SIMPLE COMODULES AND LOCALIZATION IN COALGEBRAS. Gabriel Navarro

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1 SIMPLE COMODULES AND LOCALIZATION IN COALGEBRAS Gabriel Navarro Department of Algebra, University of Granada, Avda. Fuentenueva s/n, E-18071, Granada, Spain Abstract In this article we review recent developments in representation theory of coalgebras, aiming for an extension of the classical theory for artinian algebras. The key tool is the use of the theory of localization in categories of comodules and, in particular, the behaviour of simple comodules through the action of the section functor. For that reason, a description of the localization in coalgebras is given. INTRODUCTION The representation theory of artinian algebras is a classical and fruitful theory that has provided us with many tools and results for years, see [2] and [3]. Nevertheless, due to its ambitious primary aim, that is, to describe, as a category, the (finite dimensional) modules over any artinian algebra, many of these techniques strongly require finite dimensionality over the field and it does not seem possible to generalize them to an arbitrary algebra. Recently, some authors have tried to get rid of the imposed finiteness conditions by taking advantage 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 of the freedom on choosing their dimension, as an intermediate step between finite dimensional and infinite dimensional algebras. More concretely, in [28], it is proven that the category of comodules over a coalgebra is equivalent to the category of pseudo-compact modules, in the sense 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 finite dimensional algebras. Therefore it is rather natural to discuss the development of the following points in coalgebra theory: 1. Some quiver techniques similar to the classical ones stated for algebras. For instance, a description of coalgebras and comodules by means of quivers and linear representation of 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 coalgebra (cf. [6] and [7]). 3. Define the representation types of comodules (finite, tame and wild) and find some criteria in order to determine the representation type of a coalgebra. Prove the tame-wild dichotomy for coalgebras (cf. [28] and [29]). 141

2 4. Develop a (co)tilting theory for coalgebras in order to describe coalgebras whose global dimension 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 (following the general ideas of Gabriel [11] for abelian categories) to the first and the third point described above. It is worth mentioning that the key point of most of such applications is the behaviour of 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 selfcontained and elementary, we collect some background notation and basic facts about coalgebras, localization and quivers. Section 2 is devoted to study some properties of a coalgebra by means of the geometry of some quivers associated to it, the case of a pointed coalgebra being of particular interest, since we may embed it into a path coalgebra. In Section 3 we describe the localization in path coalgebras in order to give examples of the theory. In Section 4 we study how the section functor transforms simple comodules. In particular, we give conditions in order to 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, we hihlight a theorem of Gabriel for coalgebras in the case where the quiver we are dealing with is acyclic. 1 PRELIMINARIES Throughout we fix a ground field K and we assume that all vector spaces are over K and every map 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, all C-comodules are right C-comodules. It is well-known that C has a decomposition, as right C-comodule, C C = M i I C E t i i, where {E i } i IC is a complete set of pairwise non-isomorphic indecomposable injective right C- comodules and t i is a positive integer for any i I C. This produces a decomposition of the socle of C (the sum of all its simple subcomodules), soc C, as follows: soc C = M i I C S t i i, where {S i } i IC is a complete set of pairwise non-isomorphic simple right C-comodules. It is easy to prove that dim K S i t i = dim K (End C (S i )) for any i I C, see [29]. For any right C-comodule M, we denote by soc M the socle of M and by E(M) its injective envelope. We assume that soc E i = S i, and, consequently, E(S i ) = E i, for each i I C. The coalgebra C is said to be basic if t i = 1 for any i I C, or, equivalently, if dim K S i = dim K (End C (S i )) for any i I C, or, equivalently, if S i is a simple subcoalgebra of C for any 142

3 i I C, see for instance [31]. In particular, C is called pointed if dim K S i = 1 for any i I C. Here we show that not every basic coalgebra is pointed: Example 1.1. Let C be the R-vector space of the complex numbers with comultiplication and counit given by the following formulae: (1) = 1 1 i i ; (i) = i 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 are equivalent) to a basic one [28, Proposition 5.6], throughout we assume that C is basic and there are decompositions C = M i I C E i and soc C = M i I C S i, where E i E j and S i S j for i j. Symmetrically, there exists the left-hand version of all the facts explained above. In particular, C admits a decomposition as left C-comodule CC = M i I C F i and soc C = M i I C S i. Remark 1.2. Observe that S i = S i for any i I C, since C is basic and therefore each simple (left or right) C-comodule is a simple subcoalgebra. Nevertheless, the right injective envelope E i and the left injective envelope F i of S i could be different. Following [28], for every finite dimensional right C-comodule M we consider the length vector of M, length M = (m i ) i IC Z (IC), where m i N is the number of simple composition factors of M isomorphic to S i. In [28] it is proven that the map M length M extends to a group isomorphism K 0 (C) Z (IC), where K 0 (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 E m i i satisfies that m i is a finite non-negative integer for any i I C. It is easy to prove that this is equivalent to the fact that dim K Hom C (S i,m) is finite for each i I C. Let D be another coalgebra, by [33], a (D,C)-bicomodule M is quasi-finite as right C-comodule if and only if the cotensor functor D M (see [33] for its definition) has a left adjoint functor known as the Cohom functor. In such a case, we denote it as Cohom C (M, ). There it is proven that Cohom C (M,N) = lim λ Hom C (N λ,m), where {N λ } λ is the set of all finite dimensional subcomodules of N and ( ) is the standard duality Hom K (,K). The functor Cohom C (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 the cotensor functor D M preserves direct sums and is left exact, and it is right exact as well when M is injective as left D-comodule. Throughout we denote by M C f, M C q 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 M 1 M M

