SIMPLE COMODULES AND LOCALIZATION IN COALGEBRAS

Gabriel Navarro

Department of Algebra, University of Granada, Avda. Fuentenueva s/n, E-18071, Granada, Spaine-mail: gnavarro@ugr.es

Abstract

In this article we review recent developments in representation theory of coalgebras, aimingfor an extension of the classical theory for artinian algebras. The key tool is the use of thetheory of localization in categories of comodules and, in particular, the behaviour of simplecomodules through the action of the section functor. For that reason, a description of thelocalization in coalgebras is given.

INTRODUCTION

The representation theory of artinian algebras is a classical and fruitful theory that has providedus with many tools and results for years, see [2] and [3]. Nevertheless, due to its ambitiousprimary aim, that is, to describe, as a category, the (finite dimensional) modules over any artinianalgebra, many of these techniques strongly require finite dimensionality over the field and itdoes not seem possible to generalize them to an arbitrary algebra.Recently, some authors have tried to get rid of the imposed finiteness conditions by takingadvantage of coalgebras and their categories of comodules, see [4, 15, 16, 18, 23, 28, 29, 36].The main reasons for this are, on the one hand, that coalgebras may be realized, because ofthe freedom on choosing their dimension, as an intermediate step between finite dimensionaland infinite dimensional algebras. More concretely, in [28], it is proven that the category ofcomodules over a coalgebra is equivalent to the category of pseudo-compact modules, in thesense of [11], over the dual algebra. On the other hand, because of their locally finite nature,coalgebras are a good candidate for extending many techniques and results stated for finitedimensional algebras. Therefore it is rather natural to discuss the development of the followingpoints in coalgebra theory:

1. Some quiver techniques similar to the classical ones stated for algebras. For instance, adescription of coalgebras and comodules by means of quivers and linear representationof quivers (cf. [15], [16] and [29]).

2. An Auslander-Reiten theory for coalgebras. In particular, treat the problem of existence(and calculation) of the transpose, almost split sequences and the AR-quiver of a coalge-bra (cf. [6] and [7]).

3. Define the representation types of comodules (finite, tame and wild) and find some cri-teria in order to determine the representation type of a coalgebra. Prove the tame-wilddichotomy for coalgebras (cf. [28] and [29]).

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4. Develop a (co)tilting theory for coalgebras in order to describe coalgebras whose globaldimension is greater than one (cf. [34] and [35]).

5. Describe completely the representation theory of some particular classes of coalgebras.For example, co-semisimple, pure semisimple [18], semiperfect [19], serial [9], biserial[5], hereditary [14], and others.

This article is a survey of the applications of the theory of localization in coalgebras (followingthe general ideas of Gabriel [11] for abelian categories) to the first and the third point describedabove. It is worth mentioning that the key point of most of such applications is the behaviourof simple comodules through the action of the section functor.The paper is structured as follows. In Section 1, in order to make the exposition more self-contained and elementary, we collect some background notation and basic facts about coalge-bras, localization and quivers. Section 2 is devoted to study some properties of a coalgebra bymeans of the geometry of some quivers associated to it, the case of a pointed coalgebra beingof particular interest, since we may embed it into a path coalgebra. In Section 3 we describe thelocalization in path coalgebras in order to give examples of the theory. In Section 4 we studyhow the section functor transforms simple comodules. In particular, we give conditions in orderto decide whether or not the section functor preserves finite dimensional comodules. Finally,Section 5 is devoted to some results in representation theory of coalgebras. In particular, wehihlight a theorem of Gabriel for coalgebras in the case where the quiver we are dealing with isacyclic.

1 PRELIMINARIES

Throughout we fix a ground field K and we assume that all vector spaces are over K and everymap is a K-linear map. In particular, C is a K-coalgebra. We refer the reader to the books[1], [20] and [32] for notions and notations about coalgebras. Unless otherwise stated, allC-comodules are right C-comodules. It is well-known that C has a decomposition, as rightC-comodule,

CC =Mi∈IC

Etii ,

where Eii∈IC is a complete set of pairwise non-isomorphic indecomposable injective right C-comodules and ti is a positive integer for any i ∈ IC. This produces a decomposition of the socleof C (the sum of all its simple subcomodules), soc C, as follows:

soc C =Mi∈IC

Stii ,

where Sii∈IC is a complete set of pairwise non-isomorphic simple right C-comodules. It iseasy to prove that

ti =dimKSi

dimK(EndC(Si))

for any i ∈ IC, see [29].For any right C-comodule M, we denote by soc M the socle of M and by E(M) its injectiveenvelope. We assume that soc Ei = Si, and, consequently, E(Si) = Ei, for each i ∈ IC.The coalgebra C is said to be basic if ti = 1 for any i ∈ IC, or, equivalently, if dimKSi =dimK(EndC(Si)) for any i ∈ IC, or, equivalently, if Si is a simple subcoalgebra of C for any

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i ∈ IC, see for instance [31]. In particular, C is called pointed if dimKSi = 1 for any i ∈ IC. Herewe show that not every basic coalgebra is pointed:

Example 1.1. Let C be the R-vector space of the complex numbers with comultiplication andcounit given by the following formulae:

∆(1) = 1⊗1− i⊗ i ; ∆(i) = i⊗1+1⊗ i ; ε(1) = 1 ; ε(i) = 0.

Then C is a non-pointed and basic coalgebra, in fact, it is simple.

If the field K is algebraically closed then we may prove that C is pointed if and only if C is basic(cf. [29, Corollary 2.7]).Since every coalgebra is Morita-Takeuchi equivalent (that is, their categories of comodules areequivalent) to a basic one [28, Proposition 5.6], throughout we assume that C is basic and thereare decompositions

C =Mi∈IC

Ei and soc C =Mi∈IC

Si ,

where Ei E j and Si S j for i 6= j. Symmetrically, there exists the left-hand version of all thefacts explained above. In particular, C admits a decomposition as left C-comodule

CC =Mi∈IC

Fi and soc C =Mi∈IC

S′i .

Remark 1.2. Observe that Si = S′i for any i ∈ IC, since C is basic and therefore each simple (leftor right) C-comodule is a simple subcoalgebra. Nevertheless, the right injective envelope Ei andthe left injective envelope Fi of Si could be different.

Following [28], for every finite dimensional right C-comodule M we consider the length vectorof M, length M = (mi)i∈IC ∈ Z(IC), where mi ∈ N is the number of simple composition factorsof M isomorphic to Si. In [28] it is proven that the map M 7→ length M extends to a groupisomorphism K0(C)−→ Z(IC), where K0(C) is the Grothendieck group of C.Let M be a right C-comodule, we say that M is quasi-finite if its injective envelope E(M) =⊕i∈Ic Emi

i satisfies that mi is a finite non-negative integer for any i ∈ IC. It is easy to prove thatthis is equivalent to the fact that dimKHomC(Si,M) is finite for each i ∈ IC. Let D be anothercoalgebra, by [33], a (D,C)-bicomodule M is quasi-finite as right C-comodule if and only ifthe cotensor functor −DM (see [33] for its definition) has a left adjoint functor known as theCohom functor. In such a case, we denote it as CohomC(M,−). There it is proven that

