Tame comodule type, roiter bocses, and a geometry context for coalgebras

Simson, Daniel

doi:10.1007/s11253-009-0253-6

Cited by 3 publications

(8 citation statements)

References 18 publications

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“…Nevertheless, the tame-wild dichotomy is still an open problem. These definitions are slightly changed in order to treat the category of finitely cogenerated comodules, see [34] and [35]. This is quite natural since indecomposable injective comodules are commonly infinite dimensional and then they are not considered following the classical notions.…”

Section: Applications To Representation Theorymentioning

confidence: 99%

“…Then, the coalgebra C is Morita-Takeuchi equivalent to an admissible subcoalgebra of the path coalgebra of its Gabriel quiver, see [42]. Hence we shall assume that C ⊆ kQ for certain quiver Q and kQ 1 ⊆ C. We recall from [34] and [35] that, for any finitely copresented C-comodule N with a minimal injective copresentation…”

Section: Applications To Representation Theorymentioning

confidence: 99%

“…Let us consider the "localized" coalgebra D = eCe, where e correspond to the vertices of the above subquivers. Then D is also generated by paths, semiprime (Theorem 3.4) and fc-tame [35,Proposition 5.1]. Furthermore, it contains Γ or Λ as subquiver, see [16].…”

Section: Applications To Representation Theorymentioning

confidence: 99%

“…[35, Proposition 5.1(b)] Let C be an arbitrary coalgebra. C is fc-tame if and only if any socle-finite "localized" coalgebra of C is fc-tame.…”

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confidence: 99%

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Valued Gabriel quiver of a wedge product and semiprime coalgebras

Navarro

2013

Front. Math. China

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Abstract. We make a first approach to the representation theory of the wedge product of coalgebras by means of the description of its valued Gabriel quiver. Then we define semiprime coalgebras and study its category of comodules by the use of localization techniques. In particular, we prove that, whether its Gabriel quiver is locally finite, any monomial semiprime fc-tame coalgebra is string. We also prove a weaker version of Eisenbud-Griffith theorem for coalgebras, namely, any hereditary semiprime strictly quasi-finite coalgebra is serial.

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Section: Applications To Representation Theorymentioning

confidence: 99%

Section: Applications To Representation Theorymentioning

confidence: 99%

Section: Applications To Representation Theorymentioning

confidence: 99%

“…[35, Proposition 5.1(b)] Let C be an arbitrary coalgebra. C is fc-tame if and only if any socle-finite "localized" coalgebra of C is fc-tame.…”

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confidence: 99%

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Valued Gabriel quiver of a wedge product and semiprime coalgebras

Navarro

2013

Front. Math. China

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“…We recall from [38,44,45] that there are two different notions of tameness of K I. We define C to be of tame (resp.…”

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confidence: 99%

Incidence coalgebras of interval finite posets of tame comodule type

Leszczyński

Simson

2015

Colloq. Math.

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The incidence coalgebras K I of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form q • : Z (I) → Z, where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category K I-comod of finite-dimensional left K I-modules is equivalent to the tameness of the category K I-Comod fc of finitely copresented left K I-modules. Hence, the tame-wild dichotomy for the coalgebras K I is deduced. Moreover, we prove that for an interval finite A * m -free poset I the incidence coalgebra K I is of tame comodule type if and only if the quadratic form q • is weakly non-negative. Finally, we give a complete list of all infinite connected interval finite A * m -free posets I such that K I is of tame comodule type. In this case we prove that, for any pair of finite-dimensional left K I-comodules M and N , bK I (dim M, dim N ) = ∞ j=0 (−1) j dimK Ext j K I (M, N ), where bK I : Z (I) × Z (I) → Z is the Euler Z-bilinear form of I and dim M , dim N are the dimension vectors of M and N .

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Tilting Torsion-Free Classes in the Category of Comodules

Yao

2020

Bull. Iran. Math. Soc.

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Tame comodule type, roiter bocses, and a geometry context for coalgebras

Cited by 3 publications

References 18 publications

Valued Gabriel quiver of a wedge product and semiprime coalgebras

Valued Gabriel quiver of a wedge product and semiprime coalgebras

Incidence coalgebras of interval finite posets of tame comodule type

Tilting Torsion-Free Classes in the Category of Comodules

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