manuscripta mathematica

, Volume 158, Issue 1–2, pp 55–73 | Cite as

Linear nested Artin approximation theorem for algebraic power series

  • Francisco-Jesús Castro-Jiménez
  • Dorin Popescu
  • Guillaume Rond


We give an elementary proof of the nested Artin approximation theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subring of the ring of formal power series, we clarify the relationship between this theorem and the problem of the commutation of two operations for ideals: the operation of replacing an ideal by its completion and the operation of replacing an ideal by one of its elimination ideals. In particular we prove that a Grothendieck conjecture about morphisms of analytic/formal algebras and Artin’s question about linear nested approximation problem are equivalent.

Mathematics Subject Classification

Primary: 13B40 Secondary: 13J05 14B12 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Álgebra and IMUSUniversidad de SevillaSevillaSpain
  2. 2.Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit 5, University of BucharestBucharestRomania
  3. 3.UMR 7373, I2M, Centrale Marseille, CNRSAix-Marseille UniversitéMarseilleFrance

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