Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Principal homogeneous spaces and group scheme extensions

Author: William C. Waterhouse
Journal: Trans. Amer. Math. Soc. 153 (1971), 181-189
MSC: Primary 14.50; Secondary 13.00
MathSciNet review: 0269659
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose $ G$ is a finite commutative group scheme over a ring $ R$. Using Hopf-algebraic techniques, S. U. Chase has shown that the group of principal homogeneous spaces for $ G$ is isomorphic to $ \operatorname{Ext} (G',{G_m})$, where $ G'$ is the Cartier dual to $ G$ and the Ext is in a specially-chosen Grothendieck topology. The present paper proves that the sheaf $ \operatorname{Ext} (G',{G_m})$ vanishes, and from this derives a more general form of Chase's theorem. Our Ext will be in the usual (fpqc) topology, and we show why this gives the same group. We also give an explicit isomorphism and indicate how it is related to the existence of a normal basis.

References [Enhancements On Off] (What's this?)

  • [1] S. U. Chase and Alex Rosenberg, Amitsur cohomology and the Brauer group, Mem. Amer. Math. Soc. No. 52 (1965), 34–79. MR 0195923
  • [2] Stephen U. Chase and Moss E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in Mathematics, Vol. 97, Springer-Verlag, Berlin-New York, 1969. MR 0260724
  • [3] M. Demazure, A. Grothendieck et al., Schémas en groupes. Fasc. 1, Exposés 1 à 4, Séminaire de Géometrie Algébrique, 1963, Inst. Hautes Études Sci., Paris, 1963/64. MR 34#7517.
  • [4] H. Epp, Commutative group schemes, Harrison's theorem, and Galois extensions, Thesis, Northwestern University, 1966, Dissertation Abstracts 27 B (1967). Abstract #3595.
  • [5] P. Gabriel, Groupes formels, Schémas en Groupes (Sém. Géométrie Algébrique, Inst. Hautes Études Sci., 1963/64) Inst. Hautes Études Sci., Paris, 1965, pp. 66-152+3 (French). MR 0213358
  • [6] Roger Godement, Topologie algébrique et théorie des faisceaux, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Hermann, Paris, 1958 (French). MR 0102797
  • [7] A. Grothendieck, Revêtements étales et groupe fondamental. Fase. 2, Exposé 8, Séminaire de Géometrie Algébrique, 1960/61, Inst. Hautes Études Sci., Paris, 1963. MR 36 #179b.
  • [8] F. Oort, Commutative group schemes, Lecture Notes in Mathematics, vol. 15, Springer-Verlag, Berlin-New York, 1966. MR 0213365
  • [9] Stephen S. Shatz, Principal homogeneous spaces for finite group schemes, Proc. Amer. Math. Soc. 22 (1969), 678–680. MR 249442,
  • [10] J. Verdier, Cohomologie étale des schémas, Séminaire de Géométrie Algébrique, 1963, Inst. Hautes Études Sci., Paris, 1963/64.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 14.50, 13.00

Retrieve articles in all journals with MSC: 14.50, 13.00

Additional Information

Keywords: Commutative group scheme, principal homogeneous space, line bundle, extension group, dual group scheme, Grothendieck topology, normal basis
Article copyright: © Copyright 1971 American Mathematical Society