Artin approximation theorem: Difference between revisions

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Let ''R'' be a field or an excellent discrete valuation ring, let ''A'' be the henselization of an ''R''-algebra of finite type at a prime ideal, let ''m'' be a proper ideal of ''A'', let <math> \hat{A}</math> be the ''m''-adic completion of ''A'', and let

:''<math>F'':\colon (''A''\text{-algebras}) →\to (\text{sets}), </math>

be a functor sending filtered colimits to filtered colimits (Artin calls such a functor locally of finite presentation). Then for any integer ''c'' and any <math> \overline{\xi} \in F(\hat{A})</math>, there is a <math> \xi \in F(A)</math> such that

Then for any integer ''c'' and any <math> \overline{\xi} \in F(\hat{A})</math> there is a <math> \xi \in F(A)</math> such that

:<math>\overline{\xi}</math> ≡ <math>\xi</math> mod ''m''^''c''.

:<math>\overline{\xi}</math> ≡\equiv <math>\xi</math> mod\bmod ''m''<sup>''^c''</supmath>.

== See also ==