4 in M C satisfies that M belongs to T if and only if M 1 and M 2 belong to T. Following [11] and [25], for any dense subcategory T of M C, there exists an abelian category M C /T and an exact functor T : M C M C /T, such that T (M) = 0 for each M T, satisfying the following universal 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 is called 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]) if the 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 be colocalizing if T has a left adjoint functor H, called the colocalizing functor. T is said to be perfect 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 a coalgebra 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 1 M C /T holds. (c) If T is colocalizing, then the colocalizing functor H is right exact and the equivalence T H 1 M C /T holds. From the general theory of localization in Grothendieck categories [11], it is well-known that there exists a one-to-one correspondence between localizing subcategories of M C and sets of indecomposable 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 J E j, where J I C (therefore the associated set of indecomposable injective comodules is {E j } j J ). Then M C /T M D, where D is the coalgebra of coendomorphism Cohom C (E,E), and the quotient and section functors are Cohom C (E, ) and D E, respectively. By [31], the quotient and the section functors define an equivalence of categories between M D and the category ME C of E-copresented right C-comodules, that is, the right C-comodules M which admit an exact sequence 0 M E 0 E 1, where E 0 and E 1 are direct sums of direct summands of the comodule E. In [8], [17] and [36], localizing subcategories are described by means of idempotents in the dual algebra C. In particular, it is proven that the quotient category is the category of right comodules over the coalgebra ece, where e is an idempotent associated to the localizing subcategory (that is, E = Ce, where E is the injective right C-comodule associated to the localizing subcategory). The coalgebra structure of ece (cf. [26]) is given by ece (exe) = ex (1) e ex (2) e and ε ece (exe) = ε C (x) (x) for any x C, where C (x) = (x) x (1) x (2) using the sigma-notation of [32]. Throughout we denote by T e the localizing subcategory associated to the idempotent e. For completeness, we remind from [8] (see also [17]) the following description of the localizing functors. We recall that, given an idempotent e C, for each right C-comodule M, the vector space em is endowed with a structure of right ece-comodule given by ρ em (ex) = ex (1) ex (0) e (x) where ρ M (x) = (x) x (1) x (0) using the sigma-notation of [32]. 144

5 Lemma 1.4. Let C be a coalgebra and e be an idempotent in C. Then the following statements hold: (a) The quotient functor T : M C M ece is naturally equivalent to the functor e( ). T is also naturally equivalent to the cotensor functor C ec and the Cohom functor Cohom C (Ce, ). (b) The section functor S : M ece M C is naturally equivalent to the cotensor functor ece Ce. As a consequence, T is perfect localizing if and only if Ce is injective as left ece-comodule. (c) If T e is a colocalizing subcategory of M C, the colocalizing functor H : M ece M C is naturally equivalent to the functor Cohom ece (ec, ). As a consequence, T is perfect colocalizing 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: T =e( )= C ec T =e( )= M C C ec M ece and M C M ece. S= ece Ce H=Cohom ece (ec, ) Lastly, for completeness, we remind some points about quivers and path (co)algebras. Following Gabriel [12], by a quiver, Q, we mean a quadruple (Q 0,Q 1,h,s) where Q 0 is the set of vertices (points), Q 1 is the set of arrows and, for each arrow α Q 1, 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 formal composition of arrows p = α m α 2 α 1 where h(α 1 ) = i, s(α m ) = j and s(α k 1 ) = h(α k ), for k = 2,...,m. To any vertex i Q 0 we attach a stationary path of length 0, say e i, starting and ending at i such that αe i = α (resp. e i β = β) for any arrow α (resp. β) with h(α) = i (resp. s(β) = i). We identify the set of vertices and the set of stationary paths. An (oriented) cycle is a path in Q which starts and ends at the same 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 endowed with the structure of a (non necessarily unitary) K-algebra with multiplication induced by concatenation of paths, that is, (α m α 2 α 1 )(β n β 2 β 1 ) = { αm α 2 α 1 β n β 2 β 1 if s(β n ) = h(α 1 ), 0 otherwise; KQ is the path algebra of the quiver Q. The algebra KQ can be graded by KQ = KQ 0 KQ 1 KQ m, where Q m is the set of all paths of length m. An ideal Ω KQ is called an ideal of relations or a relation ideal if Ω KQ 2 KQ 3. We denote by KQ m the ideal of KQ generated by the paths of length greater or equal than m. By a quiver with relations we mean a pair (Q,Ω), where Q is a quiver and Ω a relation ideal of KQ. For more details and basic facts from representation theory of algebras the reader is referred to [2] and [3]. 145

