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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?

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