Spec of $\mathfrak{p}$-adic completion of number ring

Question

Suppose $\mathcal{O}_K$ is a number ring and $\mathfrak{p}\in \operatorname{Spec}\mathcal{O}_K$ is a prime ideal. Let $\mathcal{O}_{\mathfrak{p}}$ be the $\mathfrak{p}$-adic completion of $\mathcal{O}_K$ then consider $\operatorname{Spec}\mathcal{O}_{\mathfrak{p}}$, what would be the relationship between $\operatorname{Spec}\mathcal{O}_{\mathfrak{p}}$ and $\operatorname{Spec}\mathcal{O}_K$? I am just having trouble to connect these two in my mind, and just looking for some intuitive descriptions such as "one is embedded into the others" along with some explanations.

Answer 1

In order for this to make sense I'm going to assume that $\mathfrak{p} \ne (0)$ as localizing and completing at the generic point of $\operatorname{Spec}\mathcal{O}_K$ is silly and kind of degenerate. The embedding $\operatorname{Spec} \mathcal{O}_{\mathfrak{p}} \to \operatorname{Spec}\mathcal{O}_K$ picks out the subspace $\lbrace 0, \mathfrak{p}\rbrace$ of $\operatorname{Spec}\mathcal{O}_{K}$ and has sheaf morphism induced from a witness of the fact that $$ \mathcal{O}_{\mathfrak{p}} \cong \lim_{\substack{\longleftarrow \\ n \in \mathbb{N}}} \frac{\mathcal{O}_K}{\mathfrak{p}^n} \cong \lim_{\substack{\longleftarrow \\ n \in \mathbb{N}}} \frac{\mathcal{O}_{K,(\mathfrak{p})}}{\mathfrak{p}^n\mathcal{O}_{K,(\mathfrak{p})}} $$ where $\mathcal{O}_{K,(\mathfrak{p})}$ is the localization of $\mathcal{O}_K$ at the set $(\mathfrak{p}) = \mathcal{O}_K \setminus \mathfrak{p}$.

Here is how I like to think of this embedding and what it's doing. First, there is a strong analogue of the scheme $\operatorname{Spec}\mathcal{O}_K$ as some sort of curve; it a collection of points (the underlying topological space) and functions on those points which can be at least locally ''analytically'' continued in ways that make sense (this is the information stored and captured by the structure sheaf $\mathcal{O}_K$). In this formal analogy, we can associate to elements $x$ of $\mathcal{O}_K$ the prime ideals (points) $\mathfrak{p}$ of $\mathcal{O}_K$ for which $x$ vanishes within some neighborhood of $\mathfrak{p}$ and the prime ideals (points) $\mathfrak{q}$ of $\mathcal{O}_K$ for which $x$ is invertible in a neighborhood of $\mathfrak{q}$.

Let's go through this in some more detail. Geometrically, the map $\operatorname{Spec} \mathcal{O}_{\mathfrak{p}} \to \operatorname{Spec}\mathcal{O}_K$ is seeing that $\operatorname{Spec}\mathcal{O}_{\mathfrak{p}}$ can be thought of as the space of functions on $\lbrace (0), \mathfrak{p}\rbrace$ which have a ''Taylor expansion'' of the form $$ x = \sum_{n=0}^{\infty}a_n\pi^n $$ around the prime $\mathfrak{p}$ (for a uniformizer $\pi$ of the maximal ideal $\mathfrak{p}\mathcal{O}_{\mathfrak{p}}$ of $\mathcal{O}_{\mathfrak{p}}$); the embedding tells us that every element of $\mathcal{O}_K$ admits such an expansion. The fact that the image of the spectral map picks out only the generic point of $\mathcal{O}_K$ and the prime $\mathfrak{p}$ is saying that within an infinitesimal neighborhood of $\mathfrak{p}$, the element $x \in \mathcal{O}_K$ is either in some higher infinitesimal neighborhood of $\mathfrak{p}\mathcal{O}_{\mathfrak{p}}$ (and hence belongs to the prime ideal $\mathfrak{p}$) or is invertible within an infinitesimal neighborhood of $\mathfrak{p}$ (in the sense that the $\mathfrak{p}$-adic power series representation $$ x = \sum_{n=0}^{\infty}a_n\pi^n $$ has constant term $a_0$ a unit in $\mathcal{O}_{\mathfrak{p}}$). This lets us intuit the maximal ideal $\mathfrak{p}\mathcal{O}_{\mathfrak{p}}$ as the germs of those functions on $\operatorname{Spec}\mathcal{O}_{\mathfrak{p}}$ which vanish at $\mathfrak{p}\mathcal{O}_{\mathcal{p}}$ and the maximal ideal $\mathfrak{p}\mathcal{O}_{K,(\mathfrak{p})}$ as the germs of functions on $\operatorname{Spec}\mathcal{O}_K$ which vanish at $\mathfrak{p}$.