## Unipotent algebraic groups |

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### Contents

Introduction 1 4 | 1 |

il Notations conventions and some basic preliminary facts5 | 17 |

commutative group schemes application to the two | 29 |

Copyright | |

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### Common terms and phrases

abelian additive group affine algebraic affine k-group scheme affine k-scheme algebra map algebraic closure arithmetic genus assume automorphism biadditive canonical central extension closed subgroup coaction cohomology commutative k-group scheme defined denote DG-III DG-IV divisor divisor class group Edited equation exact sequence extension of G faithfully flat field extension finite dimensional function functor G-module Grothendieck ground field Hence homomorphism Hopf algebra Horn hy(G induced integer invertible sheaf isomorphic k-algebra k-closed k-derivation k-form k-form of G k-G-module k-gr k-group of Russell k-group scheme k-homomorphism k-isomorphic k-morphism k-normal completion k-rational point k-smooth k-wound left k[F]-module LEMMA Let G module morphism nonconstant one-dimensional unipotent group polynomial ring Proof PROPOSITION purely inseparable quotient Russell type scheme G scheme of G Seminaire Spec Spec(A structure subgroup scheme Suppose Theorem Theory two-dimensional unipotent unipotent algebraic k-group unipotent k-group unique factorization domain vector group