On the Structure of Ideals of the Dual Algebra of a Coalgebra

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*. Here we examine the interplay between the topology and the structure of ideals of 0*. 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 Rad C* can be lifted. This technique is particularly effective when C=Cj. The strongest results we obtain concern ideals of CJ. Using the properties of idempotents we show that Cj = £x y cx A Cy where Cx and Cy run over the simple subcoalgebras of C. Our first theorem states that a coalgebra C is locally finite and C0 is reflexive if and only if every cofinite ideal of C* contains a finitely generated dense ideal. We show in general that a cofinite ideal / 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=Cj, or more importantly if / D Rad C* and C*// is algebraic. The second theorem characterizes the closure of an ideal with cofinite radical which also contains a finitely generated dense ideal. In this paper we examine some of the basic aspects of the relationship between the weak-* topology and the structure of ideals of the dual algebra of a coalgebra over a field. In [2] and [4] the weak-* topology is seen to play an important role in the study of various finiteness conditions that one may place on a coalgebra. Here we try to put the topological ideas developed in these two references into a more general context. Regarding a coalgebra C as a C-bicomodule we turn it into a left C = C ® Cop-comodule. This makes C* a left =>¿?-module.The cyclic submodules are the closures of the principal (two-sided) ideals of C*. Thus ¿?*-submodules are ideals, and the finitely generated submodules are the closed ideals of C* which contain a finitely generated dense ideal (see [2]). If JV( is a cofinite maximal ideal of C* or more generally an algebraic ideal which contains the Jacobson radical Rad C*, then ¿M, is a ¿?*-submodule (see [4] ). A minor theme of this paper is the connection between the ideals and ¿T"-submodules. If all ideals of the C* are closed then C must be of finite type. For any coalgebra C the finite-dimensional rational C*-modules are those which have closed cofinite annihilator. If C is reflexive (all cofinite ideals closed) then C is locally finite and C0 is reflexive (see [2]). We show that the latter condition is Presented to the Society, April 21, 1973 under the title Ideals and the weak-* topology of the dual algebra of a coalgebra over a field k; received by the editors February S, 1973. AMS (MOS) subject classifications (1970). Primary 16A66; Secondary 16A32, 16A24, 16A21.