An integrality theorem of Grosshans over arbitrary base ring

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring k. One considers a split reductive group scheme G acting on a k-algebra A and leaving invariant a subalgebra R. Let U be the unipotent radical of a split Borel subgroup scheme. If R ^U = A ^U then the conclusion is that A is integral over R.