Exercise 5, § 5

*Let $K$ be a field and $S = K[X_1, \ldots, X_n]$ a graded polynomial algebra over $K$, generated by algebraically independent elements $X_i$, homogeneous of degrees $> 0$.

a) Let $Y_1, \ldots, Y_n$ be homogeneous elements of $S$ of degrees $> 0$, and let $R = K[Y_1, \ldots, Y_n]$ be the subalgebra of $S$ generated by these elements. Prove the equivalence of the following properties:

(i) $(Y_1, \ldots, Y_n)$ is an $S$-regular sequence.

(ii) $S$ is integral over $R$.

(iii) The ideal of $S$ generated by $Y_1, \ldots, Y_n$ is of finite codimension in $S$.

(iv) For any extension $\overline{K}$ of $K$, the system of equations $Y_i(x_1, \ldots, x_n) = 0$ ($1 \leq i \leq n, x_i \in \overline{K}$) has only the trivial solution $(0, \ldots, 0)$.

((i) $\Longleftrightarrow$ (iii)) follows from a theorem of Macaulay (Commutative Algebra); (iii) $\Longleftrightarrow$ (iv) follows from the Zeros Theorem (Commutative Algebra, Chap. V, §3, no. 3, Prop. 2); (ii) $\Longleftrightarrow$ (iii) is easy.)

If these properties are satisfied, show that the $Y_i$ are algebraically independent over $K$, and that $S$ is a free $R$-module of rank equal to

b) Let $G$ be a finite group of automorphisms of the graded algebra $S$, let $S^G$ be its subalgebra of invariants, and let $Y_1, \ldots, Y_n$ be elements of $S^G$ satisfying conditions (i),..., (iv) above. Show that $S^G = K[Y_1, \ldots, Y_n]$ if and only if