By Rockswold G. K.

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**Example text**

It is easy to see that the invariant ring RGa is generated by (the images of) xyn for all n. For every n, xyn is not in the RGa -ideal generated by x; xy; : : : ; xyn 1 . This shows that RGa is not Noetherian and not finitely generated. Although this example is very simple, it does not quite fit into the general setting, since usually we consider actions on affine varieties which are by definition reduced. 6 Let K be an algebraically closed field of characteristic 0. Roberts [15] found a (nonlinear) action of the additive group Ga on K 7 such that the invariant ring is not finitely generated.

6, we can also express the fj in terms of the gi . The choice of the fj and gi is equivalent to giving homomorphisms Rk ! Rl 0 and Rk ! 1. The following lemma tells us how to compute Syz. Rk ! Rl /. 2 Let A be a commutative ring and 0 ✲ N ✲ M1 ✲M ✻ ❄ M2 a commutative diagram (both along ' and exact. Then we have an exact sequence 0 ! M1 / ! M2 / ! M2 ! M1 / ! M2 / ! M2 /; m 7! n; m/ 7! M2 /. We show the exactness at M2 . M2 /. Â/ we have m D '. m/ 2 N. n; m2 '. n/ C m2 '. m2 // D 0. M1 /; '. n//.

The polynomial ring KŒV D 1 dD0 KŒVd is graded with respect to the total degree. The G-action on KŒV preserves degree and KŒVG Â KŒV inherits the grading. This grading of KŒV and KŒVG is very useful and it makes polynomial invariants of representations easier to deal with than regular invariant functions of affine varieties. A fundamental problem in invariant theory is to find generators of the invariant ring KŒVG as a K-algebra. So a basic question is: Can one always find finitely many generators f1 ; : : : ; fr such that KŒVG D KŒ f1 ; : : : ; fr ?