Journal of Algebra 324 (2010) Contents lists available at ScienceDirect. Journal of Algebra.

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1 Journal of Algebra 324 (2) ontents lists available at ScienceDirect Journal of Algebra wwwelseviercom/locate/jalgebra oxeter transformation and inverses of artan matrices for coalgebras William hin a,, Daniel Simson b, a Department of Mathematics, DePaul University, hicago, IL 664, USA b Faculty of Mathematics and omputer Science, Nicolaus opernicus University, 87- Toruń, ul hopina 2/8, Poland article info abstract Article history: Received 2 May 29 ommunicated by Nicolás Andruskiewitsch MS: 6G2 6G6 6G7 6W3 Keywords: oalgebra omodule Pseudocompact algebra artan matrix oxeter transformation Auslander Reiten quiver Let be a coalgebra and let Z, I Z I Z I be the Grothendieck groups of the category op -inj and -inj of the socle-finite injective right and left -comodules, respectively One of the main aims of the paper is to study the oxeter transformation Φ : Z I Z I and its dual Φ : ZI Z I of a pointed sharp Euler coalgebra, and to relate the action ofφ and Φ on a class of indecomposable finitely cogenerated -comodules N with the ends of almost split sequences starting with N or ending at N By applying hin, Kleiner, and Quinn (22) [5], we also show that if is a pointed K -coalgebra such that the every vertex of the left Gabriel quiver Q of has only finitely many neighbours then for any indecomposable non-projective left -comodule N of finite K -dimension, there exists a unique almost split sequence τ N N N in the category -omod fc of finitely cogenerated left -comodules, with an indecomposable comodule τ N We show that dim τ N = Φ (dim N), if is hereditary, or more generally, if injdim DN = andhom (, DN) = 2 Elsevier Inc All rights reserved Throughout we fix an arbitrary field K and D( ) = ( ) = Hom K (, K ) is the ordinary K -linear duality functor We recall that a K -coalgebra is said to be pointed if all simple -comodules are one-dimensional Let be a pointed K -coalgebra and = Hom K (, K ) the K -dual (pseudocompact [7]) K -algebra with respect to the convolution product, see [9,4] We denote by -omod and -comod the category of left -comodules and finite-dimensional left -comodules, respectively The * orresponding author addresses: wchin@condordepauledu (W hin), simson@matunitorunpl (D Simson) Supported by Polish Research Grant P3A N N2/2692/35/ /$ see front matter 2 Elsevier Inc All rights reserved doi:6/jjalgebra2629

2 2224 W hin, D Simson / Journal of Algebra 324 (2) corresponding categories of right -comodules are denoted by op -omod and op -comod The socle of a comodule M in -omod is denoted by soc M We recall from [5] and [6] that, for a class of coalgebras (including left semiperfect ones), given an indecomposable non-injective -comodule M in -comod and an indecomposable non-projective -comodule N in -comod there exist almost split sequences M M τ M and τ N N N ( ) in -omod, where -omod τ fc -comod f P are the Auslander Reiten translate operators (7) On the other hand, for a class of computable coalgebras, aartanmatrixc M I (Z), its τ inverse c, and a corresponding oxeter transformation Φ : K () K () is defined and studied in [9] (see also [4] and []), where K () = K (-comod) = Z (I ) is the Grothendieck group of -comod One of the main aims of this paper is to construct an inverse c (left and right) of the artan matrix c and oxeter transformations Φ : Z I Z I, Φ : ZI Z I, for any computable coalgebra such that any simple left (and right) -comodule admits a finite and socle-finite injective resolution We prove that, under a suitable assumption on indecomposable -comodules N and M, there exist almost split sequences ( ) in -omod and dimτ (M) = Φ (dim M) and dimτ ( N) = Φ (dim N), where dim X Z I is the dimension vector (2) of the comodule X, compare with [, orollary IV29] and [9, p 67] We recall from [2] that a coalgebra is said to be left locally artinian if every indecomposable injective left -comodule is artinian Recall also that a coalgebra over an algebraically closed field is pointed if and only if it is basic, see [7] Preliminaries on comodule categories Let be a K -coalgebra We collect in this section basic facts concerning -comodules, pseudocompact -modules, the existence of almost split sequences in -omod, duality and injectives in the category of comodules We recall from [9,4,7] that any left -comodule M is viewed as a rational (= discrete) right module over the pseudocompact algebra and M = D(M) = Hom K (M, K ) is a pseudocompact left -module The functor D( ) defines a duality D : -omod -P, where -P is the category of pseudocompact left -modules The quasi-inverse is the functor ( ) = hom K (, K ) that associates to any Y in -P the left -comodule Y = hom K (Y, K ) consisting of all continuous K -linear maps Y K It follows from [] that the algebra is left (and right) topologically semiperfect, that is, every simple left -module admits a projective cover in -P (see also [7]); equivalently, admits a decomposition = j I e j in -P, where {e j } j I is a topologically complete set of pairwise orthogonal primitive idempotents such that e j e j is a local algebra, for every i I The decomposition is unique up to isomorphism and permutation The coalgebra (or more generally, any --bicomodule) can be viewed as a bimodule over the algebra with respect to the right and the left hit actions of on, usually denoted by the symbols, as in [9] and [4] Here we omit these symbols and simply use juxtaposition, eg, e = e and e = e, foranye Notice that e is an injective right -comodule and e is an injective left -comodule, for any idempotent e The following two simple lemmata are often used in the paper Lemma Assume that is a coalgebra, e = e 2 is an idempotent in,andd( ) = Hom K (, K ) (a) There is an isomorphism D(e) = eofleft -modules