CohomC(M,N) = lim−→λ

HomC(Nλ,M)∗,

where Nλλ is the set of all finite dimensional subcomodules of N and (−)∗ is the standardduality HomK(−,K). The functor CohomC(M,−) preserves direct sums and is right exact.Moreover, it is left exact if and only if M is injective as right C-comodule. We remind that thecotensor functor −DM preserves direct sums and is left exact, and it is right exact as wellwhen M is injective as left D-comodule.Throughout we denote by M C

f , M Cq f and M C the category of finite dimensional, quasi-finite

and all right C-comodules, respectively.A full subcategory T of M C is said to be dense (or a Serre class) if each exact sequence

0 // M1 // M // M2 // 0

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in M C satisfies that M belongs to T if and only if M1 and M2 belong to T . Following [11]and [25], for any dense subcategory T of M C, there exists an abelian category M C/T and anexact functor T : M C → M C/T , such that T (M) = 0 for each M ∈ T , satisfying the followinguniversal property: for any exact functor F : M C → C such that F(M) = 0 for each M ∈ T ,there exists a unique functor F : M C/T → C verifying that F = FT . The category M C/T iscalled the quotient category of M C with respect to T , and T is known as the quotient functor.Let now T be a dense subcategory of the category M C, T is said to be localizing (cf. [11]) ifthe quotient functor T : M C → M C/T has a right adjoint functor S, called the section functor.If the section functor is exact, T is called perfect localizing. Dually, see [22], T is said to becolocalizing if T has a left adjoint functor H, called the colocalizing functor. T is said to beperfect colocalizing if the colocalizing functor is exact.Let us list some properties of the localizing functors (cf. [11] and [22]).

Lemma 1.3. Let T be a dense subcategory of the category of right comodules M C over acoalgebra C. The following statements hold:(a) T is exact.(b) If T is localizing, then the section functor S is left exact and the equivalence T S ' 1M C/Tholds.(c) If T is colocalizing, then the colocalizing functor H is right exact and the equivalenceT H ' 1M C/T holds.

From the general theory of localization in Grothendieck categories [11], it is well-known thatthere exists a one-to-one correspondence between localizing subcategories of M C and sets ofindecomposable injective right C-comodules, and, as a consequence, sets of simple right C-comodules. More concretely, a localizing subcategory is determined by an injective right C-comodule E =⊕ j∈JE j, where J ⊆ IC (therefore the associated set of indecomposable injectivecomodules is E j j∈J). Then M C/T ' M D, where D is the coalgebra of coendomorphismCohomC(E,E), and the quotient and section functors are CohomC(E,−) and −DE, respec-tively. By [31], the quotient and the section functors define an equivalence of categories betweenM D and the category M C

E of E-copresented right C-comodules, that is, the right C-comodulesM which admit an exact sequence 0 // M // E0 // E1 , where E0 and E1 are directsums of direct summands of the comodule E.In [8], [17] and [36], localizing subcategories are described by means of idempotents in thedual algebra C∗. In particular, it is proven that the quotient category is the category of rightcomodules over the coalgebra eCe, where e is an idempotent associated to the localizing sub-category (that is, E = Ce, where E is the injective right C-comodule associated to the localizingsubcategory). The coalgebra structure of eCe (cf. [26]) is given by

∆eCe(exe) = ∑(x)

ex(1)e⊗ ex(2)e and εeCe(exe) = εC(x)

for any x ∈C, where ∆C(x) = ∑(x) x(1)⊗ x(2) using the sigma-notation of [32]. Throughout wedenote by Te the localizing subcategory associated to the idempotent e. For completeness, weremind from [8] (see also [17]) the following description of the localizing functors. We recallthat, given an idempotent e∈C∗, for each right C-comodule M, the vector space eM is endowedwith a structure of right eCe-comodule given by

ρeM(ex) = ∑(x)

ex(1)⊗ ex(0)e

where ρM(x) = ∑(x) x(1)⊗ x(0) using the sigma-notation of [32].

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Lemma 1.4. Let C be a coalgebra and e be an idempotent in C∗. Then the following statementshold:

(a) The quotient functor T : M C →M eCe is naturally equivalent to the functor e(−). T is alsonaturally equivalent to the cotensor functor−CeC and the Cohom functor CohomC(Ce,−).

(b) The section functor S : M eCe →M C is naturally equivalent to the cotensor functor−eCeCe.As a consequence, T is perfect localizing if and only if Ce is injective as left eCe-comodule.

(c) If Te is a colocalizing subcategory of M C, the colocalizing functor H : M eCe → M C isnaturally equivalent to the functor CohomeCe(eC,−). As a consequence, T is perfect colo-calizing if and only if eC is injective as right eCe-comodule.

For the convenience of the reader we summarize the functors obtained in the situation of(co)localization by means of the diagrams:

M CT=e(−)=−CeC //

M eCeS=−eCeCe

oo and M CT=e(−)=−CeC //

M eCeH=CohomeCe(eC,−)

oo .

Lastly, for completeness, we remind some points about quivers and path (co)algebras. Follow-ing Gabriel [12], by a quiver, Q, we mean a quadruple (Q0,Q1,h,s) where Q0 is the set ofvertices (points), Q1 is the set of arrows and, for each arrow α ∈Q1, the vertices h(α) and s(α)are the source (or start or head point) and the sink (or end point) of α, respectively (see [2] and[3]). If i and j are vertices in Q, an (oriented) path in Q of length m from i to j is a formalcomposition of arrows

p = αm · · ·α2α1

where h(α1) = i, s(αm) = j and s(αk−1) = h(αk), for k = 2, . . . ,m. To any vertex i ∈ Q0 weattach a stationary path of length 0, say ei, starting and ending at i such that αei = α (resp.eiβ = β) for any arrow α (resp. β) with h(α) = i (resp. s(β) = i). We identify the set of verticesand the set of stationary paths. An (oriented) cycle is a path in Q which starts and ends at thesame vertex. Q is said to be acyclic if there is no oriented cycle in Q.Let KQ be the K-vector space generated by the set of all paths in Q. Then KQ can be en-dowed with the structure of a (non necessarily unitary) K-algebra with multiplication inducedby concatenation of paths, that is,

(αm · · ·α2α1)(βn · · ·β2β1) =

αm · · ·α2α1βn · · ·β2β1 if s(βn) = h(α1),0 otherwise;

KQ is the path algebra of the quiver Q. The algebra KQ can be graded by

KQ = KQ0⊕KQ1⊕·· ·⊕KQm⊕·· · ,

where Qm is the set of all paths of length m. An ideal Ω⊆ KQ is called an ideal of relations or arelation ideal if Ω ⊆ KQ2⊕KQ3⊕·· · . We denote by KQ≥m the ideal of KQ generated by thepaths of length greater or equal than m. By a quiver with relations we mean a pair (Q,Ω), whereQ is a quiver and Ω a relation ideal of KQ. For more details and basic facts from representationtheory of algebras the reader is referred to [2] and [3].

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Following [36], the path algebra KQ can be viewed as a graded K-coalgebra with comultiplica-tion induced by the decomposition of paths, that is, if p = αm · · ·α1 is a path from the vertex ito the vertex j, then

∆(p) = e j⊗ p+ p⊗ ei +m−1

∑i=1

αm · · ·αi+1⊗αi · · ·α1 = ∑ητ=p

η⊗ τ

and for a stationary path, ei, we have ∆(ei) = ei⊗ei. The counit of KQ is defined by the formula

ε(α) =

1 if α ∈ Q0,0 if α is a path of length ≥ 1.

