2 3 Ju n 20 06 AN EXPLICIT FORMULA FOR A STRONG CONNECTION

An explicit formula for a strong connection form in a principal extension by a coseparable coalgebra is given. 1. In the studies of geometry of non-commutative principal bundles or coalgebraGalois extensions (cf. [7]) an important role is played by the notion of a strong connection (for the universal differential structure) first introduced in the context of Hopf-Galois extensions in [10]. The existence of a strong connection guarantees that a bundle associated to a coalgebra-Galois extension is a (finitely generated and) projective module, hence it is (a module of sections on) a vector bundle in the sense of non-commutative geometry (cf. [9]). Furthermore, a strong connection form gives rise to a Chern-Galois character [6], a mapping from the Grothendieck group of isomorphism classes of finite dimensional corepresentations of the structure coalgebra to the cyclic homology of the base algebra (see [3] for the most general, relative formulation). The existence of a strong connection in a Hopf-Galois extension is assured by the classical Schneider Theorem I [15]. This states that a free coaction of a Hopf algebra with bijective antipode on its injective comodule algebra determines a HopfGalois extension with a strong connection. This theorem has been extended to coalgebra (entwined) extensions with a coseparable colagebra [4, Theorem 4.6] [14, Theorem 5.9]. In all these cases the proof of existence is not a constructive proof: the existence follows by general arguments, but no explicit form of connection is given. On the other hand, the knowledge of this form is needed for construction and calculation of Levi-Civitá connections and projectors for associated bundles, and the Chern-Galois characters. Recently, several examples of strong connections have been constructed (cf. [8], [11], [13], [12]) or their form conjectured [1], but no general procedure has been established. The aim of this note is to give a direct proof of a Schneider type theorem for coalgebra extensions in which the connection is explicitly given. We work over a field k, unadorned tensor product is over k. For a vector space V , the identity map is denoted by the same symbol V . All algebras are associative and unital. In an algebra A, 1 denotes the unit both as an element and as a klinear map k → A and μ : A⊗A → A denotes the product. In a coalgebra C, the coproduct is denoted by ∆ and counit by ε. We denote coactions of C on a vector space A by ̺ (the right coaction) and ̺ (the left coaction). The following Date: June 2006. 1991 Mathematics Subject Classification. 16W30; 58B34.