  Our starting point is Mumford's conjecture, on representations of Chevalley
groups over fields, as it is phrased in the preface of "Geometric Invariant
Theory". After extending the conjecture appropriately, we show that it holds
over an arbitrary commutative base ring. We thus obtain the first fundamental
theorem of invariant theory (often referred to as Hilbert's fourteenth problem)
over an arbitrary Noetherian ring. We also prove results on the Grosshans
graded deformation of an algebra in the same generality. We end with tentative
finiteness results for rational cohomology over the integers.
