A functor to ringed spaces

Abstract

With the set of orders O \mathcal {O} on a field and the Harrison topology induced from the set of all primes as a base space we define a ringed space ( O , F ) (\mathcal {O},\mathcal {F}) . For each field homomorphism we find an associated ringed space morphism producing a contravariant functor from the category of fields to the category of ringed spaces. An equivalence relation ∼ \sim is defined on the set of orders and again a ringed space ( O / ∼ , F ¯ ) (\mathcal {O}/\sim , \bar {\mathcal {F}}) and a contravariant functor from fields to ringed spaces is obtained along with a natural transformation from the first to the second functor. Finally, we obtain a ringed space morphism ( O / ∼ , F ¯ ) → ( Y , O Y ) (\mathcal {O}/\sim , \bar {\mathcal {F}}) \to (Y, \mathcal {O}_Y) where Y is the spectrum of the ring of bounded elements and O Y {\mathcal {O}_Y} is the structure sheaf.