Irreducible morphisms, the Gabriel‐valued quiver andcolocalizations for coalgebras

Section: Proposition 24 Let C Be a Basic Coalgebra The Right Valuementioning

confidence: 99%

Section: The Valued Gabriel Quivermentioning

confidence: 99%

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Serial coalgebras and their valued Gabriel quivers

Gómez-Torrecillas

Navarro

2008

“…If i = j , then rad C (E(j ), E(i)) = Hom C (E(j ), E(i)) and, by the convexity assumption, the functor H U induces isomorphisms, see [28, of F i -F j -bimodules. It follows that the Gabriel valued quiver of the K-algebra R U is the valued quiver (Q , d ), because the modules E (j ) = H U E(j ), with j ∈ U , form a complete set of pairwise non-isomorphic indecomposable injective right R U -modules, see [28,Theorem 3.2].…”

Section: Proposition 63mentioning

confidence: 99%

Hom-computable coalgebras, a composition factors matrix and the Euler bilinear form of an Euler coalgebra

Simson

2007

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A composition factors matrix C F is studied for any basic Hom-computable K-coalgebra C over an arbitrary field K, in connection with a Cartan matrix C F of C. Left Euler K-coalgebras C are defined. They are studied by means of an Euler integral bilinear form b C :the Euler characteristic χ C (M, N ) of Euler pairs of C-comodules M and N , and an Euler defect ∂ C : K 0 (C) × K 0 (C) → Z of C. It is shown that b C (lgth M, lgth N) = χ C (M, N ) + ∂ C (M, N ), for all M, N in C-comod, and ∂ C = 0, if all simple C-comodules are of finite injective dimension. A diagrammatic characterisation of representationdirected hereditary Hom-computable coalgebras is given.

“…It is easily seen that I C ( a) is the set of immediate predecessors [1] of the vertex a in the left Gabriel quiver ( C Q, C d) of C, where C Q = I C , see [12] and [30]. We say that a subset U ⊆ I C of I C is predecessor closed, if I C ( a) ⊆ U , for any a ∈ U .…”

Section: Lemma 44 Assume That C Is a K-coalgebra Such That The Numbmentioning

confidence: 99%

Localising embeddings of comodule categories with applications to tame and Euler coalgebras

Simson

2007

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Given a K-coalgebra C and an injective left C-comodule E, we construct a coalgebra C E and fully faithful left exact embedding E : C E -Comod → C-Comod of comodule categories such that the image of E is the subcategory C-Comod E consisting of the comodules M with an injective presentation 0 → M → E 0 → E 1 , where E 0 and E 1 are direct sums of direct summands of the comodule E. The functor E preserves the indecomposability, the injectivity, and is right adjoint to the restriction functor res E : C-Comod → C E -Comod. Applications to the study of tame coalgebras, Betti numbers, and cosyzygy comodules of simple comodules over a left Euler coalgebra C are given. A localising reduction to countably dimensional Euler coalgebras is presented.