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, O X ) and prove that Γ(X, O X ) ≃ R for any integral affine hyperring scheme X = Spec R.

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We construct a full embedding of the category of hyperfields into Dress's category of fuzzy rings and explicitly characterize the essential image -it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker's theory of matroids over hyperfields with Dress's theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker's partial demifields.

This paper examines the category Mat • of pointed matroids and strong maps from the point of view of Hall algebras. We show that Mat • has the structure of a finitary proto-exact category -a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We define the algebraic K-theory K * (Mat • ) of Mat • via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections π s n (S) ֒→ K n (Mat • ) from the stable homotopy groups of spheres for all n. Finally, we show that the Hall algebra of Mat • is a Hopf algebra dual to Schmitt's matroidminor Hopf algebra.

Abstract. A semiring scheme generalizes a scheme in such a way that the underlying algebra is that of semirings. We generalizeČech cohomology theory and invertible sheaves to semiring schemes. In particular, when X = P n M , a projective space over a totally ordered idempotent semifield M , we show thať H m (X, O X ) is in agreement with the classical computation for all m. Finally, we classify all invertible sheaves on X = P n M by computing Pic(X) explicitly.

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