The fundamental theorem for coalgebras over the Dedekind ring and application

Abstract

In this paper, we study the local finiteness of coalgebras, known as the fundamental theorem for coalgebras over the Dedekind ring. First, we give proof of this property for coalgebras which are projective as modules over a principal ideal domain, without using the fundamental theorem for coalgebras over a field. Next, we give a version of the theorem for flat coalgebras over the Dedekind ring that extends the certainty of the theorem over a field. Finally, we apply these results to the coordinate ring of flat affine group schemes.