The coalgebra (KQ,∆,ε) (shortly KQ) is called the path coalgebra of the quiver Q. A sub-coalgebra C of a path coalgebra KQ of a quiver Q is said to be admissible if it contains thesubcoalgebra of KQ generated by all vertices and all arrows.A K-linear representation (cf. [2] and [3]) of a quiver Q is a system

X = (Xi,ϕα)i∈Q0,α∈Q1,

where Xi is a K-vector space for each i∈Q0 and ϕα : Xi → X j is a K-linear map for any α : i→ j.Given two K-linear representations of Q, (Xi,ϕα) and (Yi,ψα), a morphism

f : (Xi,ϕα)−→ (Yi,ψα)

of representations of Q is a system f = ( fi)i∈Q0 of K-linear maps fi : Xi −→ Yi for any i ∈ Q0such that, for any α : i → j in Q1, the following diagram is commutative

Xiϕα //

fi

X j

f j

Yiψα // Yj

We denote by RepK(Q) the Grothendieck K-category of K-linear representations of Q, and byRepl f

K (Q) the full Grothendieck K-subcategory of RepK(Q) formed by locally finite-dimensionalrepresentations, that is, directed unions of representations of finite length. A linear representa-tion X of Q is said to be of finite length if Xi is a finite dimensional vector space for all i ∈ Q0

and Xi = 0 for almost all indices i. Finally, we denote by repK(Q) ⊇ repl fK (Q) the full subcat-

egories of RepK(Q) formed by finitely generated representations and representations of finitelength, respectively.A linear representation X of Q is nilpotent if there exists an integer m≥ 1 such that the composedlinear map

Xi0ϕα1 // Xi1

ϕα2 // Xi2//___ · · · //___ Xim−1

ϕαm // Xim

is zero for any path αmαm−1 · · ·α1 in Q of length m. We denote by nilrepl fK (Q) the full subcate-

gory of repK(Q) formed by all nilpotent representations of finite length, and by Replnl f (Q) thefull subcategory of RepK(Q) of all locally nilpotent representations that are locally finite, thatis, directed unions of representations from nilrepl f

K (Q).Given a quiver with relations (Q,Ω), a linear representation of (Q,Ω) is a linear representationX = (Xi,ϕα) of Q which verifies that if p = ∑

ni=1 λiα

imi· · ·αi

1 is in Ω then ∑ni=1 λiϕαi

mi· · ·ϕ

αi1= 0.

As above, we may define the categories RepK(Q,Ω), Repl fK (Q,Ω), repK(Q,Ω), repl f

K (Q,Ω),Replnl f

K (Q,Ω) and nilrepl fK (Q,Ω), see [28], [29] or [36] for details.

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2 QUIVERS ASSOCIATED TO A COALGEBRA

Associating a graphical structure to a certain mathematical object is a very common strategy inthe literature. Sometimes, it provides us a nice method for replacing the object with a simplerone and improving our intuition about its properties. In our case, the quiver-theoretical ideasdeveloped by Gabriel and his school during the seventies have been the origin of many advancesin Representation Theory of Algebras for years. Moreover, many of the present developmentsof the theory use, up to some extent, these techniques and results. Among these tools, it ismostly accepted that the main one is the famous Gabriel Theorem:

Theorem 2.1 (Gabriel Theorem). Let K be an algebraically closed field. Then every basic finitedimensional algebra A is isomorphic to a quotient KQA/Ω, where QA is the Gabriel quiver ofA, and Ω is an ideal of KQA such that

K(QA)≥n ⊆ Ω ⊆ K(QA)≥2

for some integer n ≥ 2.Moreover, there exists a K-linear equivalence of categories

F : MA → RepK(Q,Ω)

between the category of right A-modules and the category of linear representations of the quiverwith relations (Q,Ω). This equivalence restricts to an equivalence

F : M fA → repK(Q,Ω)

between the category of finitely generated right A-modules and the one of finite dimensionallinear representations of (Q,Ω).

In the literature, we may find different efforts trying to extend this theorem to a wider context.As we mentioned in the Introduction, coalgebras seem to be a good candidate in order to gobeyond the classical ideas for algebras, therefore, it is a natural question to ask about obtaininga Gabriel theorem for pointed (basic, if K is algebraically closed) coalgebras in order to classifythem according their category of comodules. By [36], to any pointed coalgebra C we mayassociate its so-called Gabriel quiver QC as follows:

Vertices: The set of vertices (QC)0 is the set of group-like elements of C, i.e., the set of ele-ments c ∈C such that ∆(c) = c⊗ c. Observe that this is also the set of simple subcoalge-bras (and then, left or right simple comodules) since C is pointed.

Arrows: Given two group-like elements c and d, the number of arrows from c to d is the K-dimension of the K-vector space of non-trivial (c,d)-primitive elements of C, P′(c,d) =P(c,d)/PT (c,d), where

P(c,d) = x ∈C such that ∆(x) = d⊗ x+ x⊗ c,

and PT (c,d) = K〈c−d〉 (this elements are called the trivial (c,d)-primitive elements).

Then, in [36], Woodcock proves the following result:

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Theorem 2.2. [36] Let C be a pointed coalgebra. Then C is isomorphic to an admissiblesubcoalgebra of the path coalgebra of its Gabriel quiver.

Proof. The proof of the theorem is based on the universal property of the cotensor coalgebragiven by Nichols in [24, Proposition 1.4.2]. Choosing suitable coalgebras, bicomodules andmorphisms, this yields that any pointed coalgebra C is a subcoalgebra of the cotensor coalgebra

CTC0(M) = C0⊕M⊕ (MCM)⊕ (MCMCM)⊕·· · ,

where C0 is the coradical of C (that is, the vector space generated by the group-like elements)and M = C1/C0, where C1 is the piece of degree one of the coradical filtration of C. Since Mis the vector space of non-trivial primitive elements of C, it is easy to see that CTC0(M)∼= KQCand the result follows.

Therefore, by the previous theorem, when studying pointed coalgebras, we may bound theattention to path coalgebras and its subcoalgebras. Then, the manipulation of the elements of apointed coalgebra or the calculation of certain comodules is now much easier.The next step is given by Simson in [28]. There, the author tries to get a better approximationby means of the notion of path coalgebra of a quiver with relations. Indeed, let (Q,Ω) be aquiver with relations, the path coalgebra of (Q,Ω) is defined by the subspace of KQ,

C(Q,Ω) = a ∈ KQ | 〈a,Ω〉= 0

where 〈−,−〉 : KQ×KQ−→K is the bilinear map defined by 〈v,w〉= δv,w (the Kronecker delta)for any two paths v and w in Q. That is, in other words, C(Q,Ω) is the standard orthogonal ofthe ideal Ω contained in the path algebra (non necessarily unitary) of Q, see [1] or [15]. Wemay show the situation by means of the following picture (for clarity, here we denote by CQ thepath coalgebra of Q):

_____________

_ _ _ _ _ _ _ _ _ _ _ _ _

CQ KQC⊥KQ=p ∈ KQ | 〈C, p〉= 0 // KQCQ

Ω⊥=C(Q,Ω)ooCQ

C⊥

33I⊥

zz

KQ

C⊥∩KQ

(CQ)∗

This definition is congruent with the classical theory for finite dimensional algebras since thereis a K-linear equivalence of the category M C

f of finite dimensional right C-comodules with

the category nilrepl fK (Q,Ω) of nilpotent K-linear representations of finite length of the quiver

with relations (Q,Ω) (see [28, p. 135] and [29, Theorem 3.14]). Then Simson [28] raises thefollowing question:

Question 2.3. Is any admissible subcoalgebra C of a path coalgebra KQ isomorphic to the pathcoalgebra C(Q,Ω) of a quiver with relations (Q,Ω)?

