Algebraic Geometry Over Hyperrings

Abstract

We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is `multi-valued'. This paper largely consisits of two parts; algebraic aspects and geometric aspects of hyperrings. We first investigate several technical algebraic properties of a hyperring. In the second part, we begin by giving another interpretation of a tropical variety as an algebraic set over the hyperfield which canonically arises from a totally ordered semifield. Then we define a notion of an integral hyperring scheme $(X,{\mathcal{O}}_X)$ and prove that $\Gamma (X,{\mathcal{O}}_X)\simeq R$ for any integral affine hyperring scheme $X=SpecR$.

37 pages

References in corpus (5)

• Les variétés sur le corps à un élément
• Mapping F_1-land:An overview of geometries over the field with one element
• Absolute algebra and Segal's Gamma sets
• Hyperfield extensions, characteristic one and the Connes-Consani plane connection
• Hyperstructures of Affine Algebraic Group Schemes

Cited by in corpus (4)

• ${\mathbb{F}}_1$ for everyone
• Realization spaces of matroids over hyperfields
• von Neumann regular Hyperrings and applications to Real Reduced Multirings
• Association schemes and Hypergroups