Seminar in Real and Complex Geometry

Thursday, February 13, 2020, 16:00-18:00, Schreiber building, room 209

Dmitry Kerner (Ben-Gurion University)

Surjectivity of the completion map for rings of C^\infty-functions, necessary and sufficient conditions


Borel's lemma reads: any power series with real coefficients is realizable as the Taylor series of a smooth function. Algebraically this means the surjectivity of the completion map, C^\infty-> R[[x]], for the completion with respect to the powers of the maximal ideal. For various applications (e.g. in Singularities) one needs the surjectivity of completion for more general filtrations, not necessarily of the form {I^j}. We prove the necessary/sufficient conditions for this surjectivity. This can be regarded as an algebraic version of the classical Whitney's extension theorem.