# The Zariski Riemann Space, but with Local Rings

The Zariski Riemann space, while an abandoned approach, has lead to later developments and generalizations, including $$\text{Spv}$$ (the space of valuations) and Huber's work. In studying it, I would like to know what fails if we erase the assumption that points are valuation rings, and instead make a topological space on the local subrings.

Suppose we wanted to make a topological space on the local subrings of a field. Which desirable properties of the Zariski Riemann space $$\text{Zar}(K)$$ fail when we try to make a topological space on the local rings contained in a field instead of the valuation rings contained in a field Specifically, suppose we let $$\text{Loc}(K)$$ of a field $$K$$ be the topological space whose points are local rings, and whose open sets are unions of sets of the form $$V(f_1, ..., f_n) = \{ R \subset K \text{ a local ring } : f_1, ..., f_n \in R \}$$. This is completely analogous to the construction of the Zariski Riemann space. Also, the Zariski Riemann space will be a subspace of this one.

One approach here would be the historical one: Zariski had Riemann surfaces in mind when creating the Riemann-Zariski space, and using local rings does not generalize this situation, while the Zariski-Riemann space does. Nevertheless, there is perhaps something more to say here about why it is better to use the Zariski-Riemann space instead of $$\text{Loc}$$. One might wonder if there is an intrinsic motivation of the Zariski-Riemann space.

We can make $$\text{Zar}(K)$$ into a sheaf whose stalks at each point are the corresponding valuation rings. To do this, we create an étale topology on $$\amalg_R R$$, where the disjoint union is taken over the valuation subrings of $$K$$, and let the sheaf $$\mathcal{O}_{\text{Zar}(K)}$$ be its sheaf of sections.

So we might try to make $$\text{Loc}(K)$$ into a sheaf whose stalks at each point are the corresponding local rings. To do this, define for each $$a \in K$$ a function $$f_a : V(a) \rightarrow \amalg_{R} R$$, where the disjoint union is taken over local subrings of $$K$$, sending $$R \in V(a)$$ to $$(a, R) \in \amalg_{R} R$$. We put a topology on $$\amalg_{R} R$$ where a sub-basis consists of images of functions of the form $$f_{a}$$ for $$a \in K$$. Then define $$\mathcal{O}_{\text{Loc}(K)}$$ to be the sheaf of sections of this étale cover.

We can then identify $$\mathcal{O}_{X} (U)$$, where $$X = \text{Loc}(K)$$, with the intersection of local rings in $$U$$. The stalk of $$X$$ at $$R$$ is just $$R$$.

Does something go wrong here? What makes the Zariski-Riemann space more desirable topological space (or sheaf) to work with than $$\text{Loc}(K)$$?

Some observations:

1. The global sections of $$\text{Zar}(K)$$ is the integral closure of the prime subfield in $$\text{Zar}(K)$$. For $$\text{Loc}$$ it is the prime subring of $$K$$. So we can recover $$K$$ from the global sections as the fraction field, while the same is not true of $$\text{Loc}(K)$$.

2. $$\text{Zar}(K)$$ is a spectral space. I have my doubts that $$\text{Loc}(K)$$ is a spectral space.

3. The valuation rings are the maximal local rings under the relation of dominance. Maybe this means that $$\text{Zar}(K)$$ is universal in some sense?