3 W hin, D Simson / Journal of Algebra 324 (2) (b) If is of finite dimension, then there is an isomorphism D( e) = e of right -modules (c) Hom (, e) = Hom (, e) for every subcoalgebra of Proof See [6] and [8] Lemma 2 Let be a K -coalgebra Given a left comodule M in -comod, the K -dual space D(M) = Hom K (M, K ) admits a natural structure of right -comodule and D( ) = Hom K (, K ) defines the pair of dualities D -comod op -comod (3) D Proof See [25] An injective copresentation of a comodule M in -omod is an exact sequence M E g E, (4) where E and E are injective comodules We call a comodule M in -omod (socle) finitely copresented if M admits a socle-finite injective copresentation, that is, the injective comodules E and E have finite-dimensional socle We denote by -omod fc the full subcategory of -omod whose objects are the finitely copresented comodules, and by -inj the full subcategory of -omod fc whose objects are the socle-finite injective comodules We set -comod fc = -comod -omod fc Finally, we denote by -omod fc = omod fc /I the quotient category of -omod fc modulo the two-sided ideal I =[-inj] consisting of all f Hom (N, N ),withn and N in -omod fc, that have a factorisation through a socle-finite injective comodule, see [9] and [22] It is observed in [2] that -omod fc is an abelian category if and only if is left cocoherent In this case -omod fc is closed under extensions in -omod and contains minimal injective resolutions of comodules M in -omod fc, see [9, Section 3] We recall that a comodule M is quasi-finite if dim K Hom (X, M) is finite, for any X in -comod; equivalently, if the simple summands of soc M have finite (but perhaps unbounded) multiplicities [25] It follows that every socle-finite comodule is quasi-finite Hence all comodules in -omod fc are quasi-finite Givenaleftquasi-finite-comodule M, the covariant cohom functor h (M, ) : -omod Mod(K) is defined by associating to any comodule N in -omod the vector space h (M, N) = lim λ D Hom (N λ, M), where {N λ } is the family of all finite-dimensional subcomodules of N [25] Denote by op -omod fp the full subcategory of op -omod whose objects are the (injectively) finitely presented op -comodules, that is, the op -comodules L that admit a short exact sequence E g E L in op -omod, with socle-finite injective comodules E and E, called a soclefinite injective presentation of L Following [5, Section 3], we define a pair of contravariant left exact functors -omod fc op -omod fp (5)

4 2226 W hin, D Simson / Journal of Algebra 324 (2) to be the composite functors making the following diagrams commutative -omod fc D -P fp -omod fc ( ) -P fp ( ) + ( ) + (6) op ( ) -omod fp op -P fc, op -omod fp D op -P fc, where -P fp (resp op -P fc ) is the category of pseudocompact (top-) finitely presented (resp (top-) finitely copresented) modules (see [9,7]), D = HomK (, K ), ( ) + = hom (, ) : -P fp op -P fc is a contravariant functor that associates to any X in -P fp, with the top-finite pseudocompact projective presentation P P X, where P, P are finite direct sums of inde- f composable projective -modules, the right -module X + = hom (X, ) of all continuous - homomorphisms X, with the top-finite pseudocompact projective copresentation X + P + f + P + Finally, Y = hom K (Y, K ) consists of all continuous K -linear maps Y K and ( ) associates to X + the right -comodule (X + ) in op -omod fp, with the socle-finite injective presentation ( P + ) ( f + ) ( P + ) ( X + ) The functors in the right-hand diagram of (6) are defined analogously Sometimes, for simplicity of the notation, we write instead of Following [5] and the classical construction of Auslander [2], we define the Auslander transpose operator Tr = Tr : -omod fc op -omod fc (7) (on objects only!) that associates to any comodule M in -omod fc, with a minimal socle-finite injective copresentation (4), the comodule Tr M = Ker [ E (g) E ] in op -omod fc Basic properties of Tr are listed in [5, Proposition 32] The existence of almost split sequences in -omod fc essentially depends on the following theorem slightly extending some of the results in [5] and [6] Theorem 8 Let be a K -coalgebra and the functor (5) (a) There are functorial isomorphisms M = Hom (, M) = h (M, ), for any comodule M in -omod fc (b) The functors, are left exact and restrict to the dualities -inj op -inj (9)

5 W hin, D Simson / Journal of Algebra 324 (2) that are quasi-inverse to each other Moreover, given an idempotent e, the comodule e lies in op -inj, the comodule e lies in -inj, and there is an isomorphism (e) = e of left -comodules (c) For any comodule M in -omod fc, with a minimal socle-finite injective copresentation (4),thecomodules Tr M, E, E lie in op -omod fc, Mliesin op -omod fp, and the following sequence Tr M E (g) E M () is exact in op -omod (d) The transpose operator Tr, together with the functor, induces the equivalence of quotient categories Tr : -omod fc op -omod fc Proof For our future purpose and the convenience of the reader, we outline the proof (a) Let { λ } is the family of all finite-dimensional subcoalgebras of and let M be a comodule in -omod fc ThenM is quasi-finite, = lim λ λ, and we get isomorphisms M = ( + ( DM) ) = hom ( DM, ) = Hom (, M) [ = lim Hom ( λ, M) λ = lim λ Hom ( λ, M) ] = lim λ D Hom ( λ, M) = h (M, ) One can easily see that the composite isomorphism is functorial at M (b) Apply the definition of To prove (c) and (d), we note that the exact functors D : -omod fp -P fp and ( ) : op -P fc op -omod fc defining the functor are equivalences of categories carrying injectives to projectives and projectives to injectives, respectively Recall that is a topologically semiperfect algebra Now, given an indecomposable comodule M in -omod fc, with a minimal socle-finite injective copresentation (4), we get a pseudocompact minimal top-finite projective presentation Dg DE DE DM, in -P, with DE = E, DE = E finite direct sums of indecomposable projective -modules, of the right pseudocompact -module DM Hence, by applying the left exact functor hom (, ), and the definition of the Auslander transpose Tr ( DM) of the pseudocompact left -module DM, we get the exact sequence ( DM) + ( DE ) + ( Dg) + ( DE ) + Tr ( DM) () in op -P and the projective copresentation ( DM) + (DE ) + ( Dg) + (DE ) + of the right pseudocompact -module ( DM) +, where ( DE ) + and ( DE ) + are finitely generated projective top-finite right -modules The sequence () induces the sequence () and (c) follows The statement (d) follows from the corresponding properties of the Auslander transpose operator Tr : -P fp op -P fp on the pseudocompact finitely presented top-finite modules over Herewe can follow the proof of [, Proposition IV22] or [3, Section IV2]

