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Upper ramification groups for arbitrary valuation rings

Kazuya Kato and Vaidehee Thatte
Vol. 6 (2024), No. 4, 589–646
DOI: 10.2140/tunis.2024.6.589
Abstract

T. Saito established a ramification theory for ring extensions locally of complete intersection. We show that for a Henselian valuation ring A with field of fractions K and for a finite Galois extension L of K‘, the integral closure B of A in L is a filtered union of subrings of B which are of complete intersection over A. By this, we can obtain a ramification theory of Henselian valuation rings as the limit of the ramification theory of Saito. Our theory generalizes the ramification theory of complete discrete valuation rings of Abbes–Saito. We study “defect extensions” which are not treated in these previous works.

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Keywords
ramification filtration, Swan conductor, refined Swan conductor, defect, valuation rings
Mathematical Subject Classification 2010
Primary: 11G99
Secondary: 14G99
Milestones
Received: 21 September 2019
Revised: 14 May 2024
Accepted: 31 May 2024
Published: 18 December 2024
Authors
Kazuya Kato
Department of Mathematics
University of Chicago
Chicago, IL
United States
Vaidehee Thatte
Department of Mathematics
King’s College London
London
United Kingdom
© 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers).