Theorem 2

Retain the assumptions and notation of Theorem 1.

(i) There exists a graded vector subspace of S forming a complement to $R_+ S$ in S and stable under G.

(ii) Let U be such a complement. The canonical homomorphism from $U \otimes_K R$ to S is an isomorphism of G-modules, and the representation of G in U (resp. S) is isomorphic to the regular representation of G on K (resp. R).

Indeed, for any integer $n \geq 0$, the K-vector spaces $S_n$ and $(R_+ S) \cap S_n$ are stable under G, and it follows from Maschke’s theorem (Appendix, Prop. 2) that there exists a G-stable complement $U_n$ of $(R_+ S) \cap S_n$ in $S_n$. Then $\sum_{n \geq 0} U_n$ is a G-stable complement of $R_+ S$ in S, hence (i).

Let U be a graded vector subspace of S forming a complement of $R_+ S$ in S and stable under G. By Remark 1, every basis of the K-vector space U is also a basis of the R-module S, and consequently is a basis of the field of fractions N of S over the field of fractions L of R. Thus, the L-vector space N can be identified with $U \otimes_K L$. Since U is stable under G, this identification is compatible with the action of G. The group algebra $L[G]$ of G over L can be identified with the algebra $K[G] \otimes_K L$. The Galois extension N of L admits a normal basis (Algebra, Chap. V, § 10, no. 9, Th. 6), which can be interpreted as saying that N, considered as an $L[G]$-module, is isomorphic to the module of the regular representation of G over L. Since U is finite dimensional over K, it follows from the Appendix, Prop. 1, that the $K[G]$-module U is isomorphic to the regular representation of G over K. Our assertions follow immediately from this.