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The paper contains a construction of an analogue of the Fontaine-Wintenbergerfield-of-norms functor for higher dimensional local fields. This constructionis done completely in terms of the ramification theory of such fields. It isapplied to deduce the mixed characteristic case of a local analogue of theGrothendieck Conjecture for these fields from its characteristic p case, whichwas proved earlier by the author.

An analogue of the field-of-norms functor and of the Grothendieck conjecture