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Unfortunately, the answer is negative as the following counterexample shows:

Example 2.4. [15] Let Q be the quiver

rrrrrrrrrrr

99α1

LLLLLLLLLLL

%%

β1iiiiiiiiii

44α2

UUUUUUUUUU

**

β2

//αn //βn

UUUUU

**αi

iiiii

44

βi

γi = βiαi for all i ∈ N

and let H be the relation subcoalgebra of KQ generated (as vector space) by the set of vertices,the set of arrows and Σ = γi− γi+1i∈N.Assume that x = ∑i≥1 aiγi belongs to H⊥ and ai = 0 for i ≥ n we have some n ∈ N. Then〈γi− γi+1,x〉 = ai−ai+1 = 0 for all i ∈ N, so ai = ai+1 for all i ∈ N. But an = 0 and it followsthat x = 0. Hence H⊥KQ = 0.By a similar argument H⊥ = 〈 f 〉, where f (γi) = 1 for all i ∈N. That is, f ≡∑i≥1 γi. Obviously,H⊥KQ is not dense in H⊥ in the weak* topology (see [15] for the definition and some properties),then [15, Lemma 4.6] yields that H is not the path coalgebra of a quiver with relations.

Moreover, in [15], a criterion in order to decide whether or not a coalgebra is the path coalgebraof a quiver with relations is given:

Theorem 2.5. Let C be an admissible subcoalgebra of a path coalgebra KQ. Then C is not thepath coalgebra of a quiver with relations if and only if there exist an infinite number of differentpaths γii∈N in Q such that:

(a) All of them have common source and common sink.

(b) None of them is in C.

(c) There exist elements anj ∈ K for all j,n ∈ N such that the set γn + ∑ j>n an

jγ jn∈N is con-tained in C.

As a consequence, if Q is intervally finite, i.e., there is at most a finite number of paths betweenany two vertices, we may give a positive answer to Question 2.3 (cf. [29, Theorem 3.14]). InSection 5 we shall extend this result to a wider context.Let us now suppose that C is an arbitrary coalgebra over an arbitrary field K. We may associateto C a quiver ΓC known as the right Ext-quiver of C, see [21]. We recall that the set of verticesof ΓC is the set of pairwise non-isomorphic simple right C-comodules Sii∈IC and, for twovertices Si and S j, there exists a unique arrow S j → Si in ΓC if and only if Ext1C(S j,Si) 6= 0.This quiver admits a generalization by means of the notion of right Gabriel-valued quiver of C,(QC,dC), i. e., following [18], the valued quiver whose set of vertices is Sii∈IC and such thatthere exists a unique valued arrow

S j(d1

ji,d2ji) // Si

if and only if Ext1C(S j,Si) 6= 0 and

d1ji = dimEndC(Si)Ext1C(S j,Si) and d2

ji = dimEndC(S j)Ext1C(S j,Si),

as left EndC(S j)-module and as right EndC(Si)-module, respectively.

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The (non-valued) Gabriel quiver of C is obtained taking the same set of vertices and the numberof arrows from a vertex S j to a vertex Si is dimEndC(S j)Ext1C(S j,Si) as right EndC(S j)-module.Obviously, if C is pointed (or K is algebraically closed) then it is isomorphic to the one used byWoodcock and C is a subcoalgebra of the path coalgebra of its (non-valued) Gabriel quiver.We may proceed analogously with left C-comodules and obtain the left-hand version of all ofthem. Throughout we assume that these quivers are connected, i.e., C is indecomposable ascoalgebra.In [23] and [30], the geometry of these quivers is linked with some properties of the coalgebra,more concretely, with the morphisms between indecomposable injective C-comodules. Let usremind that, by [13], any right C-comodule M has a filtration

0 ⊂ soc M ⊂ soc2M ⊂ ·· · ⊂ M

called the Loewy series, where, for n > 1, socnM is the unique subcomodule of M satisfyingthat socn−1M ⊂ socnM and

socnMsocn−1M

= soc(

Msocn−1M

).

Given two simple right C-comodules Si and S j, we say that the vertex S j is an n-predecessorof Si if Ext1C(S j,socnEi) 6= 0 or, equivalently, if S j ⊆ soc (Ei/socnEi) ∼= socn+1Ei/socnEi. Thefollowing result is proven in [23]:

Lemma 2.6. Let Si and S j be two simple C-comodules. The following assertions are equivalent:

(a) S j is a n-predecessor of Si.

(b) There exists a non-zero morphism f : socn+1Ei → E j such that f (socnEi) = 0.

(c) There exists a morphism g : Ei → E j such that g(sociEi) = 0 for all i = 1, . . . ,n andg(socn+1Ei) 6= 0

Observe that the set of 1-predecessors of a vertex Si is in one-to-one correspondence with theset of vertices S j such that there is an arrow S j → Si in (QC,dC), that is, the set of vertices S jsuch that there is a path of length one in (QC,dC) from S j to Si. This correspondence fails forn-predecessors and paths of length n, when n > 1.

Example 2.7. Let us consider the quiver Q

'&%$ !"#1α //'&%$ !"#2

β //'&%$ !"#3

and the subcoalgebra C of KQ generated, as vector space, by 1,2,3,α,β. Then the Ext-quiverΓC is

S1 −→ S2 −→ S3.

Obviously, there is a path from S1 to S3, but there is no non-zero morphisms

f : E3 =< 3,β >−→ E1 =< 1 > .

On the other hand, if C is the coalgebra KQ, the Ext-quiver of KQ is also the previous quiverbut, in this case, we may obtain a map

f : E3 =< 3,β,βα >−→ E1 =< 1 >

defined by f (βα) = 1 and zero otherwise.

150

The fact that the former path coalgebra satisfies the bijective correspondence between n-predecessorsand paths of length n is not a happy accident. Using Lemma 2.6, the following result relatesthe paths in the Gabriel quiver of C and the morphisms between indecomposable injective C-comodules.

Theorem 2.8. [23, Theorem 2.9] Let Si and S j be two simple C-comodules and n be a positiveinteger. If S j is an n-predecessor of Si then there exists a path in (QC,dC) of length n from S j toSi. The converse also holds if C is hereditary.

Remark 2.9. Reviewing the proof of the former theorem in [23], one may observe that the sec-ond statement admits a weaker version if we only assume that all non-zero morphisms betweenindecomposable injective comodules are surjective. In such a case, we may only prove that ifthere is a path in (QC,dC) of length n from S j to Si, then S j is a t-predecessor of Si for someinteger t ≥ n.

