Forms of the affine line and its additive group

Abstract

Let k be a field, X o 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 X o . If k is not perfect, both the affine line A 1 and its additive group G tt 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 => k y where, in the case of forms of G , R has a group (or co-algebra) structure s\R->R k R such that (Ks)(t) = 1 + 1 . A complete classification of forms of G and their principal homogeneous spaces is given and the behaviour of the set of forms under base field extension is studied.