6 2228 W hin, D Simson / Journal of Algebra 324 (2) We denote by -omod fc and by -omodν fc the full subcategory of -omod fc consisting of the comodules M such that dim K Tr (M) is finite and dim K ( DM) + is finite, respectively Following the representation theory of finite-dimensional algebras, we define the (covariant) Nakayama functor ν : -omod ν fc -comod (2) by the formula ν ( ) = D ( ) A coalgebra is said to be left semiperfect [3] if every simple left comodule has a projective cover, or equivalently, the injective envelope E(X) of any finite-dimensional right -comodule X is finite-dimensional It is easy to see that, for a left semiperfect coalgebra, the functor ν restricts to the equivalence of categories ν : -inj -proj, (3) where -proj is the category of top-finite projective comodules in -comod We denote by -comod f P the full subcategory of -comod consisting of the left comodules N that, viewed as rational right -modules, have a minimal top-finite projective presentation P P N in op -P = P-, that is, P and P are top-finite projective modules in P- Here we make the identification -comod rat- = dis- P-, in the notation of [7, Section 4], where rat- is the category of finite-dimensional rational right -modules Finally, we denote by -comod f P = -comod f P /P the quotient category of -comod f P modulo the two-sided ideal P of -comod f P consisting of all f Hom (N, N ),withn and N in -comod f P, that have a factorisation through a projective right -module, when f is viewed as a -homomorphism between the rational right -modules N and N If is left semiperfect then, in view of the exact sequence () in op -omod, we have -comod f P = -comod, -omod fc = -omod fc, -omod ν fc = -omod fc and, by applying ν to the sequence () we get the exact sequence in -comod The following lemma is very useful ν (M) ν (E ) ν (g) ν (E ) D Tr (M) (4) Lemma 5 Let be a pointed K -coalgebra and let QbetheleftGabrielquiverof (a) The duality D : -comod op -comod (3) restricts to the duality D : -comod f P op -comod fc = op -comod op -omod fc In particular, a left -comodule N lies in -comod f P if and only if the right -comodule D(N) is finitely copresented (b) The following four conditions are equivalent: (b) -comod f P = -comod,

7 W hin, D Simson / Journal of Algebra 324 (2) (b2) op -comod op -omod fc, (b3) every simple comodule in op -omod is finitely copresented, (b4) the quiver Q is right locally bounded, that is, for every vertex a of Q there is only a finite number of arrows a jin Q (c) If is right locally artinian, we have -comod f P = -comod Proof (a) Since we make the identification -comod rat- = dis- P- (in the notation of [7, Section 4]), there is a commutative diagram -comod id dis- P- D = ( ) = ( ) = op -comod id -dis op -omod Then a left -comodule N lies in -comod f P if and only if there is an exact sequence P P N inp-, where P and P are top-finite projective modules in P-,orequivalently,N lies in op -P fp = P fp - By applying the duality ( ) : op -P op -omod (6), we get an exact sequence N P P in op -omod Since dim K N is finite, we have N = D(N) This shows that D(N) lies in op -comod fc,becausep and P are socle-finite injective right -comodules It follows that the duality (3) restricts to the duality D : -comod f P op -comod fc (b) By (a), the equality -comod f P = -comod holds if and only if the equality op -comod fc = op -comod holds, that is, the conditions (b) and (b2) are equivalent The implication (b2) (b3) is obvious To prove the inverse implication (b3) (b2), we assume that the simple right -comodules lie in op -omod fc and let X be a comodule in op -comod By standard arguments and the induction on the K -dimension of X, we show that X lies in -omod fc (apply the diagram in [5, p 3]) (b3) (b4) Let soc = j I Ŝ( j) be a direct sum decomposition of the right socle soc of, where I is an index set and {Ŝ( j)} j I is a set of pairwise non-isomorphic simple right -coideals Denote by Ê( j) = E(Ŝ( j)) the injective envelope of Ŝ( j) It follows from the dualities (9) and [8, Theorem 23(a)] that the quiver Q is dual to the right Gabriel quiver Q of Hence, by the assumption (b2), for every vertex a of the quiver Q, thereisonlyafinitenumberofarrows j a in Q In other words, dim K Exp (Ŝ( j), Ŝ(a)) is finite, for all j I, and dim K Ext (Ŝ( j), Ŝ(a)) =, for all but a finite number of indices j I,see[7, Definition 86] Fix a I and let Ŝ(a) Ê(a) Ê be a minimal injective resolution of Ŝ(a) in op -omod, with Ê = E(soc(Ê(a)/Ŝ(a))) Given j I, we denote by μ (Ŝ( j), Ŝ(a)) the number of times the comodule Ê( j) appears as a direct summand in Ê Since is assumed to be pointed, dim K End Ŝ( j) = and μ ( Ŝ( j), Ŝ(a) ) = dim K Ext ( Ŝ( j), Ŝ(a) ), by [2, (423)] Thus the injective op -comodule Ê is socle finite, by the observation made earlier, and it follows that the simple right -comodule Ŝ(a) is finitely copresented This shows that (b3) implies (b4) Since the inverse implication follows in a similar way, the proof of (b) is complete (c) Apply (b) and the easily seen fact that simple right comodules over any right locally artinian coalgebra are finitely copresented Following [5], we get the following result

8 223 W hin, D Simson / Journal of Algebra 324 (2) Proposition 6 Let be a K -coalgebra and D the duality functors (3) (a) The transpose equivalence of Theorem 8(d), defines the equivalence Tr : -omod fc op -comod fc, and together with the duality D : op -comod fc -comod f P defined by (3), induces the translate operator τ = D Tr : -omod fc -comod f P, (7) and an equivalence of quotient categories τ = D Tr : -omod fc Minomod fc, with a presentation (4), the following sequence -comod f P Moreover,forany ( DM) + ( DE ) + ( Dg) + ( DE ) + τ M (8) is exact in op -P and the comodule τ Mliesin-comod f P op -rat fp op -P (b) The duality (3) restricts to the duality D : -comod f P op -comod fc and together with the transpose operator Tr op : op -comod fc -omod fc defines the translate operator τ = Tr op D : -comod f P -omod fc, (9) and induces the equivalence of quotient categories τ = Tr op D : -comod f P -omod that is fc quasi-inverse to the equivalence τ = D Tr : -omod -comod fc f P in (a) (c) Let M be an indecomposable comodule in -omod Thenτ fc M = if and only if M is injective If τ M then τ M is indecomposable, non-projective, of finite K -dimension, and there is an isomorphism M = τ τ M (d) Let N be an indecomposable comodule in comod f P Thenτ N = if and only if N is projective If τ N then τ N is indecomposable, non-injective, finitely copresented, and there is an isomorphism N = τ τ N Proof By Lemma 5(a), the duality D : op comod comod (3) restricts to the duality D : op -comod fc -comod f P One also shows, by applying foregoing definitions, that a homomorphism f : X X in op -comod fc has a factorisation through a socle-finite injective comodule if and only if the homomorphism D( f ) : D(X) D(X ) in -comod f P belongs to P(D(X), D(X )) This shows that the duality D : op -comod fc -comod f P induces an equivalence of quotient categories D : op -comod fc -comod f P It follows from the definition of the category -omod that the fc transpose equivalence of Theorem 8(d), defines the equivalence Tr : -omod op -comod fc fc This together with the earlier observation implies (a) and (b) The statements (c) and (d) are obtained by a straightforward calculation and by using the definition of translates τ and τ, see [5] and consult [3] The details are left to the reader Following the terminology of representation theory of finite-dimensional algebras (see [,3]), we call the operators τ = Tr op D (9) and τ = D Tr (7), the Auslander Reiten translations of It follows from Theorem 6 that the image of τ is the subcategory -comod f P of the category -comod

