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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On the structure of ideals of the dual algebra of a coalgebra


Author: David E. Radford
Journal: Trans. Amer. Math. Soc. 198 (1974), 123-137
MSC: Primary 16A24
DOI: https://doi.org/10.1090/S0002-9947-1974-0346002-4
MathSciNet review: 0346002
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Abstract: The weak-* topology is seen to play an important role in the study of various finiteness conditions one may place on a coalgebra $ C$ and its dual algebra $ {C^ \ast }$. Here we examine the interplay between the topology and the structure of ideals of $ {C^ \ast }$. The basic theory has been worked out for the commutative and almost connected cases (see [2]). Our basic tool for reducing to the almost connected case is the classical technique of lifting idempotents. Any orthogonal set of idempotents modulo a closed ideal of $ \operatorname{Rad} {C^\ast }$ can be lifted. This technique is particularly effective when $ C = {C_1}$. The strongest results we obtain concern ideals of $ C_1^ \ast $. Using the properties of idempotents we show that $ {C_1} = \Sigma_{x,y} {{C_x}\Lambda {C_y}} $ where $ {C_x}$ and $ {C_y}$ run over the simple subcoalgebras of $ C$. Our first theorem states that a coalgebra $ C$ is locally finite and $ {C_0}$ is reflexive if and only if every cofinite ideal of $ {C^ \ast }$ contains a finitely generated dense ideal. We show in general that a cofinite ideal $ I$ which contains a finitely generated dense ideal is not closed. (In fact either equivalent condition of the theorem does not imply $ C$ reflexive.) The preceding statement is true if $ C = {C_1}$, or more importantly if $ I \supset \operatorname{Rad} {C^\ast }$ and $ {C^ \ast }/I$ is algebraic. The second theorem characterizes the closure of an ideal with cofinite radical which also contains a finitely generated dense ideal.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0346002-4
Keywords: Coalgebra, structure of ideals, idempotents, weak-* topology
Article copyright: © Copyright 1974 American Mathematical Society