# Algebra of invariants is finitely generated

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?

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

• Why the downvote? – Martin Brandenburg Dec 17 '13 at 7:17
• I did not downvote:) – 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.