Serial coalgebras

Journal of Pure and Applied Algebra 189 (2004) 89 – 107 www.elsevier.com/locate/jpaa Serial coalgebras J. Cuadraa;∗ , J. Gomez-Torrecillasb a Dpt.  Algebra y Analisis Matematico, Universidad de Almera, Almera E-04120, Spain b Dpt. Algebra  Universidad de Granada, Granada E-01870, Spain Received 28 June 2003; received in revised form 1 October 2003 Communicated by C.A. Weibel Abstract In this paper we extend the theory of serial and uniserial nite dimensional algebras to coalgebras of arbitrary dimension. Nakayama–Skorniakov Theorems are proved in this new setting and the structure of such coalgebras is determined up to Morita–Takeuchi equivalences. Our main structure theorem asserts that over an algebraically closed eld k the basic coalgebra of a serial indecomposable coalgebra is a subcoalgebra of a path coalgebra k where the quiver  is either a cycle or a chain ( nite or in nite). In the uniserial case,  is either a single point or a loop. For cocommutative coalgebras, an explicit description is given, serial coalgebras are uniserial and these are isomorphic to a direct sum of subcoalgebras of the divided power coalgebra. c 2003 Elsevier B.V. All rights reserved.  MSC: 16W30; 16G20 0. Introduction In the last years di erent types of coalgebras have been investigated in connection with some properties of their categories of comodules. For example, semiperfect coalgebras, quasi-coFrobenius coalgebras, hereditary coalgebras, or pure-semisimple coalgebras; see [16,12,19,21] respectively. This line of research is continued in a very natural way with the study of serial coalgebras, which is done in this paper. A coalgebra C is said to be right serial if every indecomposable injective right C-comodule has a unique composition series ( nite or in nite). We prove that Nakayama–  ∗ Research supported by Grants BFM2002-02717 and BFM2001-3141 from the MCYT and FEDER. Corresponding author. Tel.: +34-950-015716; fax: +34-950-015480. E-mail addresses: jcdiaz@ual.es (J. Cuadra), torrecil@ugr.es (J. Gomez-Torrecillas). c 2003 Elsevier B.V. All rights reserved. 0022-4049/$ - see front matter
doi:10.1016/j.jpaa.2003.11.005 90 J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 Skorniakov Theorem holds in this setting, that is, C is serial (left and right serial) if and only if the category MC of all right C-comodules is a uniserial category in the sense of [2] (i.e., every indecomposable object of nite length has a unique composition series); see Theorem 1.8. Consequently, every nite-dimensional comodule is a direct sum of uniserial comodules. It is shown in Proposition 1.13 that such a decomposition holds for any comodule over a serial coalgebra with nite coradical ltration. Examples of serial coalgebras are provided in Corollary 1.11 as nite dual coalgebras of either serial or hereditary prime noetherian algebras. In particular, A0 is serial for every Dedekind domain A. In Proposition 2.3, we prove that every right serial coalgebra with coseparable coradical is isomorphic to a subcoalgebra of a cotensor coalgebra over a right serial bicomodule. In the right semiperfect case, the coalgebra may be represented in the cotensor coalgebra by an admissible sequence associated to its diagram; see Theorem 2.5. Therefore, the corresponding theory for nite dimensional right serial algebras (see, e.g., [8]) is extended. Indecomposable serial coalgebras C are then characterized in terms of its diagram D(C), which, a fortiori, turns out to be its Ext-quiver. Concretely, we prove in Theorem 2.9 that a coalgebra C with coradical decomposition C0 = ⊕i∈I M c (Di ; ni ) (the Di∗ ’s are division algebras) is serial if and only if I is countable, D(C) is a chain or a cycle and Di ∼ = Dj for every i; j ∈ I . In the algebraically closed case, indecomposable serial coalgebras are recognized, up to Morita–Takeuchi equivalence, as subcoalgebras of the path coalgebra k, where  is a cycle or a chain. If, in addition, C is hereditary, then it is isomorphic to k, Theorem 2.10. A serial coalgebra C is said to be uniserial if each uniserial comodule has a homogeneous composition series. Uniserial coalgebras are described as direct sums of matrix coalgebras with coecients in colocal serial coalgebras. It is also shown that a coalgebra C is uniserial if and only if every right coideal is coprincipal. Equivalently, its diagram D(C) consists of isolated points and loops, see Theorem 3.1. In the cocommutative case, the notion of serial and uniserial coincide, as shown in Theorem 3.2. In such a case, the dual of each irreducible component is a local noetherian algebra whose lattice of ideals consists of the powers of the principal maximal ideal. Under the hypothesis of the eld k being algebraically closed, we prove that cocommutative coalgebras are isomorphic to a direct sum of subcoalgebras of the divided power coalgebra. Let us x some notation and present some preliminaries. Throughout this paper k is a xed ground eld. All algebras, coalgebras, vector spaces and ⊗, Hom, etc. are over k. Every map is a k-linear map. The reader is expected to be familiar with coalgebra theory. Basic references are [1,7,18,22]. In the sequel C always stands for a coalgebra and
;  will denote its comultiplication and counit, respectively. The category of right (resp. left) C-comodules is denoted by MC (resp. CM); for an object M of MC , its comodule structure map is denoted by M . The fundamental properties of the categories of comodules can be found in several places, see e.g. [13,14,23]. We emphasize that MC is a locally nite category and every injective indecomposable comodule is given as the injective envelope E(S) of a simple comodule S. We will use that CC = ⊕i∈I E(Si )(ni ) where {E(Si )}i∈I is a full set of injective indecomposable right comodules, and the J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 91 n′i s are nite cardinals. For aspects of equivalences between comodule categories, in particular for the de nition and properties of the coendomorphism coalgebra, we refer to [23]. We recall from [5] the de nition of a path coalgebra. Let  denote a quiver, the path coalgebra k is the k-vector space generated by the paths in  with comultiplication and counit given by 
0 if | | ¿ 0; ⊗ ; ( ) =
( ) = 1 if | | = 0; = where is the concatenation of paths and | | the length of . 1. Serial coalgebras: de nition and rst properties Every right C-comodule M has a ltration {0} ⊂ soc(M ) ⊂ soc2 (M ) ⊂ · · ·, called the Loewy series of M , de ned as follows: soc(M ) is the socle of M , and for n ¿ 1, socn (M ) is the unique subcomodule of M satisfying socn−1 (M ) ⊂ socn (M ) and soc(M=socn−1 (M )) = socn (M )=socn−1 (M ), see [14, 1.4]. There is an alternative description of this series. Let {Cn }n∈N be the coradical ltration of C, then socn+1 (M ) = −1 M (M C Cn ). De nition 1.1. A right C-comodule M is called uniserial if its lattice of subcomodules is a chain ( nite or in nite). Lemma 1.2. The following assertions are equivalent for M ∈ MC : (i) M is uniserial. (ii) The Loewy series is a composition series for M (and each term is nite dimensional). (iii) Every nite dimensional subcomodule of M is uniserial. Proof. (i) ⇒ (ii) Suppose that there is a non simple factor socn (M )=socn−1 (M ) = soc(M=socn−1 (M )). It contains two simple comodules S1 ; S2 . The subcomodules T1 ; T2 of M such that Si = Ti =socn (M ); i = 1; 2 would be incomparable, contrary to the hypothesis. Recall that a simple comodule is nite dimensional. Since the Loewy series is a composition series, and soc(M ); soc2 (M )=soc(M ) are nite dimensional, soc2 (M ) is nite dimensional. By induction it follows that socn (M ) is nite dimensional for all n ∈ N. (ii) ⇒ (iii) Let N be a nite dimensional subcomodule of M . There is n ∈ N such that N ⊂ socn (M ). Let r =max{s ∈ N : socs (M ) ⊆ N } and assume that N = socr+1 (M ). Then, N ∩ socr+1 (M ) = socr (M ) and thus (N=socr (M )) ∩ soc(M=socr (M )) = {0}. Hence N = socr (M ). (iii) ⇒ (i) For m ∈ M , let (m) be the subcomodule generated by m which is nite dimensional. Notice that M is uniserial if and only if either (m) ⊂ (n) or (n) ⊂ (m) 92 J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 for all m; n ∈ M . For any n; m ∈ M we may compare (n); (m) in (n; m) which is nite dimensional, and consequently uniserial by hypothesis. De nition 1.3. Let C be a coalgebra. (i) C is said to be a right serial coalgebra if its right injective indecomposable comodules are uniserial. A left serial coalgebra is de ned in a symmetric way. (ii) C is called serial if it is right and left serial. (iii) C is uniserial if it is serial and the composition factors of each indecomposable injective comodule are isomorphic (homogeneous uniserial). Example 1.4. (1) Any cosemisimple coalgebra is uniserial. (2) Let C be the path coalgebra associated to the quiver A∞ ∞ ··· −→ • −→ • −→ • −→ ··· : Denote by {gi : i ∈ Z} the set of vertices. The family {kgi }i∈Z is a representative family of simple right (left) comodules. Let ei be the idempotent in C ∗ de ned as ei ; gi  = 1 and zero elsewhere. The injective hull of kgi (as a right comodule) is Ei = Cei and it consists of all paths starting at gi . The family {Ei }i∈Z is a representative set of injective indecomposable right comodules. Every proper subcomodule of Ei is spanned by the paths connecting the vertex gi and some xed vertex gj with j ¿ i. Therefore, the subcomodules of Ei are clearly totally order and thus C is right serial. Similarly, C is left serial (the injective hull of kgi as a left C-comodule is Fi = ei C consisting of all paths ending at gi ). (3) Let C = k{c0 ; c1 ; c2 ; : : :} be the divided power coalgebra. Its comultiplication and counit are given by:
(cn ) = n
ci ⊗ cn−i ; (cn ) = n; 0 i=0 for all n ∈ N, where n; 0 denotes the Kronecker’s delta. This coalgebra is just the path coalgebra associated to the quiver consisting of a unique vertex with a loop. We see that it is uniserial. Since C is cocommutative, every right (resp. left) subcomodule of C is a subcoalgebra. It is known that Ci = k{c0 ; c1 ; : : : ; ci } are the only subcoalgebras of C. The only simple comodule is C0 and E(C0 ) = C. Hence CC =C C is a uniserial comodule with composition series {0} ⊂ C0 ⊂ C1 ⊂ · · · : Proposition 1.5. Let C be a coalgebra and D a subcoalgebra of C. If C is right serial (resp. uniserial), then D is right serial (resp. uniserial). Proof. Let D be a subcoalgebra of C and S a simple right D-comodule. Let i : S → ED (S); j : S → EC (S) be the canonical embeddings. There is a C-comodule map g : ED (S) → EC (S) such that gi = j. Since i is essential and j injective, g is injective. Hence ED (S) is uniserial. For the uniserial case, notice that socn (ED (S))=socn−1 (ED (S)) is a subcomodule of socn (EC (S))=socn−1 (EC (S)). J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 93 Proposition 1.6. A nite dimensional coalgebra C is right serial (resp. uniserial) if and only if its dual algebra C ∗ is right serial (resp. uniserial). Proof. This fact follows from the duality between the categories of nite dimensional right C-comodules and nite dimensional right C ∗ -modules. It is de ned as follows: ∗ For a nite dimensional right C-comodule N , its dual space C ∗ -module  N ∗ is a right ∗ ∗ ∗ ∗ ∗ ∗ via the map · : N → C ⊗ N given by n · c ; n = (n) n ; n(0) c ; n(1)  for all n∗ ∈ N ∗ ; c∗ ∈ C ∗ and n ∈ N . Moreover, the lattice of C-subcomodules of N is isomor∗ phic to the lattice of C ∗ -submodules of N ∗ via the orthogonal space (−)⊥(N ) . It is proved in [14, 1.3a] and [16, Lemma 15] that S is a simple C-comodule if and only if S ∗ is a simple C ∗ -module, and I is the injective hull of S if and