ �′.

Proof of 6.6. Let g ∈ G and write �′ = X1 � � � Xn�. So there exist a permutationi �→ g · i of 1 � � � n� and integers l1 � � � ln such that gXi =

liA′Xg�i for every i.

Clearly, the modules gX1 � � � g Xn form a complete slice g��′� in �A′ . This implies

that l1 = l2 = · · · = ln. We write l = l1. Therefore, g��′� = lA′��′�. On the otherhand, g has finite order and �A′ has no oriented cycles. So l = 0 and g��′� = �′. �

Let � be the full subquiver of � generated by F�X �X ∈ �′�. Hence � isconvex in � , has no oriented cycle and intersects each A-orbit of � exactly oncebecause �′ is a G-stable complete slice in �A′ . Moreover, the arguments used inthe proof of 2.3 show that � is faithful because so is �′. Finally, given X Y ∈ �′,we have HomA�F�X AF�Y� �

⊕g∈G HomA′�X A′ gY� = 0 because of the covering

property of F�, 2.1(d) and the fact that �′ is a G-stable slice in �A′ . Thus � is acomplete slice and A is tilted with � as a connecting component. This proves thesufficiency (a) and finishes the proof of Theorem C. �

Remark 6.7. The reader may find similar equivalences to those of Theorem Cabout skew-group algebras (instead of Galois coverings) under the additionalassumption that car�k� does not divide the order of the group G (see [6]).

ACKNOWLEDGMENT

The author gratefully acknowledges Ibrahim Assem for his encouragementsand useful comments.


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