The approach of [30] differs slightly. Given two indecomposable injective comodules Ei and E j,the radical of HomC(Ei,E j) is the K-subspace radC(Ei,E j) of HomC(Ei,E j) generated by allnon-isomorphisms. Observe, that if i 6= j, then radC(Ei,E j) = HomC(Ei,E j). Also, by Lemma2.6, if radC(Ei,E j) 6= 0 then S j is a predecessor of Si. The square of radC(Ei,E j) is defined tobe the K-subspace

rad2C(Ei,E j)⊆ radC(Ei,E j)⊆ HomC(Ei,E j)

generated by all composite homomorphisms of the form

Eif // Ek

g // E j,

where f ∈ radC(Ei,Ek) and g ∈ radC(Ek,E j). The mth power radmC (Ei,E j) of radC(Ei,E j) is

defined analogously, for each m > 2. Then it is proven the following theorem:

Theorem 2.10. [30, Theorem 2.3]

(a) There exists a unique valued arrow

S j(d1

ji,d2ji) // Si

in (QC,dC) if and only if the EndC(S j)-EndC(Si)-bimodule IrrC(Ei,E j)= radC(Ei,E j)/rad2C(Ei,E j)

is nonzero and

d1ji = dimEndC(Si)IrrC(Ei,E j) and d2

ji = dimEndC(S j)IrrC(Ei,E j)

(b) If f ∈ HomC(Ei,E j) is a nonzero and noninvertible morphism, then there exists an integerm > 0 such that f ∈ radm

C (Ei,E j)\radm+1C (Ei,E j). As a consequence, there is a sequence of

morphisms

E1 = Eif1 // E2

f2 // E3 Emfm // Em+1 = E j

such that fk ∈ IrrC(Ek,Ek+1) for each k = 1, . . . ,m and the composition fm · · · f1 is nonzero.

151

Remark 2.11. Following the Auslander-Reiten theory for finite dimensional algebras, the notionof an irreducible morphism between left C-comodules is described in [30] as follows; see [2], [3,Section 5.5] and [27, Section 11.1]. A C-comodule homomorphism f : Ei → E j is an irreduciblemorphism if f is not an isomorphism and given a factorization

Eif //

g?

????

???

E j

Eh

??

of f , where E is a injective comodule whose socle is finite dimensional, f is a section, or f isa retraction. Analogously to the case of finite-dimensional algebras, there it is proven that theset of irreducible morphism IrrC(Ei,E j) is isomorphic, as EndC(S j)-EndC(Si)-bimodule, to thequotient radC(Ei,E j)/rad2

C(Ei,E j).

3 EXAMPLES OF LOCALIZATION IN COALGEBRAS

In this section we describe the localization in pointed coalgebras stressing the case of pathcoalgebras. We provide a wide range of examples for the convenience of the reader. The resultsof this section could be found in [15], [17] and [23].Let Q = (Q0,Q1) be a quiver and KQ the path coalgebra of Q. Since the localizing subcate-gories of M KQ are parameterized by subsets of simple KQ-comodules, these are in one-to-onecorrespondence with subsets of the set of vertices Q0. On the other hand, let e be an idempotentin (KQ)∗. For each vertex x ∈ Q0, x e(x) = e · x = e · e · x = x e(x)2. Thus e(x) is zero or one.Also, it is not difficult to see that two idempotent elements f , g ∈ (KQ)∗ are equivalent (inducethe same localizing subcategory) if and only if f|Q0 = g|Q0 . So we may assume that the idempo-tent e verifies that e(p) = 0 for each path of length greater than zero. Summarizing, the subsetsof vertices of Q0 (i.e., localizing subcategories of M KQ) are in one-to-one correspondence withthe idempotents elements of (KQ)∗ which map to zero the paths of length greater than zero.Clearly, the bijection is given as follows: for each subset X ⊆ Q0, we consider the idempotenteX in (KQ)∗ given by

eX(p) =

1 if p ∈ X ⊆ Q00 otherwise

for each path p in Q. And, conversely, given e ∈ (KQ)∗ idempotent, we consider the set

Xe = x ∈ Q0 such that e(x) = 1.

Now let p be a path in Q from x to y. Then, by the above discussion,

e · p = (I⊗ e)∆(p) = ∑p=rt

r e(t) = p e(x) =

p, if x ∈ Xe0, otherwise

That is, e(KQ) is generated by the paths starting at vertices in Xe. Symmetrically, (KQ)e isgenerated by the paths ending at vertices in Xe. Therefore the localized coalgebra e(KQ)e isgenerated by the paths starting and ending at vertices in Xe. Taking into account the coalge-bra structure of e(KQ)e given in the Introduction we may prove the following theorem. Forsimplicity, we introduce the following notation: let Q be a quiver and p = αnαn−1 · · ·α1 be anon-stationary path in Q. We denote by Ip the set of vertices h(α1),s(α1),s(α2), . . . ,s(αn).

152

Given a subset of vertices X ⊆ Q0, we say that p is a cell in Q relative to X (shortly a cell)if Ip ∩X = h(p),s(p) and s(αi) /∈ X for all i = 1, . . . ,n− 1. Given x,y ∈ X , we denote byCellQ

X (x,y) the set of all cells from x to y. We denote the set of all cells in Q relative to X byCellQ

X .

Theorem 3.1. [17, Theorem 3.1] Let Q be a quiver, KQ the path coalgebra of Q and e anidempotent in (KQ)∗. Then the localized coalgebra e(KQ)e is isomorphic to the path coalgebraof the quiver Qe, where

• (Qe)0 = Xe, and

• For each two vertices x,y ∈ Xe, the number of arrows from x to y in Qe is the cardinal ofthe set CellQ

Xe(x,y).

Example 3.2. Let KQ be the path coalgebra of the quiver Q given by

β1 // •δ

##HHHHHH

α

;;vvvvvv

γ ##HHHHHH β2

;;vvvvvvη

##HHHHHH

•

µ1;;vvvvvvµ2

// ρ

;;vvvvvv

and Xe be the set formed by the white points. Then, the set of cells is

α,η,ρ,δβ1,δβ2,µ1γ,µ2γ.

Therefore the quiver Qe is the following:

δβ1

++WWWWWWWWWW

α

;;vvvvvv µ1γ //_______

µ2γ ++WWWWWWWWWW η

##HHHHHHδβ2 //_______

ρ

;;vvvvvv

where the dashed arrows are the cells of length greater than one.

Example 3.3. Let KQ be the path coalgebra of the quiver Q

•

•

OO

oo

α // •β

oo

•

and Xe be the set formed by the white point. Then the set of cells is βα, that is, the quiver Qe

is

βα

and e(KQ)e ∼= K[βα], i.e, it is the polynomial coalgebra.

153

Example 3.4. Let KQ be the path coalgebra of the quiver

'&%$ !"#1 α

((RRRRRRR

'&%$ !"#3

'&%$ !"#2 β

66lllllll

and e ∈ (KQ)∗ be the idempotent associated to the set Xe = 1,2. Then e(KQ)e is the pathcoalgebra of the quiver formed by two isolated points

'&%$ !"#1

'&%$ !"#2

Observe that T maps the indecomposable injective right C-comodule E3 =< 3,α,β > to theright eCe-comodule S1⊕S2. Thus the functor T does not preserve indecomposable comodules.

Following [16], we may generalize the former proposition to an arbitrary pointed coalgebra asfollows:

Proposition 3.5. [16, Proposition 3.2] Let C be an admissible subcoalgebra of the path coal-gebra KQ of a quiver Q. Let e be an idempotent in C∗. Then the localized coalgebra eCe isan admissible subcoalgebra of the path coalgebra KQe, where Qe is the quiver described asfollows:

• The set of vertices (Qe)0 = Xe and,

• For each x,y ∈ Xe, the number of arrows from x to y in Qe is dimKKCellQXe

(x,y)∩C.

