Approximation of Group Schemes over valuation rings

In his paper Approximation des schémas en groupes, quasi compacts sur un corps, Daniel Perrin shows that every quasi compact group scheme over a field then it is an inverse limit of group schemes of finite type.

Is there a similar result for group schemes over a valuation ring?

1 Answer

It is true if you make some mild assumption on the valuation ring $R$ and the Hopf algebra $H=R[G]$. For example, let's assume that $H$ is free over $R$, and that $R$ is a discrete valuation ring. The main ingredient in the proof in the paper you cited is the following fact about co-algebras over a field: if $C$ is a coalgebra over a field $K$, and $V\subseteq C$ is a finite dimensional subspace, then there exists a finite dimensional subcoalgebra $D\subseteq C$ such that $V\subseteq D$. The proof goes almost the same for discrete valuation rings as for fields. We need to assumption on $H$ to make sure that nothing too singular will occur.

• But $G$ may not be affine... Why do you need $R$ to be a DVR? Do you mind elaborating? Aug 5, 2015 at 5:07