Matric generators of coalgebras and bialgebras

Abstract

Takeuchi asserted that if a bialgebra [Formula: see text] over a field [Formula: see text] is finitely generated as a [Formula: see text]-algebra, then [Formula: see text] is a matric bialgebra. We introduce the notion of a matric coalgebra over a commutative ring [Formula: see text]. We show that if [Formula: see text] is faithfully projective as a [Formula: see text]-module, then [Formula: see text] is a matric coalgebra. Using this, we also show that if a bialgebra [Formula: see text] over a semihereditary ring [Formula: see text] is projective as a [Formula: see text]-module, then any finite subset of [Formula: see text] is contained in some matric subbialgebra. This result is a generalization of Takeuchi’s assertion and can be regarded as a local finiteness theorem on bialgebras.