Group schemes over fields

In the last couple of days I have added a few results on group schemes over fields to the stacks project. I mainly wanted to add the result that group schemes locally of finite type over a characteristic zero field are smooth which I hope to use later in an idea I have relating to finite groupoids in characteristic zero.

The sheaf of differentials of a group scheme over a fieldĀ  is free (this holds in any characteristic). But actually I am not sure that a scheme over a field of characteristic zero whose sheaf of differentials is free is even necessarily reduced. In fact, in a paper entitled “Algebraic group schemes in characteristic zero are reduced” (1966) Frans Oort asks: Is every group scheme over a field of characteristic zero reduced? I googled and tried mathscinet but this question seems to be still open.

Another question I have is: Does any group scheme over a field have an open subgroup scheme which is quasi-compact? It seems that this could be true… but maybe I simply do not know any of the truly enormous group schemes that exist out there?

Leave a comment if you have an idea about either of these questions.