Example 3.6. Let Q be the quiver '&%$ !"#2α2

%%KKKKKK

'&%$ !"#1

α199ssssss

β1 %%KKKKKK '&%$ !"#4

'&%$ !"#3 β2

99ssssss

and C be the admissible subcoalgebra of KQ generated, as vector space, by

1,2,3,4,α1,α2,β1,β2,α2α1 +β2β1.

Let us consider e the idempotent associated to the set Xe = 1,3,4. Then eCe is the pathcoalgebra of the quiver Qe

'&%$ !"#1β1 //'&%$ !"#3

β2 //'&%$ !"#4

Here, the element α2α1 +β2β1 corresponds to the composition of the arrows β1 and β2 of Qe.On the other hand, if C = KQ, the ”localized” quiver, say now Qe, is the following

'&%$ !"#1β1

//

β≡α2α1

%%'&%$ !"#3β2

//'&%$ !"#4

Remark 3.7. In general, if C is a proper admissible subcoalgebra of a path coalgebra KQ andX ⊆ Q0, then Qe (the quiver associated to eCe) is a subquiver of Qe (the quiver associated toe(KQ)e) and the differences appear in the set of arrows.

154

Let us introduce the following notation. Let Q be a quiver and X ⊆Q0. We say p = αn · · ·α2α1is a h(p)-tail in Q relative to X if Ip∩X = h(p) and s(αi) /∈ X for all i = 1, . . . ,n. If there isno confusion we simply say that p is a tail. Given a vertex x ∈ X , we denote by T ailQ

X (x) theset of all x-tails in Q relative to X . In [17], it is proven that

e(KQ)∼=Mx∈Xe

ECard

(T ailQ

Xe(x))+1

x ,

where Exx∈Xe is a complete set of pairwise non-isomorphic indecomposable injective righte(KQ)e-comodules. Therefore e(KQ) is always injective and is quasi-finite if and only ifT ailQ

Xe(x) is a finite set for any x ∈ Xe. Thus we have proven the following result:

Theorem 3.8. [17, Corollary 3.6] Let Q be a quiver and e ∈ (KQ)∗ be an idempotent. Then thefollowing conditions are equivalent:

(a) The localizing subcategory Te of M KQ is colocalizing.

(b) The localizing subcategory Te of M KQ is perfect colocalizing.

(c) T ailQXe

(x) is a finite set for all x ∈ Xe.

Example 3.9. Consider the quiver Q

'&%$ !"#1

α1::tttttt α2 //

α3$$JJJJJJ

αi7

77

77

where i ∈ N

and the subset X = 1. Then T ailQX (1) = αii∈N is an infinite set and the localizing subcate-

gory TX is not colocalizing.

Example 3.10. Let KQ be the path coalgebra of the quiver Q

•

• α

OO

β

oo

γ

δ // •ρ

oo

•

and X be the set formed by the white point. Then

T ailQX () = α,β,γ,δ.

Therefore TX is a colocalizing subcategory.

155

In [16] it is proven an extension of the previous Theorem to any pointed (admissible) coalgebraC⊆KQ. Let x∈Q0, M be right C-comodule and f a linear map in HomC(Sx,M). Then ρM f =( f ⊗ I) ρSx , where ρM and ρSx are the structure maps of M and Sx as right C-comodules,respectively. In order to describe f , since Sx = Kx, it is enough to choose the image for x.Suppose that f (x) = m ∈M. Since (ρM f )(x) = (( f ⊗ I)ρSx)(x), we obtain that ρM(m) = m⊗x.Therefore

Mx := HomC(Sx,M)∼= m ∈ M such that ρM(m) = m⊗ x,

as K-vector spaces, and M is quasi-finite if and only if Mx has finite dimension for all x ∈ Q0.By [16], for each x ∈ Xe,

(eC)x = KT ailQXe

(x)∩C

Therefore, the following Proposition holds:

Proposition 3.11. Let C be an admissible subcoalgebra of a path coalgebra KQ and e be anidempotent in C∗. The following assertions are equivalent:

(a) The localizing subcategory Te of M C is colocalizing.

(b) eC is a quasifinite right eCe-comodule.

(c) dimKKT ailQXe

(x)∩C is finite for all x ∈ Xe.

In particular, if T ailQXe

(x) is a finite set for each x ∈ Xe, these conditions hold.

Remark 3.12. If the coalgebra C is finite-dimensional, then any localizing subcategory is colo-calizing.

Example 3.13. Let us show a colocalizing subcategory which is not perfect colocalizing. LetQ be the quiver '&%$ !"#2

α2

%%KKKKKK

'&%$ !"#1

α199ssssss

β1 %%KKKKKK '&%$ !"#4

'&%$ !"#3 β2

99ssssss

and C be the admissible subcoalgebra of KQ generated by

1,2,3,4,α1,α2,β1,β2,α2α1 +β2β1.

Let us consider the idempotent e such that Xe = 1,2,3. Then eCe is the path coalgebra of thequiver

'&%$ !"#2 '&%$ !"#1α1oo β1 //'&%$ !"#3

and then, the indecomposable injective right eCe-comodules are

E1 = K〈1〉, E2 = K〈2,α1〉 and E3 = K〈3,β1〉.

The eCe-comodule eC = eCe⊕ eC(1− e) is injective if and only if eC(1− e) is injective. IfeC(1− e) = K〈α2,β2,α2α1 + β2β1〉 were injective then it would be a sum of indecomposableinjective right eCe-comodules. Since eC(1− e) has dimension 3, it would be isomorphic toE1⊕E1⊕E1, or E1⊕E2, or E1⊕E3. But a straightforward calculation shows that this is notpossible.

156

4 SIMPLE COMODULES

In this section, we deal with the key point of the applications of localization to representationtheory, namely the behaviour of simple comodules through the action of the localizing functors.One of the reasons of why the simple comodules play such a prominent role is the fact that thesection functor does not preserve them.

Example 4.1. Let KQ be the path coalgebra of the quiver Q

'&%$ !"#1α //'&%$ !"#2

and e be the idempotent in (KQ)∗ associated to the set Xe = 2. Then, the localized coalgebrae(KQ)e is isomorphic to the simple comodule S1 and

S(S1) = S1e(KQ)e(KQ)e = e(KQ)ee(KQ)e(KQ)e ∼= (KQ)e ∼= 〈2,α〉 6= S1.

We may modify this example in order to show that the image of a simple comodule could haveinfinite dimension.

Example 4.2. Consider the quiver Q

αn // nαn−1 // α3 //'&%$ !"#3

α2 //'&%$ !"#2α1 //'&%$ !"#1

and the idempotent e ∈ (KQ)∗ associated to the set Xe = 1. Then

S(S1) = S1e(KQ)e(KQ)e = e(KQ)ee(KQ)e(KQ)e ∼= (KQ)e ∼= 〈1,α1 · · ·αn−1αnn≥1〉.

In fact, the reader may find in [16] a proof of the following fact: S preserves finite dimensionalcomodules if and only if S(Si) is finite dimensional for each simple eCe-comodule Si. Therefore,it is rather interesting to answer the following question:

Question 4.3. Which is the image of a simple comodule through the localizing functors?

From now on, we fix an idempotent element e ∈ C∗. We will denote by Sii∈Ie⊂IC the set ofsimple comodules of the quotient category and by E ii∈Ie the set of indecomposable injectiveeCe-comodules. Let us take into consideration the quotient and the section functor associatedto Te:

M CT=e(−)=−CeC //

M eCeS=−eCeCe

oo .

