Idempotents and Morita-Takeuchi Theory

Cuadra, Juan; Gómez-Torrecillas, José

doi:10.1081/agb-120003476

Cited by 37 publications

(36 citation statements)

References 16 publications

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“…The fundamental properties of co-hom functors may be found in [13]. The counit of this adjunction is, by [4,Proposition 1.4], an isomorphism, so M eCe becomes a quotient category of M C . By [16,Proposition 3.8], every quotient category of M C is of this form.…”

Section: Localization In Coalgebrasmentioning

confidence: 99%

“…an eCe − Cbicomodule Ce), with structure maps defined in a straightforward way. This leads (see [4, Theorem 1.5, Corollary 1.6]) to an exact functor − C eC : M C → M eCe with a right adjoint − eCe eC : M eCe → M C . In fact the functor − C eC is naturally isomorphic to the co-hom functor h C (Ce, −) and also to the functor e(−) that sends M ∈ M C onto eM.…”

Section: Localization In Coalgebrasmentioning

confidence: 99%

“…Let {e i : i ∈ I} be a basic set of idempotents for C, that is, it is a set of orthogonal idempotents of C * such that E(S i ) = Ce i for every i ∈ I (see [4,Section 3]). The notation E(M) stands for an injective hull (in the category of comodules) of a comodule M. Most part of the following proposition was essentially given, with a different proof, in [16,Proposition 3.8].…”

Section: Localization In Coalgebrasmentioning

confidence: 99%

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Localization in Coalgebras: Applications to Finiteness Conditions

2007

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Section: Localization In Coalgebrasmentioning

confidence: 99%

Section: Localization In Coalgebrasmentioning

confidence: 99%

Section: Localization In Coalgebrasmentioning

confidence: 99%

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Localization in Coalgebras: Applications to Finiteness Conditions

2007

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“…of C, where e E is the idempotent of C * defined by the direct summand embedding E C. We know from [5,22,24] that the restriction functor 5) given by M → Me E , is exact and has a right adjoint…”

Section: The Gabriel-valued Quiver Of a Colocalization Coalgebra C Ementioning

confidence: 99%

Irreducible morphisms, the Gabriel‐valued quiver andcolocalizations for coalgebras

Simson

2006

International Journal of Mathematics and Mathematical Sciences

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“…The inverse F of ε * is defined by the formula [7,Theorem 1.5] for a proof). Since ε * is functorial with respect to comodule homomorphisms X → X ′ and Z → Z ′ , the functor f is the right adjoint of f • , and (b) follows.…”

Section: S(a) It Follows From the Definition Thatmentioning

confidence: 99%

Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

Kosakowska

Simson

2008

Colloq. Math.

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Abstract. Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C 1 , the second term of the coradical filtration of C. If C = C 1 , we associate with C a K-linear functor H C : C-Comod → H C -Comod that restricts to a representation equivalence, where H C is a coradical square complete hereditary bipartite K-coalgebra such that every simple H Ccomodule is injective or projective. Here H C -comod• sp is the full subcategory of H C -comod whose objects are finite-dimensional H C -comodules with projective socle having no injective summands of the form(see Theorem 5.11). Hence, we conclude that a coalgebra C with C = C 1 is left pure semisimple if and only if H C is left pure semisimple. In Section 6 we get a diagrammatic characterisation of coradical square complete coalgebras C that are left pure semisimple. Tameness and wildness of such coalgebras C is also discussed.1. Introduction. Throughout this paper we fix an arbitrary field K and we use the coalgebra representation theory notation and terminology introduced in [14] [39] for the coalgebra and comodule terminology. In particular, given a finite-dimensional K-algebra R, we denote by mod(R) the category of all finite-dimensional R-modules.Let C be a K-coalgebra with comultiplication ∆ and counit ε. We recall that a left C-comodule is a K-vector space X together with a K-linear map δ X : X → C ⊗ X such that (∆ ⊗ id X )δ X = (id C ⊗ δ X )δ X and (ε ⊗ id X )δ X is the canonical isomorphism X ∼ = K ⊗ X, where ⊗ = ⊗ K . Given a left C-comodule X, we denote by X 0 = soc X the socle of X, that is, the sum of all simple C-subcomodules of X.

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Idempotents and Morita-Takeuchi Theory

Cited by 37 publications

References 16 publications

Localization in Coalgebras: Applications to Finiteness Conditions

Localization in Coalgebras: Applications to Finiteness Conditions

Irreducible morphisms, the Gabriel‐valued quiver andcolocalizations for coalgebras

Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

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