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.


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