We recall that there exists a torsion theory on M C associated to the quotient functor T , where aright C-comodule M is a torsion comodule if T (M) = 0.Given a simple C-comodule Si, there are exactly two possibilities: on the one hand, if Si istorsion, then T (Si) = 0. And, on the other hand, if Si is torsion-free, then it is the socle ofa torsion-free indecomposable injective C-comodule Ei. By [23, Proposition 4.2], T (Ei) =E i. Therefore T (Si) is a eCe-subcomodule of E i contained in Si. Thus T (Si) = Si, see [17].Summarizing,

T (Si) =

Si if i ∈ Ie,0 if i /∈ Ie.

We remind from [23] and [31] the following properties of the section funtor:

157

Lemma 4.4. [23] The following properties hold:

(a) The functor S preserves injective, quasi-finite and indecomposable comodules.

(b) We have S(E i) = Ei for all i ∈ Ie. As a consequence, S preserves injective envelopes.

(c) The functor S : M eCe → M C restricts to a fully faithful functor S : M eCeq f → M C

q f betweenthe categories of quasi-finite comodules which preserves indecomposable comodules andrespects isomorphism classes.

(d) If Si is a simple eCe-comodule then soc S(Si) = Si.

Therefore, we may assert that S(Si) is a subcomodule of Ei which contains Si. Nevertheless, ingeneral, we cannot say anything else since it is easy to find examples of all possible cases:

Example 4.5. Let KQ be the path coalgebra of the quiver Q

'&%$ !"#1α //'&%$ !"#2 β

((RRRRRRR

'&%$ !"#4

'&%$ !"#3 γ

66lllllll

Let us consider the localizing subcategories associated to the following sets:

• If X = 4, then S4 $ S(S4) = 〈4,β,γ,βα〉= E4.

• If X = 1,2,4, then S4 $ S(S4) = 〈4,γ〉$ E4.

• If X = 2,3,4, then S4 = S(S4) $ E4.

These examples provide us a right intuition in order to study conditions to calculate S(Si) insome cases. Following them, it seems that the predecessors of the simple comodule Si play animportant role. The following Lemma agrees with this idea:

Lemma 4.6. [23, Corollary 4.3] Let Ei be a indecomposable injective C-comodule such that Siis torsion-free. S(Si) = Ei if and only if all predecessors of Si in the Gabriel-valued quiver (orin the Ext-quiver) (QC,dC) are torsion.

In opposition to the former Lemma, S preserves a simple comodule if all its 1-predecessors aretorsion-free. The following theorem is the main result of the section.

Theorem 4.7. [23, Theorem 3.7] Let S j and Si be two simple C-comodules. Then we haveS j ⊆ S(Si)/Si if and only if S j ⊆ Ei/Si and T (S j) = 0.

That is, the torsion 1-predecessors of a torsion-free vertex Si in (QC,dC) are the simple C-comodules contained in the socle of S(Si)/Si. In the following picture the torsion-free verticesare represented by white points.

???

????

??

•''OOOOOOO

//

77ooooooo

•

??

_____

_____

7w7w7w7w7w7w7w

$d$d

$d$d

$d$d

$d$d

Si

soc Ei/Si

soc S(Si)/Si

158

Corollary 4.8. Let Si be a simple eCe-comodule. The following conditions are equivalent:

(a) Ei/Si is torsion-free.

(b) There is no arrow in (QC,dC) from a torsion vertex S j to Si.

(c) S(Si) = Si.

We recall from [17] that an idempotent element e ∈C∗ is said to be right semicentral if Ce =eCe, or equivalently, if Ce is a subcoalgebra of C. In [23, Theorem 6.2] it is proven that e isright semicentral if and only if S(Si) = Si for any i ∈ Ie. Therefore we may prove the following:

Proposition 4.9. If e is right semicentral, then the Gabriel-valued quiver (QeCe,deCe) (resp.the Ext-quiver ΓeCe) of the coalgebra eCe is isomorphic to the restriction of the valued quiver(QC,dC) (resp. the Ext-quiver ΓC) of C to the subset Ie ⊆ IC.

Proof. By [23, Theorem 6.2], e is left semicentral if and only if any torsion-free vertex of(QC,dC) has no torsion predecessor. Indeed, there is no path in (QC,dC) from a torsion vertexto a torsion-free vertex. This yields that Ei/Si is E = Ce-copresented for each i ∈ IC. Thus, by[30, Theorem 3.2], the result follows.

Remark 4.10. The left semicentral idempotents are defined symmetrically. In [17], the authorsprove that e is left semicentral if and only there is no arrow from a a torsion-free vertex toa torsion vertex. These idempotents are also related to the behavior of injective and simplecomodules. In [23, Theorem 6.1] it is proven that the following statements are equivalent:

(a) e is left semicentral.

(b) T (Ei) =

E i if i ∈ Ie,0 if i /∈ Ie.

(c) Te is a stable subcategory.

If Te is also colocalizing, these are equivalent to:

(d) H(Si) = Si for any i ∈ Ie.

5 APPLICATIONS TO REPRESENTATION THEORY

One of the central points when dealing with Representation Theory of Algebras is to decide ifwe may really describe an algebra (of finite dimension over an algebraically closed field) soexhaustively as the main aim of this theory proposes (that is, describe explicitly its category offinitely generated modules). That problem leads to define two classes of algebras: tame algebrasand wild algebras. From that point of view, the category of finitely generated comodules overa wild algebra has very bad properties. This bad behavior means that this category is so bigthat it contains (via an exact representation embedding) the category of all finite dimensionalrepresentations of the noncommutative polynomial algebra K〈x,y〉. As it is well-known, thecategory of finite dimensional modules over K〈x,y〉 contains (again via an exact representationembedding) the category of all finitely generated representations for any other finite dimensionalalgebra, and thus it is not realistic aiming to give an explicit description of this category (or,by extension, of any wild algebra). The counterpart to the notion of wild algebra is the one

159

of tameness, a tame algebra being one whose indecomposable modules of finite dimensionare parameterized by a finite number of one-parameter families for each dimension vector. Aclassical result in representation theory of algebras (the Tame-Wild Dichotomy, see [10]) statesthat any finite dimensional algebra over an algebraically closed field is either of tame moduletype or of wild module type. We refer the reader to [27] for basic definitions and propertiesabout module type of algebras.Analogous concepts and results were defined by Simson in [28] and [29] for coalgebras. Weremind that given R a K-algebra. By a R-C-bimodule we mean a K-vector space L endowedwith a left R-module structure · : R⊗L → L and a right C-comodule structure ρ : L → L⊗Csuch that ρL(r · x) = r ·ρL(x). We denote by RM C the category of R-C-bimodules.Throughout this section, K will be an algebraically closed field. Following [28], a K-coalgebraC over an algebraically closed field K is said to be of tame comodule type (tame for short) iffor every v ∈ K0(C) there exist K[t]-C-bimodules L(1), . . . ,L(rv), which are finitely generatedfree K[t]-modules, such that all but finitely many indecomposable right C-comodules M withlength M = v are of the form M ∼= K1

λ⊗K[t] L(s), where s ≤ rv, K1

λ= K[t]/(t−λ) and λ ∈ K. If

there is a common bound for the numbers rv for all v ∈ K0(C), then C is called domestic.If C is a tame coalgebra, then there exists a growth function µ1

C : K0(C) → N defined as µ1C(v)

to be the minimal number rv of K[t]-C-bimodules L(1), . . . ,L(rv) satisfying the above conditions,for each v ∈ K0(C). C is said to be of polynomial growth if there exists a formal power series