9 W hin, D Simson / Journal of Algebra 324 (2) By applying Theorem 42 and orollary 43 in [5], we get the following result on the existence of almost split sequences in the category -omod fc of (socle) finitely copresented left -comodules, under some assumption on the Gabriel quiver Q of Theorem 2 Let be a K -coalgebra such that its left Gabriel quiver Q is left locally bounded, that is, for every vertex a of Q there is only a finite number of arrows j ain Q (a) The following inclusion holds -comod -omod fc (b) For any indecomposable non-injective comodule M in -omod fc, there exists a unique almost split sequence M M τ M (2) in -omod fc, with a finite-dimensional indecomposable comodule τ Mlyingin-comod f P Thesequence (2) is almost split in the whole comodule category -omod (c) For any indecomposable non-projective comodule N in -comod f P -omod fc, there exists a unique almost split sequence τ N N N (22) in -omod fc, with an indecomposable comodule τ Nlyingin-omod fc Thesequence(22) is almost split in the whole comodule category -omod (d) If, in addition, is left semiperfect then -omod fc = -omod fc, -comod f P = -comod, the Auslander Reiten translate operators act as follows τ -omod fc -comod τ and the almost split sequences (2) and (22) do exist in the category -omod fc, for any indecomposable non-injective comodule M in -omod fc and for any indecomposable non-projective comodule Nin-comod Moreover, if the comodule M lies in -comod then the almost split sequence (2) lies in -comod Proof (a) As in the proof of Lemma 5(b), we conclude from the assumption that Q is left locally bounded that every simple left -comodule admits a minimal socle-finite injective copresentation (4) Hence (a) follows as in Lemma 5(b) The statements (b) and (c) follow from Theorem 42 and orollary 43 in [5], because any comodule M lying in -omod fc is quasi-finite, Proposition 6(a) yields that τ M lies in -comod f P, for any indecomposable comodule M in -omod fc, and the following inclusions hold -comod f P -comod -omod fc,by(a) (d) Assume that is left semiperfect and let M be an indecomposable comodule in -omod fc with a minimal socle-finite injective copresentation (4) By Theorem 8, the induced sequence () is exact and the comodules (E ) and (E ) lie in op -inj Since is left semiperfect, the comodules (E ) and (E ) are finite-dimensional and, hence, dim K Tr (M) is finite, for any comodule M in -omod fc It follows that -omod fc = -omod fcsince is left semiperfect, any comodule N in -comod has a projective presentation P P N, with P, P finitedimensional projective -comodules It follows that N lies in -comod f P and, hence, the equality -comod f P = -comod holds This finishes the proof of the theorem orollary 23 Let be a pointed K -coalgebra such that the left Gabriel quiver Q of is both left and right locally bounded

10 2232 W hin, D Simson / Journal of Algebra 324 (2) (a) The inclusions -comod f P = -comod -omod fc hold and the Auslander Reiten translate operators act as follows τ -omod fc -comod (b) For any indecomposable non-injective comodule M in -omod fc, there exists a unique almost split sequence τ M M τ M in -omod fc, with an indecomposable comodule τ Mlyingin-comod (c) For any indecomposable non-projective comodule N in -comod, there exists a unique almost split sequence τ N N N in -omod fc, with an indecomposable comodule τ Nlyingin-omod fc (d) The exact sequences in (b) and (c) are almost split in the whole comodule category -omod Proof Apply Lemma 5 and Theorem 2 Remark 24 (a) Under the assumption that the left Gabriel quiver Q of is both left and right locally bounded the almost split sequences (2) and (22) lie in -omod fc If we drop the assumption then the term τ M lies in -comod f P -comod, but not necessarily lies in -omod fc (b) We are mainly interested in the existence of almost split sequences in the category -omod fc, because it is the part of -omod that plays a crucial role in the study of tameness of the coalgebra, see [22] Now we illustrate the existence of almost split sequences discussed in orollary 23 by the following example Example 25 Let Q = (Q, Q ) be the infinite locally Dynkin quiver Q : 2 s s s+ of type D and let = K Q be the hereditary path K -coalgebra of Q ThenQ ={,,, 2,} and has the Q Q matrix form K K K K K K K K K K K K K K K K K K K K K K = K K K K K K K K K K

11 W hin, D Simson / Journal of Algebra 324 (2) Fig The Auslander Reiten quiver of the category -comod = repk (Q ) and consists of the triangular Q Q square matrices with coefficients in K and at most finitely many non-zero entries Then soc = j Q S( j), where S(n) = Ke n is the simple subcoalgebra of spanned by the matrix e n with in the n n entry, and zeros elsewere Note that e n is a group-like element of Since the left Gabriel quiver Q of is the quiver Q, it follows from Lemma 5(b) that every simple right -comodule is finitely copresented and the statements (a) and (c) of orollary 23 hold for = K Q The coalgebra is right locally artinian, right semiperfect, representation-directed in the sense of [9], and left pure semisimple, that is, every left -comodule is a direct sum of finite-dimensional comodules (see [6] or [7]) It follows that -omod fc = -comod and every indecomposable nonprojective comodule N in -comod admits an almost split sequence τ N N N in -comod Under the identification -comod = rep K (Q ) of left -comodules and K -linear representations of the quiver Q (see [5] or [7], [2, (3)]), the Auslander Reiten translation quiver Γ(-comod) of -comod has four connected components (two of them are finite and two are infinite), and Γ(-comod) has the following form (see [6]) Here we use the terminology and notation introduced in [6, pp ] Recall that the vertices of the Auslander Reiten translation quiver Γ(-comod) are representatives of the indecomposable left -comodules in -comod and the existence of an arrow X Y in Γ(-comod) means that there exists an irreducible morphism f : X Y in -comod, see also [23] Each of the two finite components of Γ(-comod) contains precisely one indecomposable simple projective -comodule; namely the comodule I and I, respectively Each of the two infinite components contains no non-zero projective objects The indecomposable injective left -comodules form the right-hand section I I 6 I 5 I 4 I 3 I 2 I I ( )

