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An analogue of the field-of-norms functor and of the Grothendieck conjecture. (English) Zbl 1135.11064
The field-of-norms functor introduced by J.-M. Fontaine and J.-P. Wintenberger [C. R. Acad. Sci., Paris, Sér. A 288, 367–370 (1979; Zbl 0475.12020); ibid. 288, 441–444 (1979; Zbl 0403.12018)] allows us to identify the Galois groups of some infinite extensions of the field of $$p$$-adic numbers $${\mathbb Q}_ p$$ with those of complete discrete valuation fields of characteristic $$p$$. This functor is applied by the author [Int. J. Math. 11, No. 2, 133–175 (2000; Zbl 1073.12501)] to prove an analogue of the Grothendieck conjecture for the $$1$$-dimensional local fields.
The main goal of the paper under review is to suggest the construction of a suitable analogue of the field-of-norms functor for higher-dimensional local fields. In fact, a local analogue of the Grothendieck conjecture may allow us to recover the structure of a local field from the structure of its absolute Galois group provided with the filtration by ramification groups, if there were a suitable such functor for higher-dimensional local fields.
Here the category of infinite arithmetically profinite extensions of $${\mathbb Q}_ p$$ of Fontaine and Wintenberger is replaced by the category $${\mathcal B}^ a(N)$$ of infinite increasing field towers $$K_ 0\subset K_ 1\subset \ldots \subset K_ n\subset \ldots$$ with the restriction on the upper ramification numbers of the extensions $$K_ {n+1}/ K_ n$$ for $$n$$ large enough. One cannot use the sequence of norm compatible elements in such towers, but it is possible to work with elements $$a_ n\in {\mathcal O}_ {K_ n}$$ such that $$a_ n \equiv a_ {n+1}^ p \bmod p^ c$$ for some $$0<c\leq 1$$ independent of $$n$$.
The difficulty in the realization of this idea comes from the fact that the construction of ramification theory for an $$N$$-dimensional local field $$L$$ depends on the choice of the subfields $$L(i)$$ of $$i$$-dimensional constants, where $$1\leq i\leq N$$. Thus, the precise analogue of the Fontaine-Wintenberger functor is obtained only for a subcategory of special towers $${\mathcal B}^ {fa}(N)$$ in $${\mathcal B}^ a (N)$$. However, the construction of the functor can be extended to $${\mathcal B}^ a(N)$$. In this way, it is introduced the set of elements of the corresponding field-of-norms. This construction can be applied to deduce the mixed characteristic case of the Grothendieck conjecture from its characteristic $$p>2$$ case.
The organization of the paper is as follows. Section 1 gives the preliminaries, definitions and some properties of $$N$$-local fields. In particular the author pays special attention to the concept of $$P$$-topology, a topology on $$L$$. In Section 2, an analogue of H. P. Epp’s eliminating wild ramification [Invent. Math. 19, 235–249 (1973; Zbl 0254.13008)] is presented.
The next section presents an introduction to the ramification theory and a version of Krasner’s Lemma for higher-dimensional local fields. The author introduces and studies in Section 4 the categories of special towers of fields. Section 5 explains the construction of the family $${\mathcal X}(K_ {\bullet})$$ of local fields of characteristic $$p$$, where $$K_ {\bullet} \in {\mathcal B}^ {fa}(N)$$. The author applies Krasner’s Lemma in the next section to establish the properties of the correspondence $$K_{\bullet} \to {\mathcal K}\in {\mathcal X}(K_ {\bullet})$$ and uses these properties to define the analogue $${\mathcal X}_ {K_{\bullet}}$$, $$K_{\bullet}\in {\mathcal B}^ {fa}(N)$$ of the field-of-norms functor in Section 7. In Section 8 it is proved that the corresponding identification of the Galois group $$\Gamma _ {\widetilde{K}}$$ of the $$p$$-adic closure $$\widetilde{K}$$ of the composite of all fields from the tower $$K_{\bullet}$$ and $$\Gamma _{\tilde{K}}$$ becomes $$P$$-continuous when being restricted to their maximal abelian $$p$$-quotients. This $$P$$-continuity allows to prove in the final section the mixed characteristic case of the Grothendieck conjecture for $$p>2$$.

##### MSC:
 11S15 Ramification and extension theory 11S20 Galois theory
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##### References:
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