This is a note on Abrams＇ paper ＂Modules, Comodules, and Cotensor Products over Frobenius Algebras, Journal of Algebras＂ （1999）. With the application of Frobenius coordinates developed recently by Kadison, one ha...This is a note on Abrams＇ paper ＂Modules, Comodules, and Cotensor Products over Frobenius Algebras, Journal of Algebras＂ （1999）. With the application of Frobenius coordinates developed recently by Kadison, one has a direct proof of Abrams＇ characterization for Frobenius algebras in terms of comultiplication （see L. Kadison （1999））. For any Frobenius algebra, by using the explicit comultiplication, the explicit correspondence between the category of modules and the category of comodules is obtained. Moreover, with this we give very simplified proofs and improve Abrams＇ results on the Hom functor description of cotensor functor.展开更多
The purpose of this paper is to present some dual properties of dual comodule. It turns out that dual comodule has universal property (cf.Theorem 2). Since (( )*,()°) is an adjoint pair (cf.Theorem 3), some nice ...The purpose of this paper is to present some dual properties of dual comodule. It turns out that dual comodule has universal property (cf.Theorem 2). Since (( )*,()°) is an adjoint pair (cf.Theorem 3), some nice properties of functor ( )° are obtained. Finally Theoram 4 provides that the cotensor product is the dual of the tensor product by (M (?)A N)°≌M°□A°N°. Moreover, the result Hom(M,JV)≌ComA°(N°,M°) is proved for finite related modules M, N over a reflexive algebra A.展开更多
We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C0 of an arbitrary coalgebra C, we give an alternative definition of the Gabriel quiver of C, and then show that it...We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C0 of an arbitrary coalgebra C, we give an alternative definition of the Gabriel quiver of C, and then show that it coincides with the known Ext quiver of C and the link quiver of C. The dual Gabriel theorem for a coalgebra with a separable coradical is obtained, which generalizes the corresponding result for a pointed coalgebra. We also give a new description of C1 = C0 ∧C C0 of any coalgebra C, which can be regarded as a generalization of the first part of the well-known Taft-Wilson Theorem for pointed coal-gebras. As applications, we give a characterization of locally finite coalgebras via their Gabriel quivers, and a property of the Gabriel quiver of a quasi-coFrobenius coalgebra.展开更多
基金Project supported by AsiaLink Project "Algebras and Representations in China and Europe" ASI/B7-301/98/679-11 and the National Natural Science Foundation of China (No.10271113).
文摘This is a note on Abrams＇ paper ＂Modules, Comodules, and Cotensor Products over Frobenius Algebras, Journal of Algebras＂ （1999）. With the application of Frobenius coordinates developed recently by Kadison, one has a direct proof of Abrams＇ characterization for Frobenius algebras in terms of comultiplication （see L. Kadison （1999））. For any Frobenius algebra, by using the explicit comultiplication, the explicit correspondence between the category of modules and the category of comodules is obtained. Moreover, with this we give very simplified proofs and improve Abrams＇ results on the Hom functor description of cotensor functor.
基金the Nature Science Foundation of China(19901009),Nature Science oundation of Guangdong Province(970472000463)
文摘The purpose of this paper is to present some dual properties of dual comodule. It turns out that dual comodule has universal property (cf.Theorem 2). Since (( )*,()°) is an adjoint pair (cf.Theorem 3), some nice properties of functor ( )° are obtained. Finally Theoram 4 provides that the cotensor product is the dual of the tensor product by (M (?)A N)°≌M°□A°N°. Moreover, the result Hom(M,JV)≌ComA°(N°,M°) is proved for finite related modules M, N over a reflexive algebra A.
基金supported by the National Natural Science Foundation of China(Grant Nos.10271113&10301033)the Doctoral Foundation of the Chinese Education Ministry.
文摘We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C0 of an arbitrary coalgebra C, we give an alternative definition of the Gabriel quiver of C, and then show that it coincides with the known Ext quiver of C and the link quiver of C. The dual Gabriel theorem for a coalgebra with a separable coradical is obtained, which generalizes the corresponding result for a pointed coalgebra. We also give a new description of C1 = C0 ∧C C0 of any coalgebra C, which can be regarded as a generalization of the first part of the well-known Taft-Wilson Theorem for pointed coal-gebras. As applications, we give a characterization of locally finite coalgebras via their Gabriel quivers, and a property of the Gabriel quiver of a quasi-coFrobenius coalgebra.