12 2234 W hin, D Simson / Journal of Algebra 324 (2) of the infinite upper component of Γ(-comod), and the indecomposable left -comodules M in -comod such that dim K Tr M is infinite are the two simple projective comodules I, I and the comodules lying on the infinite right-hand section I 6 I 5 I 4 I 3 I 2 I ( ) of the infinite lower component of Γ(-comod) It follows that an indecomposable left -comodule M lies in the category -omod fc = -comod if and only if M lies in the infinite upper component of Γ(-comod) or M lies in the infinite lower component of Γ(-comod), butdoesnotlieonthe infinite section ( ) Every indecomposable comodule N lying in one of the infinite components of Γ(-comod) has an almost split sequence τ N N N in-comod and it is given by the mesh in Γ(-comod) terminating at N, compare with the examples given in Section 4 2 artan matrix of an Euler coalgebra and its inverses Throughout we assume that K is an arbitrary field and is a pointed K -coalgebra It follows that is basic and there exists a direct sum decomposition soc = j I S( j) of the left socle soc of, where I is an index set and {S( j)} j I is a set of pairwise non-isomorphic simple left -coideals, see [4] or [7] Then {S( j)} j I is a set of representatives of the isomorphism classes of simple left -comodules and dim K S( j) = dim K End S( j) =, for any j I For every j I,letE( j) = E(S( j)) denote the injective envelope of S( j) It follows that E( j) is indecomposable, = j I E( j), and there is a primitive idempotent e j such that E( j) = e j Working with right -comodules, we have the simple right -comodules Ŝ( j) = DS( j), withinjective envelopes Ê( j) = (E( j)) = e j in op -inj, see Theorem 8 Throughout we fix a set {e j } j I of primitive idempotents of such that E( j) = e j,forall j I Following the representation theory of finite-dimensional algebras, given a left -comodule M (viewed as a rational right -module), we define its dimension vector dim M =[dim K Me j ] j I, (2) where dim K Me j has values in Z { } Since is pointed, dim K Me j = dim K Hom (M, e j ) = dim K Hom (M, E( j)) and the dimension vector dim M coincides with the composition length vector lgth M =[l j (M)] j I of M (introduced in [9]), where l j (M) = dim K Hom ( M, E( j) ) = dimk Me j (22) It follows from [9, Proposition 26] that l j (M) = dim K Me j is the multiplicity the simple comodule S( j) appears as a composition factor in the socle filtration soc M soc M soc m M of M Following [9], M is said to be computable if the composition length multiplicity l j (M) = dim K Me j of S( j) in M is finite, for every j I,orequivalently,dim M Z I (the product of I copies of the infinite cyclic group Z) A pointed coalgebra is defined to be computable if the injective comodule E(i) is computable, or equivalently, if the dimension vector e(i) = dim E(i) =[dim K e i e j ] j I = [ dim K Hom ( E(i), E( j) )] j I (23) has finite coordinates, for every i I Note that the class of computable coalgebras contains left semiperfect coalgebras, right semiperfect coalgebras and the incidence coalgebras K I of intervally finite posets I, see [9] Moreover, if is computable and left cocoherent then the K -category -omod fc is abelian, has enough injective objects, and is Ext-finite, that is, dim K Ext m (M, N) is finite, for all m and all comodules M, N in -omod fc

13 W hin, D Simson / Journal of Algebra 324 (2) Given a pointed computable coalgebra, with soc = j I S( j), we define the left artan matrix of to be the integral I I matrix c =[c ij ] i, j I = e(i) M I (Z), (24) whose i j entry is the composition length multiplicity c ij = e(i) j = dim K e i e j of S( j) in E(i) In other words, the ith row of c is the dimension vector e(i) = dim E(i) of E(i), see [9, Definition 4] We say that a row (or a column) of a matrix is finite, if the number of its non-zero coordinates is finite A matrix is called row-finite (or column-finite) if each of its rows (columns) is finite We start with the following simple observations Lemma 25 Let be a pointed computable K -coalgebra, with a fixed decomposition soc = j I S( j), and let K () = K (-comod) be the Grothendieck group of -comod (a) Given a -comodule M in -omod, the dimension vector dim M has only a finite number of non-zero coordinates if and only if dim K MisfiniteIfdim K M <,thendim K M = j I dim K Me j (b) The map M dim M is an additive function on short exact sequences in -omod and induces the group isomorphism dim : K () Z (I ), [M] dim M, where Z (I ) is the direct sum of I copies of Z The group K () is free abelian with the basis {[S( j)]} j I corresponding via dim to the standard basis vectors e j = dim S( j) of Z (I ) Proof (a) To prove the sufficiency, assume that dim K M is finite Then ( M = Hom (M, ) = Hom M, ) ( ) E( j) = Hom M, E( j) = Me j j I j I j I It follows that dim K M = dim K M = j I dim K Me j Hence, the sum is finite and dim M has only a finite number of non-zero coordinates The converse implication follows in a similar way (b) The additivity of dim is obvious Hence, M dim M defines the group homomorphism dim : K () Z (I ) Sincetheset{[S( j)]} j I generates K (), theset{e j } j I is a Z-basis of Z (I ) and dim S( j) = e j then the homomorphism dim is bijective Lemma 26 Let be a pointed computable K -coalgebra and let c M I (Z) be the left artan matrix (24) of (a) c op = c tr, that is, c op M I (Z) is the transpose of the matrix c (b) The ith row of the matrix c is finite if and only if the indecomposable injective left -comodule E(i) is finite-dimensional (c) The jth column of the matrix c is finite if and only if the indecomposable injective right -comodule Ê( j) = (E( j)) is finite-dimensional (d) The left artan matrix c of is row-finite if and only if is right semiperfect (e) The left artan matrix c of is column-finite if and only if is left semiperfect Proof (a) Let c op =[c ij ] i, j I, c =[c ij ] i, j I M I (Z) be the artan matrices (24), of the coalgebra op and, respectively We recall from Theorem 8 that there is a duality : -inj op -inj and Ê( j) = (E( j)) Hence, by applying (23), we get c ij = dim K Hom op( Ê(i), Ê( j) ) = dim K Hom ( E( j), E(i) ) = c ji, for every pair of elements i, j I This yields the equality c op = c tr

