# Universal Property of the Zariski-Riemann Space

Let $$k = \mathbb{Q}$$ or $$\mathbb{C}$$. Let $$K$$ be a finitely generated field extension of $$k$$. A model of $$K$$ is a variety $$V \subset \mathbb{CP}^n$$ defined over $$k$$, such that the rational function field of $$V$$ is isomorphic to $$K$$. In this question, it is said that the Zariski-Riemann space is is the inverse limit of the underlying topological spaces of these Varieties.

My question is this: let $$k$$ be any field, and let $$K$$ be a finite transcendence degree finitely generated field extention of $$k$$. Define a generalized model of $$K$$ to be a scheme $$X$$ over $$\text{Spec}(k)$$ with $$\text{colim}_{U \subset X \text{nonempty}} \mathcal{O}_X (U)$$ is $$K$$. Is the Zariski Riemann space the universal generalized model?

Also, a reference for the first fact I mentioned would be nice. Perhaps I can work it out myself seeing the proof.