Combinatorial description of normal toric schemes over valuation rings of rank one

Normal toric varieties over a field can be described by combinatorial data, so called rational fans. Mumford did extend this description to normal toric schemes of finite type defined over a discrete valuation ring. We present a generalization of these results for normal toric schemes of finite type defined over a non-necessarily discrete valuation ring of rank one. We start with the affine case by giving a classification in terms of some admissible cones. For the non-affine case the main step is to generalize Sumihiro's theorem for normal toric varieties to this context. Since our schemes are non-noetherian in general, this is done by applying intersection theory with divisors on admissible formal schemes over rank one valuation rings. Finally, using the orbit-cone correspondence for toric varieties over a valuation ring, proved by Gubler, we get that the set of isomorphism classes of normal toric schemes of finite type over a valuation ring is in bijection with the set of fans whose cones are those obtained in the affine case.