We show that the presentation of affine $\mathbb{ T}$-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 to a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine $\mathbf{ G}$-varieties of complexity one over a field, where $\mathbf{ G}$ is a (not-nescessary split) torus, by using elementary facts on Galois descent. This class of affine $\mathbf{ G}$-varieties is described via a new combinatorial object, which we call (Galois) invariant polyhedral divisor.
Categories: math.AG