Vol. 32, No. 2, 1970

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ISSN: 0030-8730
Forms of the affine line and its additive group

Peter Russell

Vol. 32 (1970), No. 2, 527–539

Let k be a field, X0 an object (e.g., scheme, group scheme) defined over k. An object X of the same type and isomorphic to X0 over some field K k is called a form of X0. If k is not perfect, both the affine line A1 and its additive group Ga have nontrivial sets of forms, and these are investigated here. Equivalently, one is interested in k-algebras R such that K kRK[t] (the polynomial ring in one variable) for some field K k, where, in the case of forms of Ga,R has a group (or co-algebra) structure s : R R kR such that (K s)(t) = i 1 + 1 t. A complete classification of forms of Ga and their principal homogeneous spaces is given and the behaviour of the set of forms under base field extension is studied.

Mathematical Subject Classification
Primary: 14.50
Received: 16 May 1969
Published: 1 February 1970
Peter Russell
McGill University