## Algebraic geometry over semi-structures and hyper-structures of characteristic one

##### Abstract

In this thesis, we study algebraic geometry in characteristic one from the perspective of semirings
and hyperrings. The thesis largely consists of three parts:
(1) We develop the basic notions and several methods of algebraic geometry over semirings. We first construct a semi-scheme by directly generalizing the classical construction of a scheme, and prove that any semiring can be canonically realized as a semiring of global functions on an affine semischeme. We then develop Cech cohomology theory for semi-schemes, and show that the classical
isomorphism is still valid for a semi-scheme. Finally, we introduce the notion of a valuation on a semiring, and prove that an analogue of an abstract curve by using the (suitably defined) function field Qmax(T) is homeomorphic to the projective line over the field with one element. (2) We develop algebraic geometry over hyperrings. The first motivation for this study arises from the following problem posed in [9]: if one follows the classical construction to define the hyper-scheme (X = SpecR,O_X), where R is a hyperring, then a canonical isomorphism R ≃ O_X(X) does not hold in general. By investigating algebraic properties of hyperrings (which include a construction of a quotient hyperring and Hilbert Nullstellensatz), we give a partial answer for their problem as follows: when R does not have a (multiplicative) zero-divisor, the canonical isomorphism R ≃ O_X(X) holds for a hyper-scheme (X = SpecR,OX). In other words, R can be realized as a hyperring of global functions on an affine hyper-scheme.
We also give a (partial) affirmative answer to the following speculation posed by Connes and Consani
in [7]: let A = k[T] or k[T, 1/T ], where k = Q or Fp. When k = Fp, the topological space SpecA
is a hypergroup with a canonical hyper-operation ∗ induced from a coproduct of A. The similar
statement holds with k = Q and SpecA\{δ}, where δ is the generic point (cf. [7, Theorems 7.1
and 7.13]). Connes and Consani expected that the similar result would be true for Chevalley group schemes. We prove that when X = SpecA is an affine algebraic group scheme over arbitrary field, then, together with a canonical hyper-operation ∗ on X introduced in [7], (X, ∗) becomes a slightly general (in a precise sense) object than a hypergroup. (3) We give a (partial) converse of S.Henry’s symmetrization procedure which produces a hypergroup from a semigroup in a canonical way (cf. [21]). Furthermore, via the symmetrization process, we connect the notions of (1) and (2), and prove that such a link is closely related with the notion of real prime ideals.