only if I ∗ is the projective cover of S ∗ , see [14, 1.3a] and [16, Lemma 15] for further details. Given a right C-comodule M , let cf(M ) denote the coecients space of M , see [14, p. 142]. It is known that cf(M ) is the smallest subcoalgebra of C such that the structure map M : M → M ⊗ C factorises throughout M ⊗ cf(M ), making M into a right cf(M )-comodule. Note that a vector subspace N of M is a C-subcomodule of M if and only if it is a cf(M )-subcomodule of M . Then, the lattice of C-subcomodules of M coincides with the lattice of cf(M )-subcomodules of M . Hence M is uniserial as C-comodule if and only if it is uniserial as cf(M )-comodule. Finally, recall that if M is nite-dimensional, then so is cf(M ). Proposition 1.7. A coalgebra C is right serial if and only if C1 is right serial. Proof. Assume that C1 is right serial and let E = EC (S) be the injective hull of a simple right C-comodule S. Let N be a nite dimensional subcomodule of E. By Lemma 1.2, it suces to prove that N is uniserial. Consider D = cf(N ) and J the ∗ Jacobson radical of D∗ . We know that J = D0⊥(D ) and, since D is nite dimensional, ∗ J 2 = D1⊥(D ) . By hypothesis and Proposition 1.5, D1 is right serial. Proposition 1.6 yields that D1∗ ∼ = D∗ =J 2 is a right serial algebra. By [8, Corollary 10.2.1], D∗ is right serial and thus D is right serial. Since soc(N ) = S, the subcomodule N embeds in ED (S). Hence N is uniserial as a D-comodule and so uniserial as a C-comodule. Theorem 1.8. The following properties about a coalgebra C are equivalent: (i) (ii) (iii) (iv) (v) (vi) (vii) C is serial. Every nite dimensional right C-comodule is a direct sum of uniserial comodules. Every nite dimensional left C-comodule is a direct sum of uniserial comodules. Every nite dimensional indecomposable right C-comodule is uniserial. Every nite dimensional indecomposable left C-comodule is uniserial. Every nite dimensional subcoalgebra of C is serial. C1 is serial. Proof. (i) ⇒ (ii) Let M be a nite dimensional right C-comodule and D = cf(M ). Since D is a nite dimensional subcoalgebra of C, Propositions 1.5, 1.6 show that D∗ is serial. In view of the isomorphism of categories MD ∼ = D∗ M and the Nakayama– 94 J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 Skorniakov Theorem ([8, Theorem 10.1.1]), N is a direct sum of uniserial D-comodules. But a uniserial D-comodule is uniserial as a C-comodule. Note that the same argument is valid to prove (vi) ⇒ (ii). (ii) ⇒ (iv) Obvious. (iv) ⇒ (i) Let S be a simple right comodule and E = EC (S) its injective hull. It suces to check that for any m; n ∈ E, either (n) ⊂ (m) or (m) ⊂ (n). This follows by comparing (n); (m) in (n; m); which is nite dimensional and indecomposable. (i) ⇒ (vi) Straightforward from Proposition 1.5. (i) ⇔ (vii) It follows from Proposition 1.7. (i) ⇔ (iii) ⇔ (v) is symmetric to (i) ⇔ (ii) ⇔ (iv) by the Nakayama–Skorniakov Theorem for nite dimensional algebras, [8, Theorem 10.1.11]. We recall from [12] that a coalgebra C is called left quasi-coFrobenius if C as a left C ∗ -module embeds in a free left C ∗ -module. The coalgebra C is called quasicoFrobenius if it is left and right quasi-coFrobenius. It has been shown in [12, Remarks 1.5 (a)] that a nite dimensional coalgebra C is quasi-coFrobenius if and only if the dual algebra C ∗ is quasi-Frobenius. Theorem 1.9. The following assertions about a coalgebra C are equivalent: (i) C is uniserial. (ii) Every nite dimensional right C-comodule is a direct sum of homogeneous uniserial comodules. (iii) Every nite dimensional left C-comodule is a direct sum of homogeneous uniserial comodules. (iv) Every nite dimensional indecomposable right C-comodule is homogeneous uniserial. (v) Every nite dimensional indecomposable left C-comodule is homogeneous uniserial. (vi) Every nite dimensional subcoalgebra of C is quasi-coFrobenius. (vii) C1 is uniserial. Proof. Keeping in mind the characterization of uniserial algebras given in [8, Theorem 9.4.1, Corollary 9.4.5] and [10, Theorem 2.1], one may adapt the proof of Theorem 1.8 to uniserial coalgebras. Corollary 1.10. Let C be a coalgebra and {Ci }i∈I a family of subcoalgebras such that C = ⊕i∈I Ci . Then, C is serial (resp. uniserial) if and only if Ci is serial (resp. uniserial) for all i ∈ I . Proof. Assume that Ci is serial for all i ∈ I , and let M be a nite dimensional Ccomodule. Then M = ⊕nj=1 Mij with Mij = −1 M (M C Cij ) for some i1 ; : : : ; in ∈ I . Since Mij is a Cij -comodule and Cij is serial, Theorem 1.8 implies that Mij is a direct sum of uniserial Cij -comodules. Hence M is a direct sum of uniserial C-comodules. Theorem 1.8 implies that C is serial. The converse is deduced from Proposition 1.5. For uniserial coalgebras the argument is analogous using Theorem 1.9. J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 95 We recall that the nite dual coalgebra of an algebra A is A0 = {f ∈ A∗ : f(I ) = 0 for some ideal I of A with dim(A=I ) ¡ ∞}. Corollary 1.11. Let A be an algebra and assume that every nite dimensional quotient algebra of A is serial. Then A0 is serial. As a consequence (i) If A is serial, then A0 is serial. (ii) If A is an hereditary noetherian prime algebra, then A0 is serial. In particular, the nite dual of a Dedekind domain is serial. Proof. Every nite dimensional subcoalgebra of A0 is of the form (A=I )∗ where I is a co nite two-sided ideal of A. Combining the hypothesis and Proposition 1.6 we have that every nite dimensional subcoalgebra of A0 is serial. Theorem 1.8 now applies. By [3, Theorem 32.2] every quotient algebra of a serial algebra is serial. This proves (i). Assertion (ii) follows from [9, Corollary 3.2] which claims that every proper quotient algebra of an hereditary noetherian prime algebra is serial. Dedekind domains are examples of such algebras. Remark 1.12. One of the most important results