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3 results

  • IX - Group schemes

  • Skip Garibaldi, Institute for Defense Analyses, USA, Holger P. Petersson, FernUniversität in Hagen, Michel L. Racine, University of Ottawa
  • Book:
    Albert Algebras over Commutative Rings
    Published online:
    07 November 2024
    Print publication:
    21 November 2024, pp 567-619

  • Summary

    In most of this book, we have studied Albert and octonion algebras. In this chapter, we connect those with the theory of semi-simple affine group schemes, especially those of type E6, F4, and G2. As part of this effort, we give an introduction to non-abelian flat cohomology and its applications to descent. We leverage this together with known results about affine group schemes such as Gross’s mass formula to classify the Albert algebras over the integers, a recently discovered result.

  • Lefschetz theorems in flat cohomology and applications

  • Part of
  • Sean Cotner, Bogdan Zavyalov
  • Journal:
    Compositio Mathematica / Volume 160 / Issue 8 / August 2024
    Published online by Cambridge University Press:
    11 September 2024, pp. 1850-1870
    Print publication:
    August 2024
    • Article
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      • Open access
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  • We prove a version of the Lefschetz hyperplane theorem for fppf cohomology with coefficients in any finite commutative group scheme over the ground field. As consequences, we establish new Lefschetz results for the Picard scheme.

  • Periods of abelian varieties

  • J. S. Milne
  • Journal:
    Compositio Mathematica / Volume 140 / Issue 5 / September 2004
    Published online by Cambridge University Press:
    04 December 2007, pp. 1149-1175
    Print publication:
    September 2004
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  • We prove various characterizations of the period torsor of abelian varieties.