Some non noetherian $C^\infty$ quasianalytic local rings

Abstract

We give an example of a non-noetherian quasi-analytic ring constructed using a quasi-analytic Denjoy-Carleman class. If we denote by $ \mathcal{D}_n$ the ring of those $ C^\infty$ quasianalytic function germs at $0\in \mathbb{R}^n$ which are definable in a polynomially bounded o-minimal structure. We show that the system $\{ \mathcal{D}_n\,/\, n\in\mathbb{N}^*\}$ is not noetherian, i.e. there exists $m\in\mathbb{N}$, $m > 1$, such that the ring $\mathcal{D}_m$ is not noetherian.