6 Following [36], the path algebra KQ can be viewed as a graded K-coalgebra with comultiplication induced by the decomposition of paths, that is, if p = α m α 1 is a path from the vertex i to the vertex j, then m 1 (p) = e j p + p e i + i=1 α m α i+1 α i α 1 = η τ ητ=p and for a stationary path, e i, we have (e i ) = e i e i. The counit of KQ is defined by the formula { 1 if α Q0, ε(α) = 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 subcoalgebra C of a path coalgebra KQ of a quiver Q is said to be admissible if it contains the subcoalgebra 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 = (X i,ϕ α ) i Q0,α Q 1, where X i is a K-vector space for each i Q 0 and ϕ α : X i X j is a K-linear map for any α : i j. Given two K-linear representations of Q, (X i,ϕ α ) and (Y i,ψ α ), a morphism f : (X i,ϕ α ) (Y i,ψ α ) of representations of Q is a system f = ( f i ) i Q0 of K-linear maps f i : X i Y i for any i Q 0 such that, for any α : i j in Q 1, the following diagram is commutative ϕ α X i X j f i Y i ψ α Y j We denote by Rep K (Q) the Grothendieck K-category of K-linear representations of Q, and by Rep l f K (Q) the full Grothendieck K-subcategory of Rep K (Q) formed by locally finite-dimensional representations, that is, directed unions of representations of finite length. A linear representation X of Q is said to be of finite length if X i is a finite dimensional vector space for all i Q 0 and X i = 0 for almost all indices i. Finally, we denote by rep K (Q) rep l f K (Q) the full subcategories of Rep K (Q) formed by finitely generated representations and representations of finite length, respectively. A linear representation X of Q is nilpotent if there exists an integer m 1 such that the composed linear map ϕ α1 ϕ α2 ϕ X i0 X i1 X i2 α X m im 1 X im is zero for any path α m α m 1 α 1 in Q of length m. We denote by nilrep l f K (Q) the full subcategory of rep K (Q) formed by all nilpotent representations of finite length, and by Rep lnl f (Q) the full subcategory of Rep K (Q) of all locally nilpotent representations that are locally finite, that is, directed unions of representations from nilrep l f K (Q). Given a quiver with relations (Q,Ω), a linear representation of (Q,Ω) is a linear representation X = (X i,ϕ α ) of Q which verifies that if p = n i=1 λ iα i m i α i 1 is in Ω then n i=1 λ iϕ α i mi ϕ α i = 0. 1 As above, we may define the categories Rep K (Q,Ω), Rep l f K (Q,Ω), rep f K (Q,Ω), repl f (Q,Ω) and nilrepl (Q,Ω), see [28], [29] or [36] for details. Rep lnl f K K 146 f j K (Q,Ω),

7 2 QUIVERS ASSOCIATED TO A COALGEBRA Associating a graphical structure to a certain mathematical object is a very common strategy in the literature. Sometimes, it provides us a nice method for replacing the object with a simpler one and improving our intuition about its properties. In our case, the quiver-theoretical ideas developed by Gabriel and his school during the seventies have been the origin of many advances in Representation Theory of Algebras for years. Moreover, many of the present developments of the theory use, up to some extent, these techniques and results. Among these tools, it is mostly 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 finite dimensional algebra A is isomorphic to a quotient KQ A /Ω, where Q A is the Gabriel quiver of A, and Ω is an ideal of KQ A such that K(Q A ) n Ω K(Q A ) 2 for some integer n 2. Moreover, there exists a K-linear equivalence of categories F : M A Rep K (Q,Ω) between the category of right A-modules and the category of linear representations of the quiver with relations (Q,Ω). This equivalence restricts to an equivalence F : M f A rep K (Q,Ω) between the category of finitely generated right A-modules and the one of finite dimensional linear 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 go beyond the classical ideas for algebras, therefore, it is a natural question to ask about obtaining a Gabriel theorem for pointed (basic, if K is algebraically closed) coalgebras in order to classify them according their category of comodules. By [36], to any pointed coalgebra C we may associate its so-called Gabriel quiver Q C as follows: Vertices: The set of vertices (Q C ) 0 is the set of group-like elements of C, i.e., the set of elements c C such that (c) = c c. Observe that this is also the set of simple subcoalgebras (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: 147

8 Theorem 2.2. [36] Let C be a pointed coalgebra. Then C is isomorphic to an admissible subcoalgebra of the path coalgebra of its Gabriel quiver. Proof. The proof of the theorem is based on the universal property of the cotensor coalgebra given by Nichols in [24, Proposition 1.4.2]. Choosing suitable coalgebras, bicomodules and morphisms, this yields that any pointed coalgebra C is a subcoalgebra of the cotensor coalgebra CT C0 (M) = C 0 M (M C M) (M C M C M), where C 0 is the coradical of C (that is, the vector space generated by the group-like elements) and M = C 1 /C 0, where C 1 is the piece of degree one of the coradical filtration of C. Since M is the vector space of non-trivial primitive elements of C, it is easy to see that CT C0 (M) = KQ C and the result follows. Therefore, by the previous theorem, when studying pointed coalgebras, we may bound the attention to path coalgebras and its subcoalgebras. Then, the manipulation of the elements of a pointed 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 approximation by means of the notion of path coalgebra of a quiver with relations. Indeed, let (Q,Ω) be a quiver 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 = δ v,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 of the ideal Ω contained in the path algebra (non necessarily unitary) of Q, see [1] or [15]. We may show the situation by means of the following picture (for clarity, here we denote by CQ the path coalgebra of Q): CQ I C C KQ ={p KQ C, p = 0} Ω =C(Q,Ω) KQ C KQ (CQ) This definition is congruent with the classical theory for finite dimensional algebras since there is a K-linear equivalence of the category M f C of finite dimensional right C-comodules with the category nilrep l f (Q,Ω) of nilpotent K-linear representations of finite length of the quiver K with relations (Q,Ω) (see [28, p. 135] and [29, Theorem 3.14]). Then Simson [28] raises the following question: Question 2.3. Is any admissible subcoalgebra C of a path coalgebra KQ isomorphic to the path coalgebra C(Q,Ω) of a quiver with relations (Q,Ω)? 148

