Ramification theory and perfectoid spaces (Q2877503)

This interesting paper compares the ramification groups of complete discrete valuation fields of different characteristics, generalizing a theorem of Deligne to the case when the residue field is no longer assumed to be perfect.NEWLINENEWLINELet \(K_1\) and \(K_2\) be complete discrete valuation fields of residue characteristic \(p >0\), with uniformizers \(\pi_{K_1}\) and \(\pi_{K_2}\), respectively. Without assuming that the residue field is perfect, the ramification theory was introduced by \textit{A. Abbes} and \textit{T. Saito} [Am. J. Math. 124, No. 5, 879--920 (2002; Zbl 1084.11064)]. Suppose that \(\mathcal O_{K_1} / (\pi_{K_1}^m) \simeq \mathcal O_{K_2} / (\pi_{K_2}^m)\) for some positive integer \(m\) which is no more than the absolute ramification indices of \(K_1\) and \(K_2\). The main result of this paper is an equivalence between the categories of finite separable extensions of \(K_1\) and of \(K_2\) whose ramification indices are bounded by \(j\). This generalizes a theorem of \textit{P. Deligne} [in: Représentations des groupes réductifs sur un corps local, Travaux en Cours, 120--157 (1984; Zbl 0578.12014)] to the imperfect residue field case. The author also proved a logarithmic version of this result, compatibility with higher fields of norms d'après \textit{A. J. Scholl} [Doc. Math., J. DMV Extra Vol., 685--709 (2007; Zbl 1186.11070)], and some results on the integrality of Artin and Swan conductors when they are small.NEWLINENEWLINEThe novelty of the paper lies in the proof of the theorem, which makes use of Scholze's perfectoid spaces. It is enough to show that if \(L_1\) and \(L_2\) are finite separable extensions of \(K_1\) and \(K_2\), respectively such that \(\mathcal O_{L_1}/ (\pi_{K_1}^m) \simeq \mathcal O_{L_2} /(\pi_{K_2}^m)\), then if the ramification of \(L_1/K_1\) is bounded by \(j \leq m\), so is the ramification of \(L_2/K_2\). The interesting case of the theorem is when \(K_1 \cong k((u))\) is a field of equal characteristic and \(K_2\) is of mixed characteristic. The key is to show that, taking the rigid spaces \(X_{L_i/K_i}^j\) associated to the extension \(L_i/K_i\) for \(i=1,2\), there are perfectoid covers \(X_{L_i/K_i,\infty}^j\) of \(X_{L_i/K_i}^j\) that are tilt of each other in the sense of \textit{P. Scholze} [Publ. Math., Inst. Hautes Étud. Sci. 116, 245--313 (2012; Zbl 1263.14022)]. This then establishes a canonical bijection between the geometric connected components of \(X_{L_i/K_i}^j\), proving the main theorem.