Coalgebraic algebra
From a purely algebraic point of view, a Hopf algebra over a commutative ring \(R\) is a group object in the category of associative and counitary \(R\)-coalgebras. Thus, Hopf algebras are referred by the authors as coalgebraic groups. This leads to the notions of (commutative and cocommutative) coalgebraic ring and coalgebraic module, which are assumed to be bigraded for homological purposes (the paper has its motivation in Algebraic Topology). The authors establish two main results. Theorem 3.4 says that the category of finite-type coalgebraic modules over a coalgebraic ring \(A_{*,*}\) is Abelian. Theorem 5.3 establishes that the category of coalgebraic \(A_{*,*}\)-modules is a symmetric monoidal category under a tensor product functor \(\otimes_{A_{*,*}}\) which is explicitly constructed. Its derived functors are considered in Section 6. The topological background of the paper is explained in the Introduction.
scientific article; zbMATH DE number 1308199