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. We also prove some other related conjectures in invariant theory.