Franjou, Vincent; van der Kallen, Wilberd
Power reductivity over an arbitrary base.
Doc. Math. (Bielefeld) Extra Vol. Andrei A. Suslin's Sixtieth Birthday, 171-195 (2010)
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.
Mathematics Subject Classification
20G35, 14L24, 20G05, 20G10
Chevalley groups, 14th Hilbert problem, cohomology, geometric reductivity