Polyhedral divisors and torus actions of complexity one over arbitrary fields

We show that the presentation of affine TT-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra associated with a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine G-varieties of complexity one over a field, where G is a (not necessarily split) torus, by using elementary facts on Galois descent. This class of affine G-varieties is described via a new combinatorial object, which we call (Galois) invariant polyhedral divisor.