in the theory of serial rings is that for a left artinian serial ring every left and every right module decomposes as a direct sum of uniserial modules, see [3, Theorem 32.3]. We do not know whether this result holds for serial coalgebras. Although it is true for coalgebras having a nite coradical ltration, see Proposition 1.13. Note that this hypothesis assures that every comodule has nite Loewy length. In the artinian ring case this is a consequence of the nilpotency of the Jacobson radical. Proposition 1.13. Let C be a serial coalgebra and M a right C-comodule. Then socn (M ) is a direct sum of uniserial comodules for all n ∈ N. In particular, if C = Cn for some n ∈ N, every right C-comodule is a direct sum of uniserial comodules. Proof. Let M be a right C-comodule such that M = socn+1 (M ). Then M = −1 M (M C Cn ) = socn+1 (M ) and thus M is a Cn -comodule. We prove the result by induction on the length of the Loewy series {soci (M )}n+1 i=1 . It is clear that soc(M ) is a direct sum of uniserial comodules. Assume that socn (M ) is a direct sum of uniserial comodules. Let M = socn (M ), then there is m ∈ socn+1 (M ) − socn (M ). Let (m) be the subcomodule generated by m. By Theorem 1.8, (m) = ⊕sj=1 Uj where each Uj is uniserial. We claim that for some j ∈ {1; 2; : : : ; s} it holds that Uj = socn (Uj ). If Uj = socn (Uj ) for all j = 1; : : : ; n, then   s s s    Uj = socn (Uj ) = socn  Uj  = socn ((m)) ⊂ socn (M ): (m) = j=1 j=1 n j=1 From this, m ∈ soc (M ), a contradiction. Since Uj is uniserial and C is serial, EC (Uj ) is uniserial. There is h ∈ N such that Uj ∼ = ECh (Uj ). It must be h = n because Uj has length n + 1. By Zorn’s Lemma there exists a maximal independent family U of uniserial Cn -subcomodules of M of length n + 1. Let U = ⊕X ∈U X . Since U is 96 J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 injective as Cn -comodule, we can express M = U ⊕ L where U is a direct sum of uniserial comodules and L does not contain uniserial subcomodules of length n + 1. We claim that L = socn (L). If there is y ∈ socn+1 (L) − socn (L), then reasoning as in the above paragraph we would nd a uniserial subcomodule of L of length n + 1. Applying the induction hypothesis to L we are done. We recall from [16] that a coalgebra is called right semiperfect if the injective hull of a right simple comodule is nite dimensional. Proposition 1.14. Let C be a serial coalgebra. Every subcoalgebra with nite coradical ltration is semiperfect (both sides). In particular, if C has nite coradical ltration, C is semiperfect. Proof. Let S be a simple C-comodule and E = EC (S). It is not dicult to see that n+1 (EC (S)). By Lemma 1.2 we get that ECn (S) is nite ECn (S) = −1 M (E C Cn ) = soc dimensional for all n ∈ N. 2. Structure theorems for serial coalgebras Let C be a coalgebra and let {Si }i∈I be a full set of simple right C-comodules. The set {E(Si )}i∈I is a representative family of injective indecomposable right C-comodules. For each Si there is a primitive idempotent ei in C0∗ such that Si ∼ = C0 ei . Each idempotent ei may be lifted to a primitive idempotent ei ∈ C ∗ such that E(Si ) ∼ = Cei , see [6] for more details. The family {ei }i∈I is called a basic set of idempotents for C. For a xed ei the right C0 -comodule C1 ei =C0 ei is cosemisimple, then  (m ) C1 ei =C0 ei ∼ C0 ej ij ; = j∈I m′ij s where the are cardinal numbers. The right diagram of C is de ned as the directed graph D(C) with vertex set {ei }i∈I and with mij arrows between ei and ej . From the de nition it is clear that D(C) = D(C1 ). Proposition 2.1. A coalgebra C is right serial if and only if there is at most one arrow starting at each vertex in D(C). Proof. Assume that C is right serial and let ei be a vertex of D(C). Since Cei is uniserial, C1 ei =C0 ei is simple or zero. If it is non zero, then C1 ei =C0 ei ∼ = C0 ej for some ej . Thus there is an arrow from ei to ej . There is only one because {ei }i∈I is basic. Conversely, given ei such that C1 ei is not simple, there is a unique ej such that C1 ei =C0 ei ∼ = C0 ej by hypothesis. Hence C1 ei is uniserial which gives that C1 is right serial. That C is right serial follows from Proposition 1.7. We now compare the right diagram of C with the right Ext-quiver of C. We recall from [17] that the right Ext-quiver of C is the diagram (C) whose vertices are J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 97 the simples {Si }i∈I in MC and for i; j ∈ I there is an arrow Si → Sj if there is an indecomposable C-comodule P and an exact sequence 0 → Si → P → Sj → 0. Two simple right C-comodules Si and Sj are said to be connected if there is a path in (C) (as an undirected quiver) from Si to Sj . It was proved in [17] that there is a family {C } ∈
of subcoalgebras of C such that C = ⊕ ∈
C where each C is associated to a connected component of (C). Remark 2.2. When C is a right serial coalgebra then D(C) and (C) are isomorphic. Let Si and Sj be two simple right C-comodules and ei ; ej two primitive idempotents of C ∗ such that Si ∼ = C0 ej . If there is an arrow from ei to ej in D(C), = C0 ei and Sj ∼ then it is clear that there is an arrow from Si to Sj in (C). Assume that there is an arrow from Si to Sj in (C). Let P be the indecomposable C-comodule appearing in a short exact sequence with Si and Sj as extremes. Then Si = soc(P). Taking EC1 (P) ∼ = C1 ei we have that Sj embeds in EC1 (P)=soc(EC1 (P)). But this is simple by hypothesis. Hence EC1 (P)=soc(EC1 (P)) ∼ = Sj and thus there is an arrow from ei to ej in D(C). We recall from [20] the de nition of the cotensor coalgebra and some related properties. Let C be a coalgebra and M a C-bicomodule. The cotensor coalgebra of M over C, denoted by TCc (M ), is de ned as follows: as a vector space TCc (M ) = ⊕n∈N M C n where M C n denotes the cotensor product of M with itself n times. When n = 0,  M C n = C. The comultiplication
