A functor to ringed spaces

Abstract:

With the set of orders $\mathcal {O}$ on a field and the Harrison topology induced from the set of all primes as a base space we define a ringed space $(\mathcal {O},\mathcal {F})$. 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 $(\mathcal {O}/\sim , \bar {\mathcal {F}})$ 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 $(\mathcal {O}/\sim , \bar {\mathcal {F}}) \to (Y, \mathcal {O}_Y)$ where Y is the spectrum of the ring of bounded elements and ${\mathcal {O}_Y}$ is the structure sheaf.