#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Andrei A. Suslin's Sixtieth Birthday (2010), 171-195

** Vincent Franjou and Wilberd van der Kallen **
Power Reductivity over an Arbitrary Base

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.

2010 Mathematics Subject Classification: 20G35; 20G05; 20G10

Keywords and Phrases: Chevalley group; Hilbert's 14th; cohomology; Geometric reductivity.

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