Algebra & Number Theory

Lefschetz theorem for abelian fundamental group with modulus

Moritz Kerz and Shuji Saito

Full-text: Open access

Abstract

We prove a Lefschetz hypersurface theorem for abelian fundamental groups allowing wild ramification along some divisor. In fact, we show that isomorphism holds if the degree of the hypersurface is large relative to the ramification along the divisor.

Article information

Source
Algebra Number Theory, Volume 8, Number 3 (2014), 689-701.

Dates
Received: 27 April 2013
Revised: 12 November 2013
Accepted: 10 December 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730171

Digital Object Identifier
doi:10.2140/ant.2014.8.689

Mathematical Reviews number (MathSciNet)
MR3218806

Zentralblatt MATH identifier
1318.14014

Subjects
Primary: 14H30: Coverings, fundamental group [See also 14E20, 14F35]
Secondary: 14E22: Ramification problems [See also 11S15]

Keywords
fundamental group ramification Lefschetz theorem

Citation

Kerz, Moritz; Saito, Shuji. Lefschetz theorem for abelian fundamental group with modulus. Algebra Number Theory 8 (2014), no. 3, 689--701. doi:10.2140/ant.2014.8.689. https://projecteuclid.org/euclid.ant/1513730171


Export citation

References

  • H. Esnault and M. Kerz, “A finiteness theorem for Galois representations of function fields over finite fields (after Deligne)”, Acta Math. Vietnam. 37:4 (2012), 531–562.
  • A. Grothendieck, Local cohomology, Lecture Notes in Mathematics 41, Springer, Berlin, 1967.
  • A. J. de Jong, “Smoothness, semi-stability and alterations”, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93.
  • K. Kato, “Swan conductors for characters of degree one in the imperfect residue field case”, pp. 101–131 in Algebraic $K$-theory and algebraic number theory (Honolulu, 1987), edited by M. R. Stein and R. K. Dennis, Contemp. Math. 83, Amer. Math. Soc., Providence, RI, 1989.
  • M. Kerz and S. Saito, “Chow group of $0$-cycles with modulus and higher dimensional class field theory”, preprint, 2013.
  • G. Laumon, “Semi-continuité du conducteur de Swan (d'après P. Deligne)”, pp. 173–219 in The Euler–Poincaré characteristic, edited by J.-L. Verdier, Astérisque 83, Soc. Math. France, Paris, 1981.
  • S. Matsuda, “On the Swan conductor in positive characteristic”, Amer. J. Math. 119:4 (1997), 705–739.
  • W. Raskind, “Abelian class field theory of arithmetic schemes”, pp. 85–187 in $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, 1992), part 1, edited by B. Jacob and A. Rosenberg, Proc. Sympos. Pure Math. 58, Amer. Math. Soc., Providence, RI, 1995.
  • A. Schmidt and M. Spieß, “Singular homology and class field theory of varieties over finite fields”, J. Reine Angew. Math. 527 (2000), 13–36.
  • A. Grothendieck, Séminaire de Géométrie Algébrique du Bois Marie 1960–1961: revêtements étales et groupe fondamental, Lecture Notes in Mathematics 224, Springer, Berlin, 1971.
  • A. Grothendieck, Séminaire de Géométrie Algébrique du Bois-Marie 1962: cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, Advanced Studies in Pure Mathematics 2, North-Holland, Amsterdam, 1968.