Understanding the Defect via Ramification Theory

Classical ramification theory deals with complete discrete valuation fields k((X)) with perfect residue fields k. Invariants such as the Swan conductor capture important information about extensions of these fields. Many fascinating complications arise when we allow non-discrete valuations and imperfect residue fields k. Particularly in positive residue characteristic, we encounter the mysterious phenomenon of the defect (or ramification deficiency). The occurrence of a non-trivial defect is one of the main obstacles to long-standing problems, such as obtaining resolution of singularities in positive characteristic.

Degree p extensions of valuation fields are building blocks of the general case. In this talk, we will present a generalization of ramification invariants for such extensions and discuss how this leads to a better understanding of the defect. If time permits, we will briefly discuss their connection with some recent work (joint with K. Kato) on upper ramification groups.