9 Unfortunately, the answer is negative as the following counterexample shows: Example 2.4. [15] Let Q be the quiver α 1 α β 1 2 β 2 α n β n γ α i i = β i α i for all i N β i and let H be the relation subcoalgebra of KQ generated (as vector space) by the set of vertices, the set of arrows and Σ = {γ i γ i+1 } i N. Assume that x = i 1 a i γ i belongs to H and a i = 0 for i n we have some n N. Then γ i γ i+1,x = a i a i+1 = 0 for all i N, so a i = a i+1 for all i N. But a n = 0 and it follows that x = 0. Hence H KQ = 0. By a similar argument H = f, where f (γ i ) = 1 for all i N. That is, f i 1 γ i. 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 coalgebra of a quiver with relations is given: Theorem 2.5. Let C be an admissible subcoalgebra of a path coalgebra KQ. Then C is not the path coalgebra of a quiver with relations if and only if there exist an infinite number of different paths {γ i } i 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 a n j K for all j,n N such that the set {γ n + j>n a n j γ j} n N is contained in C. As a consequence, if Q is intervally finite, i.e., there is at most a finite number of paths between any two vertices, we may give a positive answer to Question 2.3 (cf. [29, Theorem 3.14]). In Section 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 associate to C a quiver Γ C known as the right Ext-quiver of C, see [21]. We recall that the set of vertices of Γ C is the set of pairwise non-isomorphic simple right C-comodules {S i } i IC and, for two vertices S i and S j, there exists a unique arrow S j S i in Γ C if and only if ExtC 1 (S j,s i ) 0. This quiver admits a generalization by means of the notion of right Gabriel-valued quiver of C, (Q C,d C ), i. e., following [18], the valued quiver whose set of vertices is {S i } i IC and such that there exists a unique valued arrow S j (d 1 ji,d2 ji ) S i if and only if ExtC 1 (S j,s i ) 0 and d 1 ji = dim EndC (S i )ExtC 1 (S j,s i ) and d 2 ji = dim EndC (S j )ExtC 1 (S j,s i ), as left End C (S j )-module and as right End C (S i )-module, respectively. 149

10 The (non-valued) Gabriel quiver of C is obtained taking the same set of vertices and the number of arrows from a vertex S j to a vertex S i is dim EndC (S j )Ext 1 C (S j,s i ) as right End C (S j )-module. Obviously, if C is pointed (or K is algebraically closed) then it is isomorphic to the one used by Woodcock 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 of them. Throughout we assume that these quivers are connected, i.e., C is indecomposable as coalgebra. 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 us remind that, by [13], any right C-comodule M has a filtration 0 soc M soc 2 M M called the Loewy series, where, for n > 1, soc n M is the unique subcomodule of M satisfying that soc n 1 M soc n M and soc n ( ) M M soc n 1 M = soc soc n 1. M Given two simple right C-comodules S i and S j, we say that the vertex S j is an n-predecessor of S i if Ext 1 C (S j,soc n E i ) 0 or, equivalently, if S j soc (E i /soc n E i ) = soc n+1 E i /soc n E i. The following result is proven in [23]: Lemma 2.6. Let S i and S j be two simple C-comodules. The following assertions are equivalent: (a) S j is a n-predecessor of S i. (b) There exists a non-zero morphism f : soc n+1 E i E j such that f (soc n E i ) = 0. (c) There exists a morphism g : E i E j such that g(soc i E i ) = 0 for all i = 1,...,n and g(soc n+1 E i ) 0 Observe that the set of 1-predecessors of a vertex S i is in one-to-one correspondence with the set of vertices S j such that there is an arrow S j S i in (Q C,d C ), that is, the set of vertices S j such that there is a path of length one in (Q C,d C ) from S j to S i. This correspondence fails for n-predecessors and paths of length n, when n > 1. Example 2.7. Let us consider the quiver Q 1 α 2 β 3 and the subcoalgebra C of KQ generated, as vector space, by {1,2,3,α,β}. Then the Ext-quiver Γ C is S 1 S 2 S 3. Obviously, there is a path from S 1 to S 3, but there is no non-zero morphisms f : E 3 =< 3,β > E 1 =< 1 >. On the other hand, if C is the coalgebra KQ, the Ext-quiver of KQ is also the previous quiver but, in this case, we may obtain a map defined by f (βα) = 1 and zero otherwise. f : E 3 =< 3,β,βα > E 1 =< 1 > 150

