On the Castelnuovo-Mumford Regularity of Rings of Polynomial Invariants

We show that, when a group acts on a polynomial ring over a field, the  
ring of invariants has Castelnuovo-Mumford regularity at most zero. As  
a consequence, we prove a well-known conjecture that the invariants  
are always generated
in degrees at most n(|G|-1), where n >1 is the number of polynomial
generators and |G|>1 is the order of the group.

The proof is a mixture of commutative algebra and representation theory.