Cosemisimple Coextensions and Homological Dimension of Smash Coproducts

Let H be a finite dimensional cosemisimple Hopf algebra, C a left H-comodule coalgebra and let C = C/C（H^＊）^＋ be the quotient coalgebra and the smash coproduct of C and H. It is shown that if C/C is a eosemisimple coextension and C is an injective right C-comodule, then gl. dim（the smash coproduct of C and H） = gl. dim（C） = gl. dim（C）, where gl. dim（C） denotes the global dimension of coalgebra C.     

Induced and Coinduced Representations of Hopf Group Coalgebras

The induction theory for a Hopf group coalgebra is outlined. Given a Hopf group coalgebra H, the notions of a quotient Hopf group coalgebra and group coisotropic quantum subgroup of H are introduced. The properties of (co)induced representations are studied and the geometric interpretation and simplicity theory of such representations are given.      

The Exchange Property in Grothendieck Categories: Applications

We introduce the concept of an object with the (finite) exchange property in an arbitrary Grothendieck category, and we present the basic properties of such an object. Applications are given for categories of graded modules and for categories of comodules over a coalgebra. Among other results, it is proved that an arbitrary coalgebra 𝒞 over a field has the finite exchange property.     

Representation-Directed Incidence Coalgebras of Intervally Finite Posets and the Tame-Wild Dichotomy

Incidence coalgebras C = K ^□ I of intervally finite posets I that are representation-directed are characterized in the article, and the posets I with this property are described. In particular, it is shown that the coalgebra C = K ^□ I is representation-directed if and only if the Euler quadratic form q _C : ?^(I) → ? of C is weakly positive. Every such a coalgebra C is tame of discrete comodule type and gl. dimC ≤ 2. As a consequence, we get a characterization of the incidence coalgebras C = K ^□ I that are left pure semisimple in the sense that every left C-comodule is a direct sum of finite dimensional subcomodules. It is shown that every such coalgebra C = K ^□ I is representation-directed and gl. dimC ≤ 2. Finally, the tame-wild dichotomy theorem is proved, for the coalgebras K ^□ I that are right semiperfect.     

On Minimal Coalgebras

We define an out-degree for F-coalgebras and show that the coalgebras of outdegree at most κ form a covariety. As a subcategory of all F-coalgebras, this class has a terminal object, which for many problems can stand in for the terminal F-coalgebra, which need not exist in general. As examples, we derive structure theoretic results about minimal coalgebras, showing that, for instance minimization of coalgebras is functorial, that products of finitely many minimal coalgebras exist and are given by their largest common subcoalgebra, that minimal subcoalgebras have no inner endomorphisms and show how minimal subcoalgebras can be constructed from Moore-automata. Since the elements of minimal subcoalgebras must correspond uniquely to the formulae of any logic characterizing observational equivalence, we give in the last section a straightforward and self-contained account of the coalgebraic logic of D. Pattinson and L. Schröder, which we believe is simpler and more direct than the original exposition.     

When is a Coalgebra a Generator?

Let C be a coalgebra over a field k. The aim of this paper is to study the following problem : (P) If C is a k-coalgebra such that C is a generator for the category of left comodules, is C a left quasi-co-Frobenius coalgebra ? The converse always holds. We show that if C has a finite coradical series, the answer is positive.     

(*)-Serial coalgebras

In this paper, we introduce the notion of (*)-serial coalgebras which is a generalization of serial coalgebras. We investigate the properties of (*)-serial coalgebras and their comodules, and obtain sufficient and necessary conditions for a basic coalgebra to be (*)-serial.     

Prime-Dimensional Hopf Algebras

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Functors for Coalgebras

Functors preserving weak pullbacks provide the basis for a rich structure theory of coalgebras. We give an easy to use criterion to check whether a functor preserves weak pullbacks. We apply the characterization to the functor which associates a set X with the set (X) of all filters on X. It turns out that this functor preserves weak pullbacks, yet does not preserve weak generalized pullbacks. Since topological spaces can be considered as -coalgebras, in fact they constitute a covariety, we find that the intersection of subcoalgebras need not be a coalgebra, and 1-generated -coalgebras need not exist. Received August 24, 1998; accepted in final form October 12, 1998.