14 2236 W hin, D Simson / Journal of Algebra 324 (2) (b) We recall from (24) that the ith row of c is the dimension vector e(i) = dim E(i) of the injective left -comodule E(i) Then (a) follows by applying Lemma 25(a) to M = E(i) (c) By (a), the jth column of c is the jth row of c op Hence, (c) follows from (b) applied to the coalgebra op (d) By (b), the matrix c is row finite if and only if dim K E(i) is finite, for any i I,orequivalently, if and only if the injective envelope of any simple left -comodule is of finite K -dimension But this property is equivalent to the right semiperfectness of, see [3] (e) According to (c), the matrix c is column finite if and only if the injective envelope of any simple right -comodule is of finite K -dimension Since this property is equivalent to the left semiperfectness of [3], the statement (e) follows and the proof is complete In [9], a class of computable coalgebras, called left Euler coalgebras, is defined in such a way that the left artan matrix c M I (Z) of such a coalgebra has a left inverse c in the nonassociative matrix algebra M I (Z) Unfortunately, usually a left inverse c is not row-finite or columnfinite Below, we introduce a class of Euler coalgebras such that the left inverse c of c is a row-finite and a column-finite matrix We would like to remark here that the multiplication in the matrix algebra M I (Z) is not associative The matrices in M I (Z) may have unequal left and right inverses and that one-sided inverse of a matrix may not be unique Moreover, the left inverse may exist, without a right inverse existing Also being invertible as a Z-linear map is not equivalent to being invertible as a matrix, see [27] We introduce a class of left Euler coalgebras as follows Definition 27 A K -coalgebra is defined to be a left (resp right) sharp Euler coalgebra if has the following two properties (a) is computable, that is, dim K Hom (E, E ) is finite, for every pair of indecomposable injective left -comodules E and E (b) Every simple left (resp right) -comodule S admits a finite and socle-finite injective resolution h h 2 h n S E E En, (28) that is, the injective comodules E,,E n are socle-finite A K -coalgebra is defined to be a sharp Euler coalgebra if it is both left and right sharp Euler coalgebra and the following condition is satisfied (c) dim K Ext m (S, S ) = dim K Ext m op(ŝ, Ŝ), forallm and all simple left -comodules S and S, where Ŝ = DS and Ŝ = DS are the dual simple right -comodules Obviously, any sharp Euler coalgebra is an Euler coalgebra in the sense of [9] Now we show that (one-sided) semiperfect coalgebras with gldim < are sharp Euler coalgebras Lemma 29 Assume that is a pointed left or right semiperfect coalgebra, with a fixed decomposition soc = j I S( j) (a) Ext m (S(a), S(b)) = Ext m op(ŝ(b), Ŝ(a)), for all m and any pair of simple left -comodules S(a) and S(b), witha, b I,whereŜ(b) = DS(b) and Ŝ(a) = DS(a) are the dual simple right -comodules corresponding to a, b I (b) If the global dimension gldim of is finite, then is a sharp Euler coalgebra Proof We prove the lemma in case is a pointed left semiperfect coalgebra The proof in case is right semiperfect follows in a similar way (a) Let S(a) and S(b) be simple left -comodules Since is left semiperfect, there is a minimal projective resolution P (a) of S(a) in -comod By Lemma 2, there is a duality D : -comod op -comod that carries P (a) to a minimal injective resolution DP (a) of Ŝ(a) in

15 W hin, D Simson / Journal of Algebra 324 (2) the category op -comod and induces an isomorphism of chain complexes Hom (P (a), S(b)) = Hom op(ŝ(b), DP (a)) Hence, we get the induced isomorphism of the cohomology K -spaces ( ) [ ( Ext m S(a), S(b) = H m Hom P (a), S(b) )] [ ( = H m Hom op Ŝ(b), DP (a) )] ( = ) Ext m Ŝ(b), Ŝ(a) op, and (a) follows (b) Assume that gldim of is finite Since is left semiperfect, the indecomposable injectives in op -omod are finite-dimensional and therefore any simple right -comodule has a finite injective resolution in op -comod Hence, is a right sharp Euler coalgebra To prove that is a left sharp Euler coalgebra, assume that S is a left simple -comodule and let h h 2 h n S E E En, be a minimal injective resolution of S in -omod We show that the injective comodules E ( j) j),,e( n are socle-finite Assume that S = S(b) Since a minimal projective resolution P (b) of S = S(b) lies in -comod and is of finite length gldim, it follows that, for m, dim K Ext m (S(a), S(b)) is finite, for all a I, and dim K Ext m (S(a), S(b)) =, for all but a finite number of simple comodules S(a) Since is pointed, dim K Ext m (S(a), S(b)) is the Bass number μ m (S(a), S(b)) of the pair (S(a), S(b)), that is, μ m (S(a), S(b)) is the multiplicity the indecomposable injective comodule E(a) appears in E m, as a direct summand, see [2, (423)] It follows that, for each m, the number μ m (S(a), S(b)) is finite, and μ m (S(a), S(b)) is non-zero, for at most finitely many m and a finite number of indices a I onsequently, the injective comodules E ( j) j),,e( n are socle-finite, and the proof is complete Next we give a description of sharp Euler path coalgebras = K Q,withQ aquiver Lemma 2 Assume that Q is a connected quiver and K Q is path K -coalgebra of a quiver Q The following three conditions are equivalent (a) K Q is a sharp Euler coalgebra (b) K Q is left and right Euler coalgebra (c) The quiver Q is locally finite, that is, every vertex of Q has at most finitely many neighbours in Q Proof The equivalence of (b) and (c) follows from [9, Theorem 5(a)] and the implication (a) (b) is obvious Since the coalgebras = K Q and op = (K Q ) op = K Q op are hereditary then to prove the inverse implication (b) (a), it is enough to show that there is a K -linear isomorphism Ext (S(a), S(b)) = Ext op(ŝ(b), Ŝ(a)), for any pair of simple left -comodules S(a) and S(b), witha, b Q, where Ŝ(b) = DS(b) and Ŝ(a) = DS(a) are the dual simple right -comodules corresponding to a, b Q Since the elements of Ext (S(a), S(b)) can be interpreted as equivalence classes of one-fold extensions S(b) N S(a) in-comod and the duality D : -comod op -comod carries S(b) N S(a) to the exact sequence Ŝ(a) DN Ŝ(b), it defines a K -linear isomorphism Ext (S(a), S(b)) = Ext op(ŝ(b), Ŝ(a)) This finishes the proof Now we give examples of non-semiperfect sharp Euler coalgebras of infinite global dimension and of arbitrary large finite global dimension Example 2 Let I be the infinite poset of the form

