Homomorphism of quasianalytic local rings

Abstract

Let C n \mathcal {C}_n be a local quasi-analytic subring of the ring of germs of C ∞ C^\infty functions on R n \mathbb {R}^n , and let C = { C n , n ∈ N } \mathcal {C}=\{ \mathcal {C}_n\,,\, n\in \mathbb {N}\} . We suppose that C \mathcal {C} is closed under composition. Consider a map φ : ( R n , 0 ) → ( R k , 0 ) \varphi : (\mathbb {R}^n, 0)\rightarrow (\mathbb {R}^k, 0) vanishing at zero, where φ \varphi is a k k -tuple ( φ 1 , … , φ k ) (\varphi _1,\ldots ,\varphi _k) and φ 1 , … , φ k \varphi _1,\ldots ,\varphi _k are in C n \mathcal {C}_n . Then φ \varphi defines uniquely a map ϕ : C k → C n \phi : \mathcal {C}_k \rightarrow \mathcal {C}_n by composition, and ϕ \phi induces a morphism ϕ ^ : C k ^ → C n ^ \hat {\phi }: \hat {\mathcal {C}_k }\rightarrow \hat {\mathcal {C}_n} between completions. We let ϕ ∗ : C k ^ C k → C n ^ C n \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 ϕ \phi and ϕ ^ \hat {\phi } in the obvious manner. In the analytic case, i.e. when each C n \mathcal {C}_n is the ring of germs of real analytic functions, M. Eakin and A. Harris give a condition under which ϕ ∗ \phi _* is injective. In this paper we prove that the same statement does not hold for a quasianalytic system unless this system is analytic.