11 The fact that the former path coalgebra satisfies the bijective correspondence between n-predecessors and paths of length n is not a happy accident. Using Lemma 2.6, the following result relates the paths in the Gabriel quiver of C and the morphisms between indecomposable injective C- comodules. Theorem 2.8. [23, Theorem 2.9] Let S i and S j be two simple C-comodules and n be a positive integer. If S j is an n-predecessor of S i then there exists a path in (Q C,d C ) of length n from S j to S i. 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 second statement admits a weaker version if we only assume that all non-zero morphisms between indecomposable injective comodules are surjective. In such a case, we may only prove that if there is a path in (Q C,d C ) of length n from S j to S i, then S j is a t-predecessor of S i for some integer t n. The approach of [30] differs slightly. Given two indecomposable injective comodules E i and E j, the radical of Hom C (E i,e j ) is the K-subspace rad C (E i,e j ) of Hom C (E i,e j ) generated by all non-isomorphisms. Observe, that if i j, then rad C (E i,e j ) = Hom C (E i,e j ). Also, by Lemma 2.6, if rad C (E i,e j ) 0 then S j is a predecessor of S i. The square of rad C (E i,e j ) is defined to be the K-subspace rad 2 C (E i,e j ) rad C (E i,e j ) Hom C (E i,e j ) generated by all composite homomorphisms of the form E i f E k g E j, where f rad C (E i,e k ) and g rad C (E k,e j ). The mth power rad m C (E i,e j ) of rad C (E i,e j ) is defined analogously, for each m > 2. Then it is proven the following theorem: Theorem [30, Theorem 2.3] (a) There exists a unique valued arrow S j (d 1 ji,d2 ji ) S i in (Q C,d C ) if and only if the End C (S j )-End C (S i )-bimodule Irr C (E i,e j ) = rad C (E i,e j )/rad 2 C (E i,e j ) is nonzero and d 1 ji = dim EndC (S i )Irr C (E i,e j ) and d 2 ji = dim EndC (S j )Irr C (E i,e j ) (b) If f Hom C (E i,e j ) is a nonzero and noninvertible morphism, then there exists an integer m > 0 such that f radc m(e i,e j )\radc m+1 (E i,e j ). As a consequence, there is a sequence of morphisms f 1 E 1 = E i f 2 E2 f m E3 E m E m+1 = E j such that f k Irr C (E k,e k+1 ) for each k = 1,...,m and the composition f m f 1 is nonzero. 151

12 Remark Following the Auslander-Reiten theory for finite dimensional algebras, the notion of 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 : E i E j is an irreducible morphism if f is not an isomorphism and given a factorization E i g f E h of f, where E is a injective comodule whose socle is finite dimensional, f is a section, or f is a retraction. Analogously to the case of finite-dimensional algebras, there it is proven that the set of irreducible morphism Irr C (E i,e j ) is isomorphic, as End C (S j )-End C (S i )-bimodule, to the quotient rad C (E i,e j )/rad 2 C (E i,e j ). E j 3 EXAMPLES OF LOCALIZATION IN COALGEBRAS In this section we describe the localization in pointed coalgebras stressing the case of path coalgebras. We provide a wide range of examples for the convenience of the reader. The results of this section could be found in [15], [17] and [23]. Let Q = (Q 0,Q 1 ) be a quiver and KQ the path coalgebra of Q. Since the localizing subcategories of M KQ are parameterized by subsets of simple KQ-comodules, these are in one-to-one correspondence with subsets of the set of vertices Q 0. On the other hand, let e be an idempotent in (KQ). For each vertex x Q 0, 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 (induce the same localizing subcategory) if and only if f Q0 = g Q0. So we may assume that the idempotent e verifies that e(p) = 0 for each path of length greater than zero. Summarizing, the subsets of vertices of Q 0 (i.e., localizing subcategories of M KQ ) are in one-to-one correspondence with the 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 Q 0, we consider the idempotent e X in (KQ) given by { 1 if p X Q0 e X (p) = 0 otherwise for each path p in Q. And, conversely, given e (KQ) idempotent, we consider the set X e = {x Q 0 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) = r e(t) = p e(x) = p=rt { p, if x Xe 0, otherwise That is, e(kq) is generated by the paths starting at vertices in X e. Symmetrically, (KQ)e is generated by the paths ending at vertices in X e. Therefore the localized coalgebra e(kq)e is generated by the paths starting and ending at vertices in X e. Taking into account the coalgebra structure of e(kq)e given in the Introduction we may prove the following theorem. For simplicity, we introduce the following notation: let Q be a quiver and p = α n α n 1 α 1 be a non-stationary path in Q. We denote by I p the set of vertices {h(α 1 ),s(α 1 ),s(α 2 ),...,s(α n )}. 152

13 Given a subset of vertices X Q 0, we say that p is a cell in Q relative to X (shortly a cell) if I p X = {h(p),s(p)} and s(α i ) / X for all i = 1,...,n 1. Given x,y X, we denote by Cell Q X (x,y) the set of all cells from x to y. We denote the set of all cells in Q relative to X by Cell Q X. Theorem 3.1. [17, Theorem 3.1] Let Q be a quiver, KQ the path coalgebra of Q and e an idempotent in (KQ). Then the localized coalgebra e(kq)e is isomorphic to the path coalgebra of the quiver Q e, where (Q e ) 0 = X e, and For each two vertices x,y X e, the number of arrows from x to y in Q e is the cardinal of the set Cell Q X e (x,y). Example 3.2. Let KQ be the path coalgebra of the quiver Q given by α γ µ 1 β 1 β 2 η µ 2 δ ρ and X e be the set formed by the white points. Then, the set of cells is Therefore the quiver Q e is the following: {α,η,ρ,δβ 1,δβ 2,µ 1 γ,µ 2 γ}. α δβ 1 µ 1 γ δβ 2 η µ 2 γ ρ 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 α β and X e be the set formed by the white point. Then the set of cells is {βα}, that is, the quiver Q e is βα and e(kq)e = K[βα], i.e, it is the polynomial coalgebra. 153