: M C n → i+j=n M C i ⊗ M C j is given by the comodule structure map of M when i = 0 or j = 0, and when i; j ¿ 0 it is induced by the map x1 ⊗ · · · ⊗ x n → i (x1 ⊗ · · · ⊗ xi ) ⊗ (xi+1 ⊗ · · · ⊗ x n ). The counit is C  where C is the counit of C and  is the projection from TCc (M ) into C. The coradical ltration of T = TCc (M ) is given by Tn = ⊕ni=0 M C i . The path coalgebra associated to a quiver is a particular case of the cotensor coalgebra, see [5, Remark 4.2]. The space P = C1 =C0 is a (C0 ; C0 )-bicomodule via the maps
+ (c(1) + C0 ) ⊗ c(2) ; P : P → P ⊗ C0 ; c + C0 → (c) − P : P → C0 ⊗ P; c + C0 →
c(1) ⊗ (c(2) + C0 ): (c) When the coradical of C is coseparable (the dual of each simple subcoalgebra is separable over k), C is isomorphic to a subcoalgebra of TCc (P), see [24, 4.6]. We say that a C0 -bicomodule M is right serial if Me is simple or zero for any primitive idempotent e ∈ C0∗ . Proposition 2.3. Let C be a coalgebra and P = C1 =C0 . (i) C is right (resp. left) serial if and only if P is right (resp. left) serial. (ii) If C is right serial and C0 is coseparable, then C is isomorphic to a subcoalgebra of TCc 0 (P) where P is a right serial C0 -bicomodule. 98 J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 (iii) If M is a right serial C0 -bicomodule, then any subcoalgebra of TCc 0 (M ) is right serial. Proof. (i) Recall from [6, Proposition 1.17] that an idempotent e ∈ C ∗ is primitive if and only if e = e|C0 is primitive. If C is right serial, then Ce is uniserial for any primitive idempotent e ∈ C ∗ . Hence soc2 (Ce)=soc(Ce) = C1 e=C0 e ∼ = (C1 =C0 )e is simple. Conversely, assume that Pe is simple or zero for any primitive idempotent e ∈ C0∗ . By lifting e to an idempotent e′ ∈ C1∗ we have that C1 e′ is uniserial. Hence C1 is right serial, and from Proposition 1.7, C is right serial. (ii) It follows from (i). (iii) Let T = TCc 0 (M ). Since T1 =T0 ∼ = M as T0 -comodules, T is right serial. Now it is just to apply Lemma 1.5. In view of the foregoing result, in order to characterize right serial coalgebras, it suces to nd the subcoalgebras of T = TCc (P) where C is a cosemisimple and coseparable coalgebra and P is a C-bicomodule that is serial as a right C-comodule. We may assume that C is basic. Then C = ⊕i∈I Ci where Ci∗ is a division algebra. For each i ∈ I let ei be the primitive and central idempotent such that Ci = Cei . We are able to characterize the right semiperfect subcoalgebras of T = TCc (P). Suppose that D is a right semiperfect subcoalgebra of T . Then, DD = ⊕j∈J ED (Cj ). Since ED (Cj ) ⊂ Tej and Tej is uniserial, ED (Cj ) = Tlj ej for some lj ∈ N. Here Tlj denotes the lj th term of the coradical ltration of T . Hence any right semiperfect subcoalgebra of T is of the form D = ⊕j∈J Dj where Dj = Tlj ej for some set of index J ⊂ I . Given a family {lj }j∈J of natural numbers we study when D = ⊕j∈J Tlj ej is a subcoalgebra of T . We need some previous facts. Lemma 2.4. Let C be a coalgebra, e ∈ C ∗ a central idempotent, and M a left Ccomodule.   (i) For any c ∈ C, (c) e; c(1) c(2) = (c) e; c(2) c(1) . (ii) Ce C M = Ce C Me. (iii) If N is a C-bicomodule, then (N C M )e = Ne C M . Proof. (i) Let C : C → C ∗∗ be the canonical embedding de ned as C (c); c∗ =c∗ ; c for all c ∈ C; c∗ ∈ C ∗ . Since e is central, ec∗ = c∗ e for all c∗ ∈ C ∗ . Then,  

C  e; c(1) c∗ ; c(2) ; e; c(1) c(2)  ; c∗ = (c) (c) =
e; c(2) c∗ ; c(1) ; (c) =  
e; c(2) c(1)  ; c∗ C  (c) : J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 The injectivity  of C yields the claim. n (ii) Let x = l=1 cl e ⊗ ml ∈ Ce C M . By de nition of the cotensor product, n
n


cl(1) e ⊗ cl(2) ⊗ ml = cl e ⊗ ml(−1) ⊗ ml(0) : l=1 (cl ) 99 (1) l=1 (ml ) Now, x= n
cl e ⊗ m l = n

e; cl(2) cl(1) ⊗ ml l=1 (cl ) l=1 = n

e; cl(1) cl(2) ⊗ ml since e is central l=1 (cl ) = n

e; cl(1) cl(2) e; cl(3)  ⊗ ml l=1 (cl ) = n

cl(1) e ⊗ e; cl(2) ml l=1 (cl ) = n

cl e ⊗ e; ml(−1) ml(0) by applying 1 ⊗ e ⊗ 1 to (1) l=1 (ml ) = n
cl e ⊗ ml e: l=1 n (iii) Let x = ( l=1 nl ⊗ ml )e ∈ (N C M )e. Then, n n

x= e; (nl ⊗ ml )(−1) (nl ⊗ ml )(0) = e; nl(−1) nl(0) ⊗ ml l=1 = n
l=1 nl e ⊗ ml : l=1 For a cosemisimple coalgebra C and a right serial (C; C)-bicomodule P one can construct a diagram D(C; P) associated to C and P in a similar manner to D(C). Indeed, if C is right serial, then D(C) ∼ = D(C0 ; P) where P = C1 =C0 . A vertex is called a sink when it is not the tail of any arrow. Theorem 2.5. Let C be a basic cosemisimple and coseparable coalgebra and {ei }i∈I  be a family of primitive central idempotents of C ∗ such that C = i∈I ei . Let P be a right serial C-bicomodule and T = TCc (P). Finally, let {li }i∈I be a family of natural numbers, Di = Tli ei and D = ⊕i∈I Di . Then, D is a subcoalgebra of T if and only if li = 0 for every sink ei in D(C; P) and li 6 lj + 1 for every arrow from ei to ej in D(C; P). 100 J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 Proof. If ei1 → ei2 → · · · → ein is a path in D(C; P), then Peis ∼ = Ceis +1 for all s = 1; : : : ; n − 1. Using the above lemma, (P C P)ei = Pei C P ∼ = Cei C P = Cei C Pei ∼ = Pei C Pei : In general, P spaces. Then,
(P Ch 2 1 1 Ch ei 1 ∼ = Pei1 ei ) ⊆
P 2 C Pei2 Ca ei ::: C CP Cb C Peih = a+b=h = Cei CP + ··· + P 1 2
2 for h 6 n. We identify these two (P Ca Cb CP )ei a+b=h Ch ei + Pei C h−1 ei CP C (h−1) C Peih−1 +P ei1 + P Ch ei C2 CC ei CP C (h−2) ei2 (∗) (⇐) Assume that i is a sink and li = 0. Then Di = Cei and it follows that
(Di ) ⊂ Di ⊗ Di . If there is an arrow ei → ej and li 6 lj + 1, then Tli −1 ej ⊂ Tlj ej . From (∗) we get for h 6 li ,
(P Ch ei ) ⊂ Cei ⊗ Th ei + T1 ei ⊗ Th−1 es1 + · · · + Th−1 ei ⊗ T1 esh −1 + Th ei ⊗ C ⊂ Cei ⊗ Di + Di ⊗ Ds1 + Di ⊗ Ds2 + · · · + Di ⊗ Dsh−1 + Di ⊗ C ⊂D⊗D where ei → es1 → · · · → esh . This gives