G(t) =∞

∑m=1

∑j1,..., jm∈IC

g j1,..., jmt j1 . . . t jm

with t = (t j) j∈IC and non-negative coefficients g j1,..., jm ∈ Z such that µ1C(v) ≤ G(v) for all v =

(v( j)) j∈IC ∈ K0(C)∼= Z(IC) such that ‖v‖ := ∑ j∈IC v( j)≥ 2. If G(t) = ∑ j∈IC g jt j, where g j ∈N,then C is called of linear growth. If µ1

C is zero we say that C is of discrete comodule type.Let Q be the quiver // //// and KQ be the path algebra of the quiver Q. Let us denote by M f

KQthe category of finite dimensional right KQ-modules. A K-coalgebra C is of wild comoduletype (wild for short) if there exists an exact and faithful K-linear functor F : M f

KQ → M Cf that

respects isomorphism classes and carries indecomposable right KQ-modules to indecomposableright C-comodules.In [29], it was proven a weak version of the Tame-Wild Dichotomy that goes as follows: overan algebraically closed field K, if C is K-coalgebra of tame comodule type, then C is not of wildcomodule type. The full version remains open:

Conjecture 5.1. Any K-coalgebra, over an algebraically closed field K, is either of tame co-module type, or of wild comodule type, and these types are mutually exclusive.

Directly from the definition we may prove the following proposition.

Proposition 5.2. Let e ∈C∗ be an idempotent which defines a perfect colocalization. If eCe iswild then C is wild.

Proof. By hypothesis, there is an exact and faithful functor F : M fKQ → M eCe

f , where Q is thequiver // //// , which respects isomorphism classes and preserves indecomposables. There-fore, by [23, Proposition 5.1], it is enough to consider the composition

M fKQ

F // M eCef

H // M Cf

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A similar result may be obtained using the section functor if S preserves finite dimensionalcomodules (for instance, if C is left semiperfect).

Proposition 5.3. Let e∈C∗ be an idempotent which defines a perfect localization and such thatS(Si) is finite dimensional for each i ∈ Ie. If eCe is wild then C is wild.

The study of the tameness using the theory of localization strongly depends of the behaviorof simple comodules. The reason comes from the fact that it determines the behavior of thelength vector. It easy to prove (cf. [23]) that, for any C-comodule L whose length vectorlength L = (li)i∈IC ∈ Z(IC), the length vector of eL is length eL = (li)i∈Ie ∈ Z(Ie), that is, it is theimage of length L through the natural projection p : Z(IC) → Z(Ie). Nevertheless, as we haveshown in the previous section, for some eCe-comodule N, the length vector length S(N) mightnot be well-defined since S(N) could be infinite dimensional. Therefore it seems that, at least,one should assume that S(Si) is finite dimensional for any i ∈ Ie. In fact, by [16], this conditionis enough.

Theorem 5.4. [16, Theorem 5.9] Let e ∈C∗ be an idempotent such that S(Si) is a finite dimen-sional right C-comodule for all i ∈ Ie. If C is of tame (discrete) comodule type then eCe is oftame (discrete) comodule type.

In particular, if e is right semicentral the conditions of the theorem hold.

Corollary 5.5 (Corollary 5.6). [16] Let C be a coalgebra and e ∈ C∗ be a right semicentralidempotent. If C is tame (of polynomial growth, of linear growth, domestic, discrete) then eCeis tame (of polynomial growth, of linear growth, domestic, discrete).

The general problem still remains open.

Question 5.6. Assume that C is a coalgebra of tame comodule type and e is an idempotent inC∗. Is the coalgebra eCe of tame comodule type?

It would be interesting to know if the localization process preserves polynomial growth, lineargrowth, discrete comodule type or domesticity.It is clear that the converse result is not true as the following example shows.

Example 5.7. Let us consider the quiver Q

// // // // // // // // Since its underlying graph is neither a Dynkin diagram nor an Euclidean graph, KQ is wild, see[29, Theorem 9.4]. But it is easy to see that eCe is of tame comodule type for each non-trivialidempotent e ∈C∗.

A partial result is given in [31]. Let Si be a simple C-comodule, we define the set

I(I i) = S j simple C-comodule | there is a path in (QC,dC) from S j to Si

In general, for some subset U ⊆ IC, we set

I(I U) =[

x∈U

I(I x)

Observe that, for any subset U ⊆ IC, the idempotent eU associated to the set of simple comodulesI(I U) is right semicentral.

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Theorem 5.8. [31, Theorem 5.7] C is of tame comodule type if and only if eUCeU is of tamecomodule type for each finite subset U ⊆ IC.

We finish the paper giving a partial answer to Question 2.3. Firstly, the reader should note thatthe counterexample given in Example 2.4 is of wild comodule type in the sense described above.This is easy to see because of the coalgebra H contains the finite dimensional subcoalgebra KΓ,where Γ is the quiver

Γ ≡

77oooooo //

''OOOOOOwwoooooo

ggOOOOOO

Since, according to Theorem 2.5, the coalgebras which are not the path coalgebra of quiver withrelations are close to be wild (in the quiver it should exist an infinite number of paths betweentwo vertices), we may reformulate Question 2.3 (stated in [28, Section 8]) as follows:

Question 5.9. Assume that K is an algebraically closed field. Is any basic tame coalgebraisomorphic to the path coalgebra C(Q,Ω) of a quiver Q with a set of relations Ω?

We may give a positive answer if the Gabriel quiver associated C is acyclic. This extends [29,Theorem 3.14 (c)] from the case in which Q is intervally finite and C ⊆ KQ is an arbitraryadmissible subcoalgebra to the case in which Q is acyclic and C ⊆ KQ is tame.

Theorem 5.10 (Acyclic Gabriel’s theorem for coalgebras). Let Q be an acyclic quiver and letK be an algebraically closed field.

(a) Any tame admissible subcoalgebra C of the path coalgebra KQ is isomorphic to the pathcoalgebra C(Q,Ω) of a quiver with relations (Q,Ω).

(b) The map Ω 7→ C(Q,Ω) defines a one-to-one correspondence between the set of relationideals Ω of the path K-algebra KQ and the set of admissible subcoalgebras H of the pathcoalgebra KQ. The inverse map is given by H 7→ H⊥.

Proof. We only sketch the proof, see [16] for details. By Theorem 2.5, if C is not the pathcoalgebra of a quiver with relations, there exist an infinite number of paths γii between twovertices x and y of Q, and elements σi = ai

1γ1 + · · ·+ainγn in C for i = 1,2,3. Let us consider the

subcoalgebra H = C∩KΓ, where Γ is the quiver formed by the paths γ1, . . . ,γn. Then H is anadmissible subcoalgebra of KΓ. Let e be the idempotent in H∗ associated to the set Xe = x,y.Then eHe is the path coalgebra of the quiver // //// , and therefore eHe is wild. Since Te is aperfect colocalizing subcategory of M H , by Proposition 5.2, H is wild. By, [29, Theorem 5.4],C is also wild. This contradicts the assumptions.

We hope that the general version of Theorem 5.10 holds.

Conjecture 5.11. Any basic tame coalgebra, over an algebraically closed field, is isomorphicto the path coalgebra of a quiver with relations.

ACKNOWLEDGEMENT

This research was supported by the Spanish MEC project MTM2004-01406.

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