14 Example 3.4. Let KQ be the path coalgebra of the quiver 1 α 3 2 β and e (KQ) be the idempotent associated to the set X e = {1,2}. Then e(kq)e is the path coalgebra of the quiver formed by two isolated points 1 2 Observe that T maps the indecomposable injective right C-comodule E 3 =< 3,α,β > to the right ece-comodule S 1 S 2. Thus the functor T does not preserve indecomposable comodules. Following [16], we may generalize the former proposition to an arbitrary pointed coalgebra as follows: Proposition 3.5. [16, Proposition 3.2] Let C be an admissible subcoalgebra of the path coalgebra KQ of a quiver Q. Let e be an idempotent in C. Then the localized coalgebra ece is an admissible subcoalgebra of the path coalgebra KQ e, where Q e is the quiver described as follows: The set of vertices (Q e ) 0 = X e and, For each x,y X e, the number of arrows from x to y in Q e is dim K KCell Q X e (x,y) C. Example 3.6. Let Q be the quiver α 2 1 α β 1 3 β 2 and C be the admissible subcoalgebra of KQ generated, as vector space, by {1,2,3,4,α 1,α 2,β 1,β 2,α 2 α 1 + β 2 β 1 }. Let us consider e the idempotent associated to the set X e = {1,3,4}. Then ece is the path coalgebra of the quiver Q e 1 β 1 3 Here, the element α 2 α 1 + β 2 β 1 corresponds to the composition of the arrows β 1 and β 2 of Q e. On the other hand, if C = KQ, the localized quiver, say now Q e, is the following β β 1 β α 2 α 1 3 β 2 4 Remark 3.7. In general, if C is a proper admissible subcoalgebra of a path coalgebra KQ and X Q 0, then Q e (the quiver associated to ece) is a subquiver of Q e (the quiver associated to e(kq)e) and the differences appear in the set of arrows. 154

15 Let us introduce the following notation. Let Q be a quiver and X Q 0. We say p = α n α 2 α 1 is a h(p)-tail in Q relative to X if I p X = {h(p)} and s(α i ) / X for all i = 1,...,n. If there is no confusion we simply say that p is a tail. Given a vertex x X, we denote by T ail Q X (x) the set of all x-tails in Q relative to X. In [17], it is proven that e(kq) = M ( E Card T ail Q )+1 Xe (x) x, x X e where {E x } x Xe is a complete set of pairwise non-isomorphic indecomposable injective right e(kq)e-comodules. Therefore e(kq) is always injective and is quasi-finite if and only if T ail Q X e (x) is a finite set for any x X e. 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 the following conditions are equivalent: (a) The localizing subcategory T e of M KQ is colocalizing. (b) The localizing subcategory T e of M KQ is perfect colocalizing. (c) T ail Q X e (x) is a finite set for all x X e. Example 3.9. Consider the quiver Q α 1 α 2 1 α 3 where i N α i and the subset X = {1}. Then T ail Q X (1) = {α i} i N is an infinite set and the localizing subcategory T X is not colocalizing. Example Let KQ be the path coalgebra of the quiver Q α δ β ρ γ and X be the set formed by the white point. Then Therefore T X is a colocalizing subcategory. T ail Q X ( ) = {α,β,γ,δ}. 155

16 In [16] it is proven an extension of the previous Theorem to any pointed (admissible) coalgebra C KQ. Let x Q 0, M be right C-comodule and f a linear map in Hom C (S x,m). Then ρ M f = ( f I) ρ Sx, where ρ M and ρ Sx are the structure maps of M and S x as right C-comodules, respectively. In order to describe f, since S x = Kx, it is enough to choose the image for x. Suppose that f (x) = m M. Since (ρ M f )(x) = (( f I)ρ Sx )(x), we obtain that ρ M (m) = m x. Therefore M x := Hom C (S x,m) = {m M such that ρ M (m) = m x}, as K-vector spaces, and M is quasi-finite if and only if M x has finite dimension for all x Q 0. By [16], for each x X e, (ec) x = KT ail Q X e (x) C Therefore, the following Proposition holds: Proposition Let C be an admissible subcoalgebra of a path coalgebra KQ and e be an idempotent in C. The following assertions are equivalent: (a) The localizing subcategory T e of M C is colocalizing. (b) ec is a quasifinite right ece-comodule. (c) dim K KT ail Q X e (x) C is finite for all x X e. In particular, if T ail Q X e (x) is a finite set for each x X e, these conditions hold. Remark If the coalgebra C is finite-dimensional, then any localizing subcategory is colocalizing. Example Let us show a colocalizing subcategory which is not perfect colocalizing. Let Q be the quiver α 2 1 α β 1 3 β 2 and C be the admissible subcoalgebra of KQ generated by {1,2,3,4,α 1,α 2,β 1,β 2,α 2 α 1 + β 2 β 1 }. Let us consider the idempotent e such that X e = {1,2,3}. Then ece is the path coalgebra of the quiver α and then, the indecomposable injective right ece-comodules are β 1 3 E 1 = K 1, E 2 = K 2,α 1 and E 3 = K 3,β 1. The ece-comodule ec = ece ec(1 e) is injective if and only if ec(1 e) is injective. If ec(1 e) = K α 2,β 2,α 2 α 1 + β 2 β 1 were injective then it would be a sum of indecomposable injective right ece-comodules. Since ec(1 e) has dimension 3, it would be isomorphic to E 1 E 1 E 1, or E 1 E 2, or E 1 E 3. But a straightforward calculation shows that this is not possible. 156

