Let $A$ be a finitely generated $k$-algebra for a field $k$, and $G$ a finite group which acts on $A$ by ring automorphisms. There is a widely known theorem which states that the algebra $A^G=\{a\in A| \forall g\in G, ga=a\}$ is finitely also generated. (Note that in general subalgebras of finitely generated algebras are not necessarily finitely generated!) Could you give me a link to a proof or give a hint to prove it?

  • $\begingroup$ You have to assume that $A$ is a finitely generated $k$-algebra for a field $k$. $\endgroup$ – Martin Brandenburg Dec 16 '13 at 21:24
  • $\begingroup$ Of course, I meant it but forgot to note it $\endgroup$ – user74574 Dec 16 '13 at 21:25

The original reference is:

Emmy Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Mathematische Annalen, vol. 77, p. 89-92, 1915, link

A more modern reference is Theorem 13.17 in

David Eisenbud, Commutative Algebra: with a view toward algebraic geometry. Vol. 150. Springer, 1995.

For more details, see chapter 7 of

David Cox, John Little and Donal O’Shea, Ideals, Varieties, and Algorithms: an introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics, Springer, 1992

  • 1
    $\begingroup$ Why the downvote? $\endgroup$ – Martin Brandenburg Dec 17 '13 at 7:17
  • $\begingroup$ I did not downvote:) $\endgroup$ – user74574 Dec 17 '13 at 16:22

Hermann Weyl, The Classical Groups: Their Invariants and Representations.

Jean Dieudonne, J.B. Carrell, Invariant Theory Old and New.

There are Russian translations of these books.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy