Ramification Theory for Arbitrary Valuation Rings in Positive Characteristic

Our goal is to develop ramification theory for arbitrary valuation fields, that is compatible with the classical theory of complete discrete valuation fields with perfect residue fields. We consider fields with more general (possibly non-discrete) valuations and arbitrary (possibly imperfect) residue fields. The "defect" case, i.e., the case where there is no extension of either the residue field or the value group, gives rise to many interesting complications. We present some new results for Artin-Schreier extensions of valuation fields in positive characteristic (\cite{V1}). These results relate the ``higher ramification ideal" of the extension with the ideal generated by the inverses of Artin-Schreier generators via the norm map. These are further related to K\" ahler differentials, which has been shown in previous work of Kato and others to offer refined information about wild ramification in the imperfect residue field case. We also introduce a generalization and further refinement of Kato's refined Swan conductor in this case. Similar results are true in the mixed characteristic case (\cite{V2}).