16 2238 W hin, D Simson / Journal of Algebra 324 (2) directed from the left to the right, where I m = G m is the garland of length m + G m : (2 m +2 vertices, m ) Obviously, I is an intervally finite poset By the results given in [23], the incidence coalgebra = K I of the poset I has the following properties (see also [9, Examples 425 and 426]): is a sharp Euler coalgebra and the global dimension gldim of is infinite 2 If S(a) is the simple left -comodule corresponding to the maximal vertex a = then the injective dimension injdim S(a) of S(a) equals m +, for any m 3 dim K Ext m (S(a), S(b)) = dim K Ext m op(ŝ(b), Ŝ(a)), for all m and all a, b I, where Ŝ(b) = DS(b) and Ŝ(a) = DS(a) are the simple right -comodules corresponding to the vertices a and b in I 4 is both left and right locally artinian, locally cocoherent, and the category -omod fc coincides with the full subcategory of -omod consisting of artinian objects 5 The coalgebra is neither left semiperfect nor right semiperfect 6 The artan Z Z square matrix c M Z (Z) of is lower triangular and has no finite rows and no finite columns 7 c has a unique left inverse c M Z(Z), which is also a unique right inverse of c Thematrix c is row-finite and column-finite 8 Let m be a fixed integer and let G m be the garland of length m + If we take I m = G m, for each m Z, in the construction of I then = K I is a sharp Euler coalgebra, gldim = m + is finite, is neither left semiperfect nor right semiperfect, and satisfies the conditions 4,6, and 7 Now, given a pointed sharp Euler coalgebra, we construct a left inverse and a right inverse of the artan matrix c M I (Z) We follow the proof of Theorem 48 in [9] and [24, Theorem 34], and we use the notation introduced there Given a left (resp right) sharp Euler coalgebra, we fix a finite minimal injective resolution h( j) S( j) E ( j) E ( j) E ( j) n, (22) h ( j) ofthesimpleleft-comodule S( j) in -omod fc,withe ( j) = E( j), and a finite minimal injective resolution ĥ( j) h ( j) 2 h( j) n Ŝ( j) Ê ( j) Ê ( j) Ê ( j), (23) ĥ ( j) of the simple right -comodule Ŝ( j) = DS( j) in op -omod fc,withê ( j) = Ê( j), respectively We fix finite direct sum decompositions E ( j) m = j) d( E(p) mp = p I p I ( j) m ĥ ( j) 2 ĥ ( j) ˆn j) d( E(p) mp, Ê ( j) m = Ê(p)ˆd ( j) mp = p I ˆn p Î ( j) m Ê(p)ˆd ( j) mp (24) of E ( j) m and Ê ( j) m,form, where I ( j) m and Î ( j) m are a finite subsets of I, d ( j) ( j) mp and ˆd mp are a positive integers, for each p I ( j) m and each p Î ( j) m, respectively, and we set d ( j) mp =, for any p I \ I ( j) m, and ˆd ( j) mp =, for any p I \ Î ( j) m Theorem 25 Let be a pointed computable K -coalgebra, with a fixed decomposition soc = j I S( j), and let c =[c ij ] i, j I M I (Z) be the left artan matrix (24) of

17 W hin, D Simson / Journal of Algebra 324 (2) (a) If is a left sharp Euler coalgebra then the matrix c =[c ij ] i, j I M I (Z), with c jp = m= ( )m d ( j) mp Z, isrow-finiteandisaleftinverseofc in M I (Z), whered ( j) mp is the integer defined by the decomposition (24) of the mth term E ( j) m of the minimal injective resolution (22) of the simple left -comodule S( j) Moreover, for each j I,wehavedim Ê( j) (c )tr = dim S( j) = e j, where dim Ê( j) is the jth column of c (b) If is a right sharp Euler coalgebra then the matrix c =[ĉ ij ] i, j I M I (Z), with ĉ jp = (p) m= ( )m ˆd Z, is column-finite and is a right inverse of c mj in M I (Z),whereˆd (p) is the integer de- mj fined by the decomposition (24) of the mth term Ê (p) of the minimal injective resolution (23) of the simple right -comodule Ŝ(p) = DS(p) Moreover, for each j I,wehavedim E( j) c = dim S( j) = e j j (c) If is a sharp Euler coalgebra then the matrix c := c = c = [ c ] M ij i, j I I (Z), (26) with c ij = ĉ ij = m= ( )m d ( j) mi = m= a left inverse of c and a right inverse of c ( )m ˆd (i) mj, is both row-finite and column-finite, and c is Proof (a) Assume that is a left sharp Euler coalgebra Then the minimal injective resolution (23) of S( j) is finite and the injective comodules E ( j) j),,e( n are socle-finite Hence the sum c jp = m= ( )m d ( j) mp is an integer, the matrix c =[c ij ] i, j I is well defined, and each of its row is finite, because the set I ( j) c jp = m= I( j) ( ) m d ( j) mp = j) I( n I is finite and n ( ) m d ( j) mp =, for all p / I ( j) m= j) j) I( I( n To prove the equality c c = E (the identity matrix), we note that, by the additivity of the function dim, the exact sequence (22) together with the decomposition (24) yields e j = dim S( j) = m= ( ) m dim E ( j) m = m= ( ) m d ( j) mp dim E(p) = c dim E(p) jp p I p I Hence the equality c c = E follows, because the pth row of the matrix c is the dimension vector e(p) = dim E(p) of E(p), see (24) By applying the matrix transpose ( ) tr : M I (Z) M I (Z) and the equality c op = c tr,weget E = E tr = c tr (c )tr = c op (c )tr and, in view of Lemma 26, the equality dim Ê( j) (c )tr = dim S( j) = e j follows (b) Assume that is a right sharp Euler coalgebra Then op is a left sharp Euler coalgebra and, by (a) with and op interchanged, the matrix c op =[ĉ ] ij i, j I M I (Z), with c jp = ( j) m= ( )m ˆd mp Z, is row-finite and is a left inverse of c op = c tr in M ( j) I (Z), where ˆd mp is the integer defined by the decomposition (24) of the mth term Ê ( j) m of the minimal injective resolution ofthesimpleright-comodule Ŝ( j) = DS( j) It follows that c jp = ĉ pj,forall j, p I, and consequently, we get (c ) tr = c op By Lemma 26, we get c op = c tr and the pth row dim Ê( j) of c op is the pth column of c Since the equality c c op op = E holds, the matrix transpose yields E = E op = c tr (c op ) tr = c op c, that is, c is a right inverse of c Hence (b) follows

