a1 Department of Mathematics, Babes-Bolyai University, Cluj, Romania, E-mail: firstname.lastname@example.org
a2 Department of Mathematics, University of Manchester, Oxford Road, Manchester. M13 9PL., Great Britain, E-mail: email@example.com
a3 Kapellestraat 10, 3080 Tervuren., Belgium, E-mail: firstname.lastname@example.org
The purpose of this paper is to establish some basic points in the model theory of comodules over a coalgebra. It is not even immediately apparent that there is a model theory of comodules since these are not structures in the usual sense of model theory. Let us give the definitions right away so that the reader can see what we mean.
Fix a field k. A k-coalgebra C is a k-vector space equipped with a k-linear map Δ: C → C ⊗ C, called the comultiplication (by ⊗ we always mean tensor product over k), and a k-linear map ε: C → k, called the counit, such that Δ⊗1 C = 1 C ⊗ Δ (coassociativity) and (1 C ⊗ ε)Δ = 1 C = (ε ⊗ 1 C )Δ, where we identify C with both k ⊗ C and C ⊗ k. These definitions are literally the duals of those for a k-algebra: express the axioms for C′ to be a k-algebra in terms of the multiplication map μ: C′ ⊗ C′ → C′ and the “unit” (embedding of k into C′), δ: k → C′ in the form that certain diagrams commute and then just turn round all the arrows. See  or more recent references such as  for more.
(Received February 13 2003)
(Revised September 05 2003)