The Group of Units on an Affine Variety

Abstract

The object of study is the group of units O^\ast(X) in the coordinate ring of a normal affine variety X over an algebraically closed field k. Methods of Galois cohomology are applied to those varieties that can be presented as a finite cyclic cover of a rational variety. On a cyclic cover X \rightarrow A^m of affine m-space over k such that the ramification divisor is irreducible and the degree is prime, it is shown that O^\ast(X) is equal to k^\ast, the nonzero scalars. The same conclusion holds, if X is a sufficiently general affine hyperelliptic curve. If X has a projective completion such that the divisor at infinity has r components, then sufficient conditions are given for O^\ast(X)/k^\ast to be isomorphic to Z^(r-1).