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manuscripta mathematica

, Volume 17, Issue 1, pp 21–54 | Cite as

Sur le produit tensoriel des groupes affines

  • Henri Gaudier
Article

Abstract

Let A and B be two commutative affine group schemes over a field. There exists an affine group A⊗B such that Hom(A⊗B,C)≃Bil(A×B,C) for any affine group C. We use technics of the commutative algebraic groups theory, in order to compute these tensor products and to characterize “flat” groups in the unipotent case. The tensor product of commutative affine groups has most properties of the usual tensor product but it is not always associative. As an application we prove a structure theorem of the category of modules over some affine connected prosmooth rings.

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Bibliographie

  1. [1]
    DEMAZURE M. et GABRIEL P. Groupes algébriques. North-Holland Publishing Company, 1970.Google Scholar
  2. [2]
    GAUDIER H. Schémas en anneaux affines. C.R. Acad. Sci. Paris 273, Série A, 1971, p. 768–771.Google Scholar
  3. [3]
    GAUDIER H. Sur les Wk-bimodules et les k-anneaux connexes. C.R. Acad. Sc. Paris, 275, Série A, 1972, p. 61–64.Google Scholar
  4. [4]
    KRAFT H. Kommutative algebraische Gruppen und Ringe. Habilitationsschrift. Bonn 1974.Google Scholar
  5. [5]
    ROSENLICHT M. Some basic theorems on algebraic groups. Amer. Jour. Math. 78 (1956) p. 401–443.Google Scholar
  6. [6]
    SCHOELLER C. Groupes affines commutatifs unipotents sur un corps non parfait. Bull. Soc. Math. France, 100, 1972, p. 241–300.Google Scholar
  7. [7]
    SCHOELLER C. Etude de la catégorie des algèbres de Hopf commutatives connexes sur un corps. Manuscripta Math., 3, 1970, p. 133–155.Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • Henri Gaudier
    • 1
  1. 1.Institut de Recherche Mathématique AvancéeUniversité Louis PasteurStrasbourg Cédex

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