Statement

Let be a polynomial algebra graded by degree and be a subalgebra for which there exists a surjective homomorphism

which preserves degree and is the identity on . That is

Then is a finitely generated algebra (and thus Noetherian).

Proof

Let be the ideal generated by all the homogeneous elements of positive degree (in ). Then let be the ideal generated by (in ).

Now that we have an ideal in , which is Noetherian, so has finite many generators, we can call them . Then, for each generator, we can write it as

where each is homogeneous (decomposing and recombining as necessary).

This means

but since are all homogeneous, then

so we can assume each is homogeneous and .

Let be the -algebra generated by . (We will prove ) Let be homogeneous. If then , so we can assume that and agree up to and including .

Then , so

Now since preserves the degree, so

since , , then we know , so .

By this theorem, the algebra of invariants is a finitely generated algebra, and is thus Noetherian.