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

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.