Exercise 10, § 2

Let $p$ be a prime number, $q$ a power of $p, G$ a finite group of order $q$, and A a commutative ring. Let $(e_g)$ be the canonical basis of the algebra A[G] of the group G over A.

a) The set $\mathfrak{a}$ of elements $\sum a_ge_g$ of A[G] such that $\sum a_g= 0$ is a two-sided ideal of A[G]. If the ring A is of characteristic $p$ (V, §1, No. 2, p. 2), then we have $x^q= 0$ for every $x\in \mathfrak{a}$ (use induction on $q$; if $q\not= 1$, choose an element $h$ of order $p$ of the center of G (I, §6, No. 5, p. 77, Corollary of Proposition 11) and deduce from the induction hypothesis that there exists a $y\in A[G]$ such that $x^{q/p}= (1-e_h)y$).

b) Suppose that A is a local ring, with maximal ideal $\mathfrak{m}$, and that the field $A/\mathfrak{m}$ is of characteristic $p$. Prove that the ring A[G] is local and that its maximal ideal is the set of elements $\sum a_ge_g\in A[G]$ such that $\sum a_g\in \mathfrak{m}$.

(Let S be a simple A[G]-module; prove that S is annihilated by $\mathfrak{m}$ and then, by applying Proposition 11 of I, §6, No. 5, p. 76, to the $\mathbf{Z}$[G]-submodule generated by a nonzero element of S, that S contains an element invariant under G.)