Purely Inseparable Extensions and Ramification Filtrations

  • Hao Yu HuEmail author


In this article, we investigate the shift of Abbes and Saito’s ramification filtrations of the absolute Galois group of a complete discrete valuation field of positive characteristic under a purely inseparable extension. We also study a functoriality property for characteristic forms.


Abbes and Saito’s ramification filtration purely inseparable extension characteristic form 

MR(2010) Subject Classification

11S15 14F20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author is grateful to T. Saito for many useful comments and thank Y. Cao and J. P. Teyssier for discussions. The author was a postdoctoral fellow at the Max-Planck Institute of Mathematics in Bonn during preparing the article and he would like to thank the institute for the hospitality.


  1. [1]
    Abbes, A., Saito, T.: Ramification of local fields with imperfect residue fields. Amer. J. Math., 124, 879–920 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Abbes, A., Saito, T.: Ramification of local fields with imperfect residue fields II. Doc. Math. Extra Volume Kato, 5–72 (2003)Google Scholar
  3. [3]
    Abbes, A., Saito, T.: Ramification and Cleanliness. Tohoku Math. J., 63(4), 775–853 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Artin, M.: Algebraic approximation of structures over complete local rings. Publ. Math. I.H.E.S., 36, 23–58 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Barrientos, I.: Log ramification via curves in rank 1. Int. Math. Res. Notices, (2016) DOI:
  6. [6]
    Bourbaki, N.: Algèbre, Chapitre 4 à 7. Éléments de Mathématique. Hermann, Paris, 1950–1952zbMATHGoogle Scholar
  7. [7]
    Esnault, H., Kerz, M.: A finiteness theorem for Galois representations of function fields over finite fields (after Deligne). Acta. Math. Vietnamica, 37(4), 531–562 (2012)MathSciNetzbMATHGoogle Scholar
  8. [8]
    Grothendieck, A., et al.: Cohomologie -adique et fonctions L. Séminaire de Géométrie Algébrique du Bois-Marie 1965–1966 (SGA 5). dirigé par A. Grothendieck avec la collaboration de I. Bucur, C. Houzel, L. Illusie, J. P. Jouanolou et J. P. Serre. Lecture Notes in Mathematics 589, Springer-Verlag, Berlin—New York, 1977Google Scholar
  9. [9]
    Hu, H.: Logarithmic ramifications of étale sheaves by restricting to curves. Int. Math. Res. Notices, (2017) DOI:
  10. [10]
    Saito, T.: Wild ramification and the characteristic cycles of an -adic sheaf. J. Inst. of Math. Jussieu, 8, 769–829 (2008)MathSciNetCrossRefGoogle Scholar
  11. [11]
    Saito, T.: Wild ramification and the contangent bundle. J. Algebraic Geom., 26, 399–473 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Saito, T.: The characteristic cycle and the singular support of a constructible sheaf. Invent. Math., 207(2), 597–695 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Saito, T.: Characteristic cycles and the conductor of direct image. Pre-print, (2017)Google Scholar
  14. [14]
    Serre, J. P.: Corps Locaux. Deuxième Édition, Publications de l’Université de Nancago, No. VIII. Hermann, Paris, 1968Google Scholar
  15. [15]
    Temkin, M.: Inseparable local uniformization. J. Alg., 373, 65–119 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Teyssier, J. B.: Nearby slopes and boundedness for -adic sheaves in positive characteristic. Pre-print. (2015)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2019

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingP. R. China

Personalised recommendations