Homomorphism of quasianalytic local rings

Abstract:

Let $\mathcal {C}_n$ be a local quasi-analytic subring of the ring of germs of $C^\infty$ functions on $\mathbb {R}^n$, and let $\mathcal {C}=\{ \mathcal {C}_n , n\in \mathbb {N}\}$. We suppose that $\mathcal {C}$ is closed under composition. Consider a map $\varphi : (\mathbb {R}^n, 0)\rightarrow (\mathbb {R}^k, 0)$ vanishing at zero, where $\varphi$ is a $k$-tuple $(\varphi _1,\ldots ,\varphi _k)$ and $\varphi _1,\ldots ,\varphi _k$ are in $\mathcal {C}_n$. Then $\varphi$ defines uniquely a map $\phi : \mathcal {C}_k \rightarrow \mathcal {C}_n$ by composition, and $\phi$ induces a morphism $\hat {\phi }: \hat {\mathcal {C}_k }\rightarrow \hat {\mathcal {C}_n}$ between completions. We let $\phi _* : \frac {\hat {\mathcal {C}_k} }{\mathcal {C}_k }\rightarrow \frac {\hat {\mathcal {C}_n }}{\mathcal {C}_n }$ be the homomorphism of groups induced by $\phi$ and $\hat {\phi }$ in the obvious manner. In the analytic case, i.e. when each $\mathcal {C}_n$ is the ring of germs of real analytic functions, M. Eakin and A. Harris give a condition under which $\phi _*$ is injective. In this paper we prove that the same statement does not hold for a quasianalytic system unless this system is analytic.