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Some properties of the ring of germs of $C^\infty$-functions

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Some properties of the ring of germs of C-functions COMPOSITIO MATHEMATICA M. VAN DER PUT Some properties of the ring of germs ofC∞-functions Compositio Mathematica, tome 34, no 1 (1977), p. 99-108. <> © Foundation Compositio Mathematica, 1977, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// implique l’accord avec les conditions générales d’utilisation ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 99 SOME PROPERTIES OF THE RING OF GERMS OF C~-FUNCTIONS M. van der Put COMPOSITIO MATHEMATICA, Vol. 34, Fasc. 1, 1977, ] Noordhoff International Publishing Printed in the Netherlands 1. Introduction and summary Let k dénote the field R or C. Let X = (XI, ..., Xn) denote n- variables and the C°°(X) = Coo(X¡, ..., Xn) denote the ring of germs of C°°-functions in the n-variables X. The ringhomomorphism 03C0 : C°°(X) ~ k[[Xll=k[[X,...,Xnll which assigns to a C°°-germ its Taylor ex- pansion, is surjective. An interesting question is whether there exists a ringhomomor- phism ~ : k [[X]] ---&#x3E; C°°(X) with 03C0~ = id. In dimension one (n = 1) the existence of many such cp’s is proved (by K. Reichard [5], M. Shiota [7] and in §3). For n &#x3E; 1 we can only show that for some subrings A of k[[X]] (e.g. A/k finitely generated or A an analytic subring) the existence of a ringhomomoprhism cp: A - C°°(X) with wç = idA. The theorems of this type have the form of the well-known Artin- approximation theorem (§4). The problem of the existence of a ringhomomorphism cp: k[[X]] ~ C°°(X) with ircp = id resembles the question whether a given local ring has a coefficient

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