(Di ) ⊆ D ⊗ D and so D is a subcoalgebra of T . (⇒) Suppose that D is a subcoalgebra of T and let ei be a sink in D(C; P). Since P is right serial, Pei = {0}. We thus get Tli ei = Cei , and hence li = 0. Let ei → ej be an arrow in D(C; P). Since D is a subcoalgebra, from (∗) it follows that li − 1 6 lj . De nition 2.6. Let C be a coalgebra and D(C) its diagram. An admissible sequence for D(C) is a map l : D(C) → N; i → li such that li = 0 if ei is a sink and li 6 lj + 1 if there is an arrow from ei to ej . Corollary 2.7. Let C be a right serial and right semiperfect basic coalgebra with coseparable coradical. Then C is determined by the following data: C0 , C1 =C0 , and an admissible sequence for D(C). Proposition 2.8. Let D = (D0 ; P; {li }i∈I ) and D′ = (D0′ ; P ′ ; {l′i }i∈I ′ ) be two right serial and right semiperfect basic coalgebras. The coalgebras D and D′ are isomorphic if and only if there is a coalgebra isomorphism  : D0 → D0′ and a bicomodule isomorphism  : P → P ′ verifying that li = l′(i) where  is the diagram isomorphism induced by   and . Proof. Assume that  : D → D′ is a coalgebra isomorphism, then  induces a coalgebra isomorphism 0 : D0 → D0′ and a bicomodule isomorphism  : D1 =D0 → D1′ =D0′ . ∗ Consider the dual map ∗ : D′ → D∗ . If {ei′ }i∈I ′ is a basic set of idempotents for ′ ∗ ′ D , then { (ei )}i∈I ′ is a basic set of idempotents for D. Moreover, for ei = ∗ (ei′ ), J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 101 (Dei ) = D′ ei′ . Thus  also induces a diagram isomorphism  : D(D) → D(D′ ). Since li is the length of Dei , it holds that li = l′(i) . Conversely, a coalgebra isomorphism ’ : D0 → D0′ and a bicomodule isomorphism f : P → P ′ induces a coalgebra isomorphism  : TDc 0 (P) → TDc ′ (P ′ ). From the hypo0 thesis it follows that (D) = D′ . Theorem 2.9. Let C be an indecomposable coalgebra and C0 = ⊕i∈I M c (Di ; ni ) be a decomposition of its coradical with {Di∗ }i∈I a family of division algebras. (i) If I is nite, then C is serial if and only if D(C) is a cycle or a nite chain and Di ∼ = Dj for any vertex ei ; ej in D(C). (ii) If I is in nite, then C is serial if and only if I is countable, D(C) is a chain, and Di ∼ = Dj for any pair of vertex ei ; ej in D(C). Proof. This proof is inspired in [8, Theorem 10.3.1] and the methods used there for nite dimensional algebras may be extended for in nite dimensional coalgebras as we next show. We can assume that C is basic since the notion of serial coalgebra is invariant under Morita–Takeuchi equivalences, and in this case D(C) is also so.  Since C is basic,  = i∈I ei where {ei }i∈I is a basic set of idempotents for C. The idempotents ei = ei |C0 are central and C0 ei = Di , see [6, Theorem 3.10, Corollary 3.12]. By hypothesis and Proposition 2.3(i), Pei and ei P are simple or zero right and left C0 -comodules respectively, where P = C1 =C0 . We claim that two di erent arrows in D(C) may not have the same head. Let eh ; ei ; ej be vertex of D(C) and ei → ej ; eh → ej be arrows. Then ej Pei = {0} and ej Peh = {0}. These would be two direct summands of the simple left C-comodule ej P yielding a contradiction. Combining this fact and Lemma 2.1 we know that there is at most one arrow starting at each vertex and a vertex can not be the head of two arrows. If I is nite, then D(C) is a cycle or a chain. Assume that I is in nite. Since C is serial, the Ext-quiver coincides with D(C). As C is indecomposable, D(C) is connected. Fix a vertex i0 . We know that for any vertex i there is a path from i to i0 or from i0 to i. Let n be the length a such path. In the rst case we label i with −n and in the second one i with n. Thus we have a map from I to Z. This map is injective in view of the above observation. Hence I is countable and D(C) has to be a chain. Let ej → ei be an arrow in D(C). Then Pij = ei Pej = {0} and ei P = Pij = Pej . As a right (resp. left) C0 -comodule, Pij is isomorphic to Di (resp. Dj ). Hence Pij is a (Dj ; Di )-bicomodule. By the universal property of the co-endomorphism coalgebra, there is a coalgebra map  : e−C0 (Pej ) → Dj . Since Pej ∼ = C0 ei as right C0 -comodules, e−C0 (Pej ) ∼ = Di . Hence we have a coalgebra map  : Di → Dj . It is = e−C0 (C0 ei ) ∼ an isomorphism because Di∗ ; Dj∗ are division algebras verifying dim(Di ) = dim(Pij ) = dim(Dj ). Conversely, assume that D(C) is a cycle or a chain and Di ∼ = Dj for any pair of vertex ei ; ej in D(C). From Lemma 2.1, C is right serial. If ei is a sink, then ei P = {0}. Let ej → ei be an arrow, then Pej ∼ = Di as right C0 -comodules. = C0 ei ∼ (T ) On the other hand, since P is semisimple as a left C0 -comodule P ∼ = ⊕l∈I Dl l . Then 102 J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 (T ) (T ) ei P ∼ = Di i as left C0 -comodules. = Di i as left C0 -comodules. Thus ei P = ei Pej ∼ (Sj ) (S ) Now, ei P ∼ = ⊕l∈I Dl l as a right C0 -comodules. Hence ei Pej = Dj . The isomorphism (T ) Pej ∼ = Dj forces Sj = 1. Summarizing, we have that Di i ∼ = Di as vector spaces. = Di ∼ So Ti = 1 and ei P is simple as a left C0 -comodule. From Proposition 2.3, C is left serial. Theorem 2.10. Let C be an indecomposable serial coalgebra over an algebraically closed eld. (i) C is Morita–Takeuchi equivalent to a subcoalgebra B of k where  is one of the following quivers: (a) A cycle or a nite chain. (b) A∞ : 0 → 1 → 2 → 3 → · · · (c) A∞ : · · · → −3 → −2 → −1 → 0. (d) A∞ · · · → −3 → −2 → −1 → 0 → 1 → 2 → 3 → · · · ∞: (ii) If C is right semiperfect, then B is determined by an admissible sequence associated to . (iii) If C is hereditary, then B ∼ = k. Proof. (i) Let B be the basic coalgebra associated to C and  the Ext-quiver of C. By [5, Theorem 4.3], B may be taken as a subcoalgebra of k containing to (k)1 . Since C is serial,  = D(C), and from Theorem 2.9,  is one of the listed quivers. (ii) Since k is algebraically closed, B0 is pointed, [5, Corollary 2.5]. Thus each