Portal de Periódicos da CAPES

Forms of the affine line and its additive group

1970; Mathematical Sciences Publishers; Volume: 32; Issue: 2 Linguagem: Inglês

10.2140/pjm.1970.32.527

1945-5844

Peter B. Russell,

Rings, Modules, and Algebras

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 α , R has a group (or co-algebra) structure s\R->R® k R such that (K®s)(t) = £ ® 1 + 1 ® ί.A complete classification of forms of G α and their principal homogeneous spaces is given and the behaviour of the set of forms under base field extension is studied.