Global degree bounds and the transfer principle for invariants (Q1764830)

For a linear algebraic group \(G\) over an algebraically closed field \(K\) the following are equivalent: (a) \(G\) has global degree bound on invariants (that is, there is a natural number \(d\) such that the ring of invariants is generated in degree \(\leq d\) for all algebraic \(G\)-modules); (b) \(G\) is finite with \(| G| \) invertible in \(K\). That (b) implies (a) was known before, and now the authors prove the reverse implication. This verifies a conjecture of \textit{H. Derksen} and \textit{G. Kemper} [in: Invariant theory in all characteristics, Proc. Workshop Invariant Theory, CRM Proc. Lect. Notes 35, 37--41 (2004; Zbl 1072.14056)], where the characteristic zero case was handled. Along the way the ``transfer principle'' of invariant theory is proved in a more general setting than usual.