simple subcoalgebra is given by a group-like element, and V = D1 =D0 consists of the non trivial skew-primitive elements. Then TBc0 (P) ∼ = k ([5, Remark 4.2]), and from Theorem 2.5, B is determined by an admissible sequence. (iii) It follows from [4, Theorem 1] which claims that an hereditary basic coalgebra is isomorphic to the path coalgebra of its Ext-quiver. Remark 2.11. The form of the Ext-quiver in Theorem 2.10 may be deduced from [2]. It is claimed without proof in page 86 that the Ext-quiver of an uniserial connected category is a cycle or a chain. For a serial indecomposable coalgebra C its category of right comodules MC is a uniserial connected category, Theorem 1.8. Remark 2.12. If the coalgebra C is assumed to be pointed, then C is isomorphic to a subcoalgebra of k and the hypothesis of k being algebraically closed in Theorem 2.10 is not necessary. Remark 2.13. In case D(C) = A∞ , the coalgebra C is left pure semisimple, see [21, Corollary 2.5]. This means that any left C-comodule decomposes as a direct sum of nite dimensional comodules. Applying Theorem 1.8 to each summand, any left C-comodule is a direct sum of uniserial comodules. This partially answers the problem proposed in Remark 1.12. Similarly, if D(C) = A∞ , then C is right pure semisimple and thus every right C-comodule is a direct sum of uniserial comodules. J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 103 We nish this section by giving a characterization of serial coalgebras in terms of its coradical being a coprincipal coideal. De nition 2.14. A right coideal I of a coalgebra C is said to be coprincipal if there is a right C-comodule map f : C → C such that ker(f) = I . Example 2.15. Let C = k{c0 ; c1 ; c2 ; : : :} be the divided power coalgebra. Consider the map f : C → C; c0 → 0; ci → ci−1 . It is easy to check that f is a C-comodule map. For i ∈ N, let fi denote the composition of f with itself i times. Then ker(fi ) = Ci and so Ci is a coprincipal coideal. Lemma 2.16. Let I be a right coideal of C. ∗ (i) I is coprincipal in C if and only if I ⊥(C ) is a principal ideal in C ∗ . (ii) If D is a subcoalgebra of C and I is coprincipal, then I ∩ D is a coprincipal coideal of D. Proof. (i) Assume that I is coprincipal and let f : C → C be a right C-comodule map such that ker(f) = I . The dual map f∗ : C ∗ → C ∗ is a right C ∗ -module map and ∗ ∗ ∗ Im(f∗ ) = Ker(f)⊥(C ) = I ⊥(C ) . Then I ⊥(C ) = C ∗ where = f∗ ().  Conversely, sup∗ pose that I ⊥(C ) = C ∗ for some ∈ C ∗ . We de ne f : C → C; c → (c)  ; c(1) c(2) . It is routine to check that f is a right C-comodule map and ker(f) = I . (ii) Let i : D → C be the inclusion map and consider i∗ : C ∗ → D∗ the dual map. ∗ ∗ ∗ We have that i∗ (I ⊥(C ) ) = i−1 (I )⊥(D ) = (I ∩ D)⊥(D ) . By hypothesis and the above ∗ ∗ item, I ⊥(C ) is a principal right ideal of C ∗ . Hence (I ∩ D)⊥(D ) is a principal right ideal of D∗ , and from (i) we conclude that I ∩ D is coprincipal in D. Theorem 2.17. Let C be a basic coalgebra. Then, C is serial if and only if C0 is a coprincipal right and a coprincipal left coideal. Proof. The socle of the right C-comodule V = C=C0 is equal to C1 =C0 . Since C is serial, each simple comodule appears only once in V . This also happens in C because C is basic. Taking injective envelopes, we get that E(V ), and hence V , embeds in C as a right comodule. This implies that C0 is a coprincipal right coideal. A symmetric argument proves that C0 is a coprincipal left ideal. Conversely, suppose that C0 is a coprincipal right coideal of C. In light of the above lemma C0 is a coprincipal coideal of C1 . Then C1 =C0 embeds in C1 as a right C1 -comodule. Since C is basic, each simple of C1 =C0 occurs with multiplicity one. Hence for each primitive idempotent e ∈ C0∗ , (C1 =C0 )e is simple or zero. From Proposition 3.4(i), it follows that C is right serial. Analogously, C is left serial. Example 2.18. Let C be the path coalgebra associated to the quiver A∞ ∞ . For each vertex i let Sin be the path of length n starting at i. We de ne f : C → C by f(Si0 ) = 0; n−1 f(Sin ) = Si+1 : 104 J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 It is not dicult to check that f is a bicomodule map whose kernel is C0 . Hence C0 is a coprincipal right and left coideal of C. Note that under the canonical isomorphism Com−C (C; C) ∼ = = C ∗ ; f → f=f∗ (), the space of bicomodule maps ComC−C (C; C) ∼ ∗ ∗ Z(C ), the center of C ∗ . Thus  = f∗ () ∈ Z(C ∗ ) and C0⊥(C ) = C ∗  = C ∗ . 3. Uniserial coalgebras characterized A coalgebra C is said to be colocal if C0 is the dual of a nite dimensional division algebra. Theorem 3.1. Let C be a coalgebra. The following assertions are equivalent: (i) (ii) (iii) (iv) C is uniserial. C∼ = ⊕ ∈
M c (D ; n ) where D is a colocal serial coalgebra. Every right (left) coideal of C is coprincipal. The diagram of C consists of isolated points and loops. Proof. (i) ⇒ (ii) Since any coalgebra is a direct sum of indecomposable coalgebras, we may assume that C is indecomposable and uniserial. Suppose that C = P (n) ⊕ P ′ where P is an indecomposable injective coideal and P ′ has no direct summands isomorphic to P. We prove that P ′ = {0}. If P ′ = {0}, then P ′ contains a simple coideal T such that T is not isomorphic to S, where S = soc(P). As C is indecomposable, its Ext-quiver its connected. There are simple comodules S1 ; : : : ; Sn with S1 =S; Sn =T and indecomposable comodules M1 ; : : : ; Mn−1 satisfying that soc(Mi ) ∼ = Si+1 . Hence E(Mi ) is = Si and Mi =Si ∼ indecomposable and, by hypothesis, homogeneous uniserial. Since soc(E(Mi )=Si ) ∼ = Si , we get Si+1 ∼ = T which is a contradiction. Thus, C = P (n) . Now, = Si . From this, S ∼ C ∼ = M c (e−C (P); n) and e−C (P) is the basic coalgebra of C with S as = e−C (P (n) ) ∼ unique simple coideal. We conclude that e−C (P) is colocal. (ii) ⇒ (iii) If J is a right coideal of C, then J = ⊕ ∈
