Ramification of local fields and Fontaine's property \((\mathrm{P}_m)\) (Q2902439)

Let \(K\) be a complete discrete valuation field with residue field \(k\) of characteristic \(p>0\), \(\mathcal O_K\) its ring of integers, and \(v_K\) the associated valuation normalized so that \(v_K(K^\times) = \mathbb Z\). Let \(G_K\) denote its absolute Galois group. The goal of this paper is to study the relation between the Fontaine property \((P_m)\) of \(K\) and the upper ramification filtration on the Galois group \(G_K\) (when the residue field \(k\) is perfect), or in general, the Abbes-Saito ramification filtration on \(G_K\) (when \(k\) is imperfect).NEWLINENEWLINEThe Fontaine property \((P_m)\) (for \(m\) a real number) is defined as follows: for \(E\) an algebraic extension of \(K\) with ring of integers \(\mathcal O_E\), we put \(\mathfrak{a}^m_{E/K} = \{x \in \mathcal O_E\,|\, v_K(x) \geq m\}\). We say that a finite Galois extension \(L/K\) satisfies the Fontaine property \((P_m)\) if,NEWLINENEWLINE\quad \quad \quad for any algebraic extension \(E/K\), if there exists an \(\mathcal O_K\)-algebraNEWLINENEWLINE\quad \quad \quad homomorphism \(\mathcal O_L \to \mathcal O_E / \mathfrak{a}^m_{E/K}\), then there exists a \(K\)-embedding \(L \hookrightarrow E\).NEWLINENEWLINEAs explained in Section 2 of the paper, the property \((P_m)\) is stable under composition of extensions \(L\) of \(K\). Consequently, the author defines two filtrations \(G_K^{> m}\) and \(G_K^{\geq m}\) on the Galois group \(G_K\) as follows: a finite Galois extension \(L\) of \(K\) is fixed by \(G_K^{> m}\) (resp. \(G_K^{\geq m}\)) if and only if \((P_{m'})\) holds for all \(m'>m\) (resp. \(m'\geq m\)).NEWLINENEWLINEWhen \(k\) is perfect, we use \(G_K^{(m)}: = G_K^{m-1}\) to denote the classical upper numbering ramification filtration shifted by one. When \(k\) may be imperfect, we have the non-logarithm ramification filtration \(G_K^{(m)}\) of Abbes-Saito. In either case, we use \(G_K^{(m+)}\) to denote the closure of \(\cup_{m'>m} G_K^{(m')}\). The main result of this paper is the following:NEWLINENEWLINE\quad \quad \quad For any \(m \in \mathbb R\), we have \(G_K^{> m} \subseteq G_K^{(m+)}\) and \(G_K^{\geq m} \subseteq G_K^{(m)}\).NEWLINENEWLINE\quad \quad \quad The two inclusions are equalities if the residue field \(k\) is perfect.NEWLINENEWLINEThe author does not know whether the equality continues to hold when the residue field fails to be perfect. The main tool of the proof is a Krasner's lemma in the setting of Abbes-Saito's construction. Such a proof is made explicit in the case of perfect residue field.NEWLINENEWLINEThe paper is clearly written and is self-contained.