Ramification of higher local fields, approaches and questions

Abstract

This is yet another attempt to organize facts, ideas and problems concerning ramification in finite extensions of complete discrete valuation fields with arbitrary residue fields. We start (§3) with a rather comprehensive description of classical ramification theory describing the behavior of ramification invariants in the case of perfect residue fields. This includes some observations that could be not published earlier, e. g., Prop. 3.3.2 or 3.5.1. We proceed in §4 with the detailed study of an example showing that almost all the classical theory breaks down if we admit inseparable extensions of residue field and this cannot be easily repaired. The remaining part of the survey describes several approaches aimed to reproduce parts of the classical theory in the non-classical setting. §5 is devoted to the description of upper numbering ramification filtration as well as analogs of Artin and Swan conductors in the general case; this description has become standard. Historically, this theory started with the appropriate definitions for abelian extensions via class field theory and cohomological duality and culminated in the general definitions in terms of rigid analytic geometry done by A. Abbes and T. Saito. In §6, we discuss two ways to realize the ramification filtration geometrically: approach using l-adic sheaves as in the work of Abbes and Saito and approach using p-adic differential equations as in the work of K. Kedlaya and the first author. Either approach has its own advantage; they are applied to prove important basic theorems on the structure of the ramification filtration. We also introduce the notion of irregularity which is related to one more analogous situation of ramification. The next section starts with the observation that we still do not have a “fully satisfactory” ramification theory since the upper ramification filtration does not give us enough information about “naive” invariants including the lower ramification filtration; we sketch some requirements for a “satisfactory theory”. We proceed to describe an approach based on the theory of elimination of wild ramification. It results in a construction bearing some properties of classical theory and giving additional information on ramification of given extension. This approach still does not fill the gap but gives some room for further development as mentioned at the end of the section. Sections 8 and 9 are devoted to the approach of Deligne who started to analyze 2-dimensional ramification problems by looking at all their 1-dimensional restrictions. This makes sense in the context of 2-dimensional schemes, and we suggest to