J , where J is a right coideal of M c (D ; n ). Since the direct sum of coprincipal coideals is coprincipal, we may assume that C = M c (D; n) where D is a colocal serial coalgebra. On the other hand, every right coideal of C is of the form M c (I; n) where I is a right coideal of D. Moreover, if I is coprincipal, then M c (I; n) is so. Hence, it suces to prove that every right coideal of D is coprincipal. The only proper right coideals of D are of the form Dn = socn (D) for n ∈ N. Since D0 is the only simple of D and D is serial, soc(D=Dn ) = Dn+1 =Dn ∼ = D and so there is an injective = D0 . Then E(D=Dn ) ∼ D-comodule map from D=Dn into D. This gives that Dn is coprincipal for all n ∈ N. (iii) ⇒ (i) Let P be an injective indecomposable C-comodule. We can write C = P (n) ⊕ P ′ , where P ′ has no direct summands isomorphic to P. By hypothesis, soc(P) ⊕ {0} is a coprincipal coideal of C. Let f : C → C be the C-comodule map such that ker(f)=soc(P)⊕{0}. Let i : P → C be the inclusion map and  : C → P the canonical projection. Consider g = fi. We claim that ker(g) = {x ∈ P : f(x) ∈ P ′ } = soc(P). Assume that there is p ∈ P with f((p; 0)) = 0. Let I be a simple coideal of C such that I ⊂ f(P) and X = {x ∈ P: f(p) ∈ I }. On the other hand, f establishes an isomorphism between the set of simple coideals non isomorphic to soc(P). There is a simple coideal J such that f(J )=I . Since f(X )= J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 105 f(J ), we may pick y ∈ J such that f((p; 0)) = f((0; y)). Then (p; −y) ∈ soc(P) ⊕ {0} yielding a contradiction. We conclude that ker(g) = soc(P) and thus soc2 (P)=soc(P) = soc(P=soc(P)) ∼ = soc(P): = soc(Im(g)) ∼ Now soc2 (P) ⊕ {0} is a coprincipal coideal of C. Let f : C → C be a C-comodule map such that ker(f) = soc2 (P) ⊕ {0}. Consider g = fi. Arguing as before, it may be proved that ker(g) = soc2 (P). Then soc3 (P)=soc2 (P) = soc(P=soc2 (P)) ∼ = soc(P): = soc(Im(g)) ∼ Continuing this process we see that the socle series is a composition series for P. (i) ⇒ (iv) Note that for each primitive idempotent ei , the comodule C1 ei =C0 ei is either zero or isomorphic to C0 ei by hypothesis. (iv) ⇒ (i) The assumption on D(C) implies that C1 is uniserial. By Theorem 1.9, C is uniserial. Theorem 3.2. Let C be a cocommutative coalgebra. The following assertions are equivalent: (i) C is serial. (ii) The dual of each irreducible component of C is noetherian and its only ideals are the powers of the maximal ideal which is principal. (iii) C is uniserial. When C is pointed and serial, each irreducible component is isomorphic to a subcoalgebra of the divided power coalgebra. Proof. (i) ⇒ (ii) Let {Ci }i∈I be a family of irreducible subcoalgebras such that C = ⊕i∈I Ci . Since Ci is serial, (Ci )1 =(Ci )0 is simple. Hence (Ci )1 is nite dimensional. By [15, Theorem 5.2.1], Ci∗ is noetherian. Any ideal I of C ∗ is nitely generated. From ∗ [15, Proposition 1.3.1] it must be of the form I = D⊥(C ) for some subcoalgebra D of ∗ ⊥(C ∗ ) C. Since Ci is serial, D = (Ci )n for some n ∈ N. Then I = D⊥(Ci ) = (Ci )n i = J n+1 ∗ ⊥(C ) where J = (Ci )0 i is the only maximal ideal, [15, Corollary 4.1.2]. From Lemma 2.16 and Theorem 2.17, J is a principal ideal. (ii) ⇒ (iii) It follows from Theorem 3.1 and Corollary 1.10. (iii) ⇒ (i) Obvious. If Ci is pointed and uniserial, its Ext-quiver  is an isolated point with a loop, Theorem 3.1. Hence k is isomorphic to the divided power coalgebra. Now Theorem 2.10(i) and Remark 2.12 apply. We give an application of this result to compute the nite dual of some Hopf algebras. It is known that a pointed cocommutative Hopf algebra H is isomorphic to the smash product H1 #kG(H ), where G(H ) is the set of group-like elements of H , H1 is the irreducible component of H containing to 1, and G(H ) acts on H1 by conjugation, see [18, Corollary 5.6.4]. If char(k)=0, then H1 ∼ = U (P(H1 )), the universal enveloping algebra of the primitive elements P(H1 ) of H1 , see [18, Theorem 5.6.5]. 106 J. Cuadra, J. Gomez-Torrecillas / Journal of Pure and Applied Algebra 189 (2004) 89 – 107 Let H be a commutative Hopf algebra such that H 0 is pointed and serial. By Theorem 3.2, P(H10 ) is one dimensional and hence U (P(H10 )) ∼ = k[x]. Then H 0 ∼ = k[x]#kG(H 0 ). 0 0 ∼ 0 If in addition H is commutative, then H = k[x] ⊗ kG(H ). Using this result we can give an alternative computation of H = k[x]0 ; k[x; x−1 ]0 or k[[x]]0 when k is algebraically closed and of characteristic zero. The group-like elements of H are the algebra maps from H to k. By Corollary 1.11(ii), H 0 is serial. Then k[x]0 ∼ = k[x] ⊗ k(k; +) where (k; +) is the additive group of k; k[x; x−1 ]0 ∼ = k[x] ⊗ k(k ∗ ; ·), where (k ∗ ; ·) is the multiplicative group of k; and k[[x]]0 ∼ = k[x]. The answer to the question posed in Remark 1.12 is negative. This will follow from the general result [11, Corollary 4.7] which asserts that a locally nitely presented Grothendieck category C is pure-semisimple if and only if every object in C is a direct sum of indecomposable objects. Applying this result to MC and CM, if C is a serial coalgebra satisfying that every left and every right C-comodule decomposes as a direct sum of uniserial comodules, then MC and CM are pure-semisimple. Hence C is right and left pure-semisimple (see [21]), that is, every right and every left C-comodule is a direct sum of nite dimensional subcomodules. In particular, each indecomposable C-comodule is nite dimensional. The counterexample to our question is the divided power coalgebra (see Example 1.4. (3)). It is serial and indecomposable as a right comodule but not nite dimensional. Acknowledgement We would like to thank Prof. J.L. Garcia for providing reference [11] and some helpful discussions. 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