17 4 SIMPLE COMODULES In this section, we deal with the key point of the applications of localization to representation theory, 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 the section functor does not preserve them. Example 4.1. Let KQ be the path coalgebra of the quiver Q 1 α 2 and e be the idempotent in (KQ) associated to the set X e = {2}. Then, the localized coalgebra e(kq)e is isomorphic to the simple comodule S 1 and S(S 1 ) = S 1 e(kq)e (KQ)e = e(kq)e e(kq)e (KQ)e = (KQ)e = 2,α = S 1. We may modify this example in order to show that the image of a simple comodule could have infinite dimension. Example 4.2. Consider the quiver Q α n n αn 1 α 3 3 α 2 2 α 1 1 and the idempotent e (KQ) associated to the set X e = {1}. Then S(S 1 ) = S 1 e(kq)e (KQ)e = e(kq)e e(kq)e (KQ)e = (KQ)e = 1,{α 1 α n 1 α n } n 1. In fact, the reader may find in [16] a proof of the following fact: S preserves finite dimensional comodules if and only if S(S i ) is finite dimensional for each simple ece-comodule S i. 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 {S i } i Ie I C the set of simple comodules of the quotient category and by {E i } i Ie the set of indecomposable injective ece-comodules. Let us take into consideration the quotient and the section functor associated to T e : T =e( )= C ec M C M ece. S= ece Ce We recall that there exists a torsion theory on M C associated to the quotient functor T, where a right C-comodule M is a torsion comodule if T (M) = 0. Given a simple C-comodule S i, there are exactly two possibilities: on the one hand, if S i is torsion, then T (S i ) = 0. And, on the other hand, if S i is torsion-free, then it is the socle of a torsion-free indecomposable injective C-comodule E i. By [23, Proposition 4.2], T (E i ) = E i. Therefore T (S i ) is a ece-subcomodule of E i contained in S i. Thus T (S i ) = S i, see [17]. Summarizing, { Si if i I T (S i ) = e, 0 if i / I e. We remind from [23] and [31] the following properties of the section funtor: 157

18 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 ) = E i for all i I e. As a consequence, S preserves injective envelopes. (c) The functor S : M ece M C restricts to a fully faithful functor S : Mq ece f Mq C f between the categories of quasi-finite comodules which preserves indecomposable comodules and respects isomorphism classes. (d) If S i is a simple ece-comodule then soc S(S i ) = S i. Therefore, we may assert that S(S i ) is a subcomodule of E i which contains S i. Nevertheless, in general, 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 1 α 2 β 4 3 γ Let us consider the localizing subcategories associated to the following sets: If X = {4}, then S 4 S(S 4 ) = 4,β,γ,βα = E 4. If X = {1,2,4}, then S 4 S(S 4 ) = 4,γ E 4. If X = {2,3,4}, then S 4 = S(S 4 ) E 4. These examples provide us a right intuition in order to study conditions to calculate S(S i ) in some cases. Following them, it seems that the predecessors of the simple comodule S i play an important role. The following Lemma agrees with this idea: Lemma 4.6. [23, Corollary 4.3] Let E i be a indecomposable injective C-comodule such that S i is torsion-free. S(S i ) = E i if and only if all predecessors of S i in the Gabriel-valued quiver (or in the Ext-quiver) (Q C,d C ) are torsion. In opposition to the former Lemma, S preserves a simple comodule if all its 1-predecessors are torsion-free. The following theorem is the main result of the section. Theorem 4.7. [23, Theorem 3.7] Let S j and S i be two simple C-comodules. Then we have S j S(S i )/S i if and only if S j E i /S i and T (S j ) = 0. That is, the torsion 1-predecessors of a torsion-free vertex S i in (Q C,d C ) are the simple C- comodules contained in the socle of S(S i )/S i. In the following picture the torsion-free vertices are represented by white points. soc S(S i )/S i soc E i /S i 158 S i

19 Corollary 4.8. Let S i be a simple ece-comodule. The following conditions are equivalent: (a) E i /S i is torsion-free. (b) There is no arrow in (Q C,d C ) from a torsion vertex S j to S i. (c) S(S i ) = S i. 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 is right semicentral if and only if S(S i ) = S i for any i I e. Therefore we may prove the following: Proposition 4.9. If e is right semicentral, then the Gabriel-valued quiver (Q ece,d ece ) (resp. the Ext-quiver Γ ece ) of the coalgebra ece is isomorphic to the restriction of the valued quiver (Q C,d C ) (resp. the Ext-quiver Γ C ) of C to the subset I e I C. Proof. By [23, Theorem 6.2], e is left semicentral if and only if any torsion-free vertex of (Q C,d C ) has no torsion predecessor. Indeed, there is no path in (Q C,d C ) from a torsion vertex to a torsion-free vertex. This yields that E i /S i is E = Ce-copresented for each i I C. Thus, by [30, Theorem 3.2], the result follows. Remark The left semicentral idempotents are defined symmetrically. In [17], the authors prove that e is left semicentral if and only there is no arrow from a a torsion-free vertex to a torsion vertex. These idempotents are also related to the behavior of injective and simple comodules. In [23, Theorem 6.1] it is proven that the following statements are equivalent: (a) e is left semicentral. { Ei if i I (b) T (E i ) = e, 0 if i / I e. (c) T e is a stable subcategory. If T e is also colocalizing, these are equivalent to: (d) H(S i ) = S i for any i I e. 5 APPLICATIONS TO REPRESENTATION THEORY One of the central points when dealing with Representation Theory of Algebras is to decide if we may really describe an algebra (of finite dimension over an algebraically closed field) so exhaustively as the main aim of this theory proposes (that is, describe explicitly its category of finitely generated modules). That problem leads to define two classes of algebras: tame algebras and wild algebras. From that point of view, the category of finitely generated comodules over a wild algebra has very bad properties. This bad behavior means that this category is so big that it contains (via an exact representation embedding) the category of all finite dimensional representations of the noncommutative polynomial algebra K x, y. As it is well-known, the category of finite dimensional modules over K x,y contains (again via an exact representation embedding) the category of all finitely generated representations for any other finite dimensional algebra, 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

