会場

埼玉大学 理学部１号館 （3階）基礎数理演習室

講演者

Vaidehee Thatte (Binghamton University)

タイトル

Arbitrary Valuation Rings and Wild Ramification

アブストラクト

Classical ramification theory deals with extensions of complete discrete valuation rings with perfect residue fields. The study of arbitrary valuation rings with possibly imperfect residue fields and possibly non-discrete valuations of rank 1 presents many fascinating complications. In particular, the defect could be non-trivial when in positive residue characteristic (i.e. we can have a non-trivial extension, such that there is no extension of the residue field or the value group). In this talk, we will present some new results for Artin-Schreier extensions of arbitrary valuation fields in positive characteristic p. 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. This ideal plays the role of the classical Swan conductor in the arbitrary case. We will also introduce a generalization and further refinement of Kato’s refined Swan conductor in this setting. Similar results are true in mixed characteristic (0; p). If time permits, we will briefly discuss the connection between these results and some recent work (joint with K. Kato) on upper ramification groups.