Semifinite bundles and the Chevalley–Weil formula

  • Shusuke OtabeEmail author


In our previous paper (Commun. Algebra, 45(8) (2017) 3422–3448), we studied the category of semifinite bundles on a proper variety defined over a field of characteristic 0. As a result, we obtained the fact that for a smooth projective curve defined over an algebraically closed field of characteristic 0 with genus \(g>1\), Nori fundamental group acts faithfully on the unipotent fundamental group of its universal covering. However, it was not mentioned about any explicit module structure. In this paper, we prove that the Chevalley–Weil formula gives a description of it.


Fundamental group schemes vector bundles Tannaka duality 

Mathematics Subject Classification

14L15 14H30 14H60 



The author would like to thank Professor Takao Yamazaki for many discussions clarifying the relation between his previous work and the Chevalley–Weil formula. The author also thanks Professor Takuya Yamauchi for suggesting many examples of projective smooth higher dimensional varieties with infinitely abelian fundamental group. The author is supported by JSPS, Grant-in-Aid for Scientific Research for JSPS fellows (16J02171).


  1. 1.
    Chevalley C, Weil A and Hecke E, Über das verhalten der integrale 1, gattung bei automorphismen des funktionenkörpers, Abh. Math. Sem. Univ. Hamburg, 10(1) (1934) 358–361CrossRefzbMATHGoogle Scholar
  2. 2.
    Deligne P and Milne J, Tannakian Categories, Lectures Notes in Mathematics 900 (1982) (Berlin-New York: Springer-Verlag)zbMATHGoogle Scholar
  3. 3.
    Grothendieck A, Représentations linéaires et compactification profinie des groupes discrets, Manuscr. Math., 2 (1970) 375–396CrossRefzbMATHGoogle Scholar
  4. 4.
    Grothendieck A, Revêtements étales et groupe fondamental, SGA1, Lecture Notes in Mathematics 224 (1971) (Berlin-New York: Springer-Verlag)Google Scholar
  5. 5.
    Hochschild G and Mostow G D, Pro-affine algebraic groups, Amer. J. Math., 91 (1969) 1127–1140MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Langer A, On the S-fundamental group scheme, Ann. Inst. Fourier (Grenoble), 61(5) (2011) 2077–2119MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Malcev A, On isomorphic matrix representations of infinite groups, Rec. Math. [Mat. Sbornik] N.S., 8(50) (1940) 405–422MathSciNetzbMATHGoogle Scholar
  8. 8.
    Mukai S, Semi-homogeneous vector bundles on an Abelian variety, J. Math. Kyoto Univ., 18(2) (1978) 239–272MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nakajima S, On Galois module structure of the cohomology groups of an algebraic variety, Invent. Math., 75(1) (1984) 1–8MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Nori M V, On the representations of the fundamental group, Compositio Math., 33 (1976) 29–41MathSciNetzbMATHGoogle Scholar
  11. 11.
    Nori M V, The fundamental group-scheme, Proc. Indian Acad. Sci. (Math. Sci.), 91 (1982) 73–122MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Otabe S, An extension of Nori fundamental group, Commun. Algebra, 45(8) (2017) 3422–3448MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Waterhouse W C, Introduction to affine group schemes, Graduate Texts in Mathematics 66 (1979) (New York-Berlin: Springer-Verlag)CrossRefGoogle Scholar
  14. 14.
    Weil A, Généralisation des fonctions abeliennes, J. de Mathématiques Pures et Appliqués, 17 (1938) 47–87zbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversitySendaiJapan

Personalised recommendations