20 of tameness, a tame algebra being one whose indecomposable modules of finite dimension are parameterized by a finite number of one-parameter families for each dimension vector. A classical result in representation theory of algebras (the Tame-Wild Dichotomy, see [10]) states that any finite dimensional algebra over an algebraically closed field is either of tame module type or of wild module type. We refer the reader to [27] for basic definitions and properties about module type of algebras. Analogous concepts and results were defined by Simson in [28] and [29] for coalgebras. We remind that given R a K-algebra. By a R-C-bimodule we mean a K-vector space L endowed with a left R-module structure : R L L and a right C-comodule structure ρ : L L C such that ρ L (r x) = r ρ L (x). We denote by R M C the category of R-C-bimodules. Throughout this section, K will be an algebraically closed field. Following [28], a K-coalgebra C over an algebraically closed field K is said to be of tame comodule type (tame for short) if for every v K 0 (C) there exist K[t]-C-bimodules L (1),...,L (rv), which are finitely generated free K[t]-modules, such that all but finitely many indecomposable right C-comodules M with length M = v are of the form M = Kλ 1 K[t] L (s), where s r v, Kλ 1 = K[t]/(t λ) and λ K. If there is a common bound for the numbers r v for all v K 0 (C), then C is called domestic. If C is a tame coalgebra, then there exists a growth function µ C 1 : K 0(C) N defined as µ C 1 (v) to be the minimal number r v of K[t]-C-bimodules L (1),...,L (rv) satisfying the above conditions, for each v K 0 (C). C is said to be of polynomial growth if there exists a formal power series G(t) = m=1 j 1,..., j m I C g j1,..., j m t j1...t jm with t = (t j ) j IC and non-negative coefficients g j1,..., j m Z such that µ C 1 (v) G(v) for all v = (v( j)) j IC K 0 (C) = Z (IC) such that v := j IC v( j) 2. If G(t) = j IC g j t j, where g j N, then C is called of linear growth. If µ C 1 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 KQ the category of finite dimensional right KQ-modules. A K-coalgebra C is of wild comodule type (wild for short) if there exists an exact and faithful K-linear functor F : M f KQ M f C that respects isomorphism classes and carries indecomposable right KQ-modules to indecomposable right C-comodules. In [29], it was proven a weak version of the Tame-Wild Dichotomy that goes as follows: over an algebraically closed field K, if C is K-coalgebra of tame comodule type, then C is not of wild comodule type. The full version remains open: Conjecture 5.1. Any K-coalgebra, over an algebraically closed field K, is either of tame comodule 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 is wild then C is wild. Proof. By hypothesis, there is an exact and faithful functor F : M f KQ M f ece, where Q is the quiver,which respects isomorphism classes and preserves indecomposables. Therefore, by [23, Proposition 5.1], it is enough to consider the composition M f KQ F M ece f H M C f 160

21 A similar result may be obtained using the section functor if S preserves finite dimensional comodules (for instance, if C is left semiperfect). Proposition 5.3. Let e C be an idempotent which defines a perfect localization and such that S(S i ) is finite dimensional for each i I e. If ece is wild then C is wild. The study of the tameness using the theory of localization strongly depends of the behavior of simple comodules. The reason comes from the fact that it determines the behavior of the length vector. It easy to prove (cf. [23]) that, for any C-comodule L whose length vector length L = (l i ) i IC Z (I C), the length vector of el is length el = (l i ) i Ie Z (I e), that is, it is the image of length L through the natural projection p : Z (I C) Z (I e). Nevertheless, as we have shown in the previous section, for some ece-comodule N, the length vector length S(N) might not be well-defined since S(N) could be infinite dimensional. Therefore it seems that, at least, one should assume that S(S i ) is finite dimensional for any i I e. In fact, by [16], this condition is enough. Theorem 5.4. [16, Theorem 5.9] Let e C be an idempotent such that S(S i ) is a finite dimensional right C-comodule for all i I e. If C is of tame (discrete) comodule type then ece is of tame (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 semicentral idempotent. If C is tame (of polynomial growth, of linear growth, domestic, discrete) then ece is 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 in C. Is the coalgebra ece of tame comodule type? It would be interesting to know if the localization process preserves polynomial growth, linear growth, 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-trivial idempotent e C. A partial result is given in [31]. Let S i be a simple C-comodule, we define the set I( i) = {S j simple C-comodule there is a path in (Q C,d C ) from S j to S i } In general, for some subset U I C, we set I( U) = [ I( x) Observe that, for any subset U I C, the idempotent e U associated to the set of simple comodules I( U) is right semicentral. x U 161