arXiv:1907.10053v1 [math.AC] 23 Jul 2019 Surjectivity of the completion map for rings of C ∞ -functions, necessary conditions and sufficient conditions Genrich Belitskii, Alberto F. Boix and Dmitry Kerner 1. Introduction 1.1. Let R be a commutative ring, filtered by a decreasing sequence of ideals R = I 0 ⊇ I 1 ⊇ I 2 ⊇ ..., set I ∞ := ∩ j≥0 I j , and consider the corresponding completion map R →  R (I•) := lim ← R / I j . Its kernel is I ∞ and thus for many rings/filtrations this map is injective. On the other hand, for many traditional (non-complete) rings of Commutative Algebra, this map is far from being surjective. Let R be a ring of smooth functions, e.g. one of (1) C ∞ (R m ,Z) / J , C ∞ (U ) / J , C ∞ ( (R m ,Z) ×U ) / J . (Here U⊂ R m is an open subset. (R m ,Z ) denotes the germ of R m along a closed subset Z ⊂ R m .) Take a filtration, {I j } j and the completion, R →  R (I•) . When is this map surjective? Borel’s lemma ensures the surjectivity of the completion C ∞ (R m ,o) → R[[x ]], for the filtration {(x ) j }. In this case, specifying an element of completion is the same as specifying all the derivatives at o. More generally, Whitney’s extension theorem gives the necessary and sufficient conditions to extend a function with prescribed derivatives on Z ⊂U to a smooth function on U . In the particular case, Z is a manifold, and the filtration is {I j = I (Z ) j }, specifying derivatives on Z is equivalent to specifying an element of  R (I•) . In this case the surjectivity follows by Whitney theorem. For more general subsets and filtrations the data of derivatives/elements of completion are essentially different objects. This case is more involved and the surjectivity of completion does not follow from Whitney extension theorem, see remark 4.4. 1.2. In this short note we address the surjectivity of the completion map for the rings of (1). In §2 we reduce the considerations to the case R = C ∞ (U ), for an open U⊂ R m . In §3 we obtain a (non-trivial) necessary condition. Our main result is Theorem 4.1: for a rather general class of filtrations the completion map is surjective, and moreover, the preimage of ˆ f ∈  R (I•) can be chosen real-analytic off the prescribed (closed) set. This surjectivity is the necessary starting point for various questions, e.g. i. Artin approximation type for C ∞ -rings, see e.g. [Bel.Boi.Ker]; ii. The study of determinacy/algebraizability of non-isolated singularities of maps and schemes, [B.K.16b], [Boi.Gre.Ker], [Bel.Ker.]. It will be interesting to extend these surjectivity results to various subclasses of smooth functions. 1.3. Notations. For any ideal I ⊂ C ∞ (U ) we take its (reduced) set of zeros, Z = V (I ) ⊂U . For any subset Z ⊂U denote by I (Z ) ⊂ C ∞ (U ) the set of functions vanishing on Z . Thus I (V (I )) ⊇ I . Not much can be said about the converse inclusion, because of the flat functions. We denote the derivatives by multi-indices, g (k ) . We abbreviate the condition (2) “any derivative ∂ i1 x1 ...∂ im xm g, with  i j = |k | satisfies |∂ i1 x1 ...∂ im xm g|≤ ..” by writing: “|g (k ) | < ...”. For a closed subset Z ⊂ R m we denote by C ∞ (R m ,Z ) the ring of germs of smooth functions at Z . These are functions defined on (small) neighborhoods of Z , with equivalence relation: f 1 ∼ f 2 if for some open neighborhood Z ⊂U holds f 1 | U = f 2 | U . Date : May 14, 2020 filename: surj.completion.2.3.tex. 2010 Mathematics Subject Classification. Primary 13J10. Secondary 13B35, 16W70, 26E10. A.F. Boix was partially supported by Israel Science Foundation (grant No. 844/14) and Spanish Ministerio de Econom´ ıa y Com- petitividad MTM2016-7881-P. D.Kerner was partially supported by Israel Science Foundation (grant No. 844/14). We thank M.Sodin for the highly useful reference to [H¨ ormander]. 1