Haboush's Theorem - Applications

Applications

Haboush's theorem can be used to generalize results of geometric invariant theory from characteristic 0, where they were already known, to characteristic p>0. In particular Nagata's earlier results together with Haboush's theorem show that if a reductive group (over an algebraically closed field) acts on a finitely generated algebra then the fixed subalgebra is also finitely generated.

Haboush's theorem implies that if G is a reductive algebraic group acting regularly on an affine algebraic variety, then disjoint closed invariant sets X and Y can be separated by an invariant function f (this means that f is 0 on X and 1 on Y).

C.S. Seshadri (1977) extended Haboush's theorem to reductive groups over schemes.

It follows from the work of Nagata (1963), Haboush, and Popov that the following conditions are equivalent for an affine algebraic group G over a field K:

• G is reductive (its unipotent radical is trivial).
• For any non-zero invariant vector in a rational representation on G, there is an invariant homogeneous polynomial that does not vanish on it.
• For any finitely generated K algebra acted on rationally by G, the algebra of fixed elements is finitely generated.