Exercise 7, § 1

Let A be the ring of germs at 0 of functions of class C^{\infty} on \mathbf{R}_+. We propose to prove that the ring A is of infinite dimension. We denote by C the algebra of functions of class C^{\infty} on \mathbf{R}_+, F the ideal formed by the functions which are zero in a neighbourhood of 0, so that A = C/F. a) Let (x_n)_{x \in \mathbf{N}} be a strictly decreasing sequence, tending to 0, of elements of \mathbf{R}_+, and let \mathscr{U} be an ultrafilter on \mathbf{N} finer than the filter of the complements of finite subsets. Prove that the set J_1 of the f \in C such that the set of the n \in \mathbf{N} such that f(x_n) = 0 belongs to \mathscr{U}, is a prime ideal of C. b) We denote by \tau_n(f) the function x \mapsto f(x + x_n). Prove that if J is a prime ideal of C, the set \varphi(J) of the f such that the set of the n \in \mathbf{N} with \tau_n f \in J belongs to \mathscr{U} is a prime ideal of C. c) We define a sequence I_n of ideals of C by I_0 = J_1, I_{n+1} = \varphi(I_n). Prove that (I_n)_{n \in \mathbf{N}} is a strictly decreasing sequence of prime ideals of C containing F.