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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Homomorphism of quasianalytic local rings


Author: Abdelhafed Elkhadiri
Journal: Proc. Amer. Math. Soc. 138 (2010), 1433-1438
MSC (2010): Primary 26E10, 32B05; Secondary 58C10
Published electronically: December 8, 2009
MathSciNet review: 2578536
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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.


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Additional Information

Abdelhafed Elkhadiri
Affiliation: Department of Mathematics, Faculty of Sciences, University Ibn Tofail, BP 133 Kénitra, Morocco
Email: kabdelhafed@hotmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10176-4
PII: S 0002-9939(09)10176-4
Keywords: Weierstrass division theorem, quasianalytic local rings
Received by editor(s): May 19, 2008
Received by editor(s) in revised form: August 23, 2009, and August 25, 2009
Published electronically: December 8, 2009
Additional Notes: This work was partially supported by PARS MI 33
Communicated by: Ted Chinburg
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.