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Generalized ring of norms and generalized \((\varphi ,\Gamma )\)-modules. (English) Zbl 1123.13007
The paper under review generalises the Fontaine-Wintenberger theory of fields of norms to higher dimension. The construction relies on the theory of almost étale coverings, using the fact that one explicit deeply ramified extension suffices to make all remaining ramification infinitesimally small. The ring of norms is obtained as the projective limit of the reductions modulo \(p\) of the stages of the deeply ramified cover. The transition maps are Frobenius (and no more the norms). This requires some amplification of the known theory. Unfortunately the ring of norms depends on many choices.
It is applied to show an isomorphism of étale fundamental groups between rings in characteristic \(p\) and characteristic zero, and to describe \(p\)-adic representations of such groups.

MSC:
13B40 Étale and flat extensions; Henselization; Artin approximation
14F30 \(p\)-adic cohomology, crystalline cohomology
14F20 Étale and other Grothendieck topologies and (co)homologies
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