18 224 W hin, D Simson / Journal of Algebra 324 (2) (c) Assume that is a sharp Euler coalgebra, that is, is left and right sharp and the equality dim K Ext m (S(a), S(b)) = dim K Ext m op(ŝ(b), Ŝ(a)) holds, for all a, b I We show that c = c Since is pointed, we have d ( j) ( ) = dim mi K Ext m S(i), S( j) see [2, (423)], and therefore d ( j) = mi and (i) ˆd mj = ( ) dim K Ext m Ŝ(i), Ŝ( j) op, (27) ˆd (i) mj It follows that, given i, j I,wehavec ji = m= ( )m d ( j) mi = (i) m= ( )m ˆd mj = ĉ ji This shows that c = c and, according to (a) and (b), the matrix c := c = c (26) is row-finite and column-finite, and is both left and right inverse of c This finishes the proof of the theorem orollary 28 Assume that is a pointed sharp Euler coalgebra as in Theorem 25, with the artan matrix c and its inverse c (26) (a) The matrix c is row-finite and column-finite, and, given a I,wehave: dim E(a) c = dim S(a) and dim S(a) c = dim E(a), dim Ê(a) (c )tr = dim S(a) and c (dim S(a)) tr = (dim Ê(a)) tr (b) The subsets {dim E(a)} a I, {dim Ê(a)} a I of the group Z I are Z-linearly independent (c) For each j I, the vector e j = dim S( j) belongs to the subgroup generated by the set {dim E(a)} a I, and to the subgroup generated by the set {dim Ê(a)} a I Proof The equalities in (a) follow from Theorem 25, and (b) is a consequence of (a), because the vectors dim S(a) = e a Z I,witha I,areZ-linearly independent (c) We recall that the ath row of c is the vector dim E(a) Sincethematrixc is row-finite, the dim E(a) and the first part of (c) follows The equality c c = E yields e j = dim S( j) = a I c ja second one follows in a similar way from the equality c c = E orollary 29 If is pointed and left semiperfect (resp right semiperfect) of finite global dimension then the artan matrix c of is column-finite (resp row-finite) and the matrix c Moreover, c (26) is a two-sided inverse of c is column-finite and row-finite, and the equalities of orollary 28(a) hold Proof By Lemma 26 (d) and (e), the artan matrix c of is column-finite (resp row-finite), if is left semiperfect (resp right semiperfect) Since, according to Lemma 29, is a sharp Euler coalgebra, the corollary follows from orollary 28 3 oxeter transformation for a sharp Euler coalgebra We study in this section the properties of the oxeter transformations defined in [9, Definition 427] for pointed Euler coalgebras Here, we also follow [, Definition III34] We modify [9, Definition 427] as follows Definition 3 Assume that is a pointed sharp Euler K -coalgebra with fixed decomposition soc = j I S( j) Letc M I (Z) be the artan matrix of and let c be the two-sided inverse (26) of c (a) The oxeter matrix of is the I I (c )tr = (c tr ) square matrix Φ = c tr c, where we set c tr =

19 W hin, D Simson / Journal of Algebra 324 (2) (b) The oxeter transformations of are the group homomorphisms Z I Φ Z I (32) Φ (y) = (y c ) ctr,for y Z I, where Z I Z I is the subgroup of Z I generated by the subset {ê(a) = dim Ê(a)} a I defined by the formulas Φ (x) = (x c tr ) c,forx Z I, and Φ and Z I Z I is the subgroup of Z I generated by the subset {e(a) = dim E(a)} a I By orollary 28, the sets {dim E(a)} a I and {dim Ê(a)} a I are Z-linearly independent in Z I and therefore they form Z-bases of Z (I ) and Z(I ), respectively Note also that M dim M defines the group isomorphism of the Grothendieck group K () = K ( op -inj) and Z (I ), and the group isomorphism of the Grothendieck group K () = K (-inj) and Z I Note that, by orollary 28, ( Φ dim E(a) ) (ê(a) ) (ê(a) ) = Φ = c tr c = e a c = e(a) = dim E(a), Φ ( ) dim E(a) = Φ ( ) e(a) = ea c = ê(a) dim E(a) It follows that the transformations (32) are well-defined and mutually inverse The following theorem is the main result of this section (compare with [, orollary IV29]) Theorem 33 Assume that is a pointed sharp Euler K -coalgebra with fixed decomposition soc = j I S( j)letφ and Φ be the oxeter transformations (32) of (a) Let M be an indecomposable left -comodule in -omod fc such that injdim M = and Hom (, M) = If M M τ M is the unique almost split sequence (2) in -omod fc, with an indecomposable comodule τ Mlying in -comod f P then dim τ M = Φ (dim M) (b) Assume that N is an indecomposable non-projective left -comodule in comod f P -omod fc such that injdim DN = and Hom (, DN) = If τ N N N is the unique almost split sequence (22) in -omod fc, with an indecomposable comodule τ Nlyingin -omod fc,then dimτ N = Φ (dim N) Proof (a) Assume that M is an indecomposable left -comodule in -omod fc such that injdim M = Then M admits a minimal injective copresentation M E g E in -omod fc where E and E are socle-finite injective comodules It follows that dim M = dim E dim E SinceΦ (dim E(a)) = dim E(a) c = dim Ê(a), foreverya I and the comodules E

20 2242 W hin, D Simson / Journal of Algebra 324 (2) and E are finite direct sums of the comodules E(a), witha I,wegetΦ (dim E ) = dim (E ), Φ (dim E ) = dim (E ) and, by applying Φ,theequalitydim M = dim E dim E yields Φ (dim M) = Φ (dim E ) Φ (dim E ) = dim (E ) dim (E ) On the other hand, the exact sequence () in op -omod, induced by the injective copresentation of M, has the form Tr (M) (E ) (g) (E ), because the assumption Hom (, M) = yields (M) = Hom (, M) =, see Theorem 8(a) Since dim K Tr (M) is finite, we have dim DTr (M) = dim Tr (M) and the exact sequence yields dim τ M = dim D Tr (M) = dim Tr (M) = dim (E ) dim (E ) = Φ (dim M) and (a) follows (b) By Proposition 6(b), there is a duality D : -comod f P op -comod fc that carries the indecomposable left -comodule in -comod f P to the indecomposable right -comodule in op -comod fc Since we assume injdim DN =, the comodule DN is not injective and there is a minimal socle-finite injective copresentation DN E E g of DN in op -omod fc Byanobvious op version of Theorem 8, there is a short exact sequence Tr op(dn) op ( E ) op (g ) ( op E ) op(dn) in -omod fc The assumption Hom (, DN) = yields op(dn) = Hom (, DN) = Then, by applying the arguments used in the proof of (a), we get ( dim Tr op(dn) = dim op E = Φ ( dim E ( ) ) dim op E ) ( ) Φ dim E = Φ (dim DN) = Φ (dim N), because dim K N is finite This finishes the proof Remark 34 If is a sharp Euler coalgebra such that gldim = and M (resp N) is an indecomposable non-injective comodule (resp non-projective comodule), we have injdim M = and Hom (, M) = (respinjdim DN = and Hom (, DN) = ), and Theorem 33 applies to M (resp to N) 4 Examples In this section we illustrate previous results by concrete examples Example 4 Let Q be the infinite locally Dynkin quiver Q :