Forms of the affine line and its additive group

Abstract

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