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Let $X$ be a complete smooth irreducible variety over a field $K$ and consider a closed point $x \in X$. Moreover let $\widehat{\mathcal O}_{X,x}$ be the $\mathfrak m_x$-adic completion of the local ring at $x$.

What is the geometrical meaning of $\operatorname{Spec}\widehat{\mathcal O}_{X,x}$? many sources say very vaguely that this spectrum carries pieces of information about very small neighbourhoods around $x$, but I'd like a more effective explanation. I know that maybe one should have a knowledge of formal schemes in order to fully understand this object, but unfortunately I only know "standard" scheme theory.

There is obviuously a morphism $f:\operatorname{Spec} \widehat{\mathcal O}_{X,x}\longrightarrow \operatorname{Spec}\mathcal O_{X,x}$, so I would like to know what kind of geometric properties one can evince respectively from $\operatorname{Spec}\widehat{\mathcal O}_{X,x}$ and $\operatorname{Spec}\mathcal O_{X,x}$: the latter is already a "local object" so why do we need the completion? In particular I'm interested in the minimal prime ideals of $\widehat{\mathcal O}_{X,x}$ which are also called "formal branches at $x$"....

Many thanks in advance.

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  • $\begingroup$ Well, I will let someone more competent than me answer your question fully, but do you not agree that in arithmetic, considering the completion $\mathbb{Z}_p$ instead of just the localization $\mathbb{Z}_{(p)}$ has pretty strong advantages ? $\endgroup$ Commented Mar 11, 2016 at 12:53
  • $\begingroup$ Yes of course, but I'm looking at the geometric picture for varieties over fields. $\endgroup$ Commented Mar 11, 2016 at 13:00

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I don't think $\mathcal O_{X,x}$ is as local an object as you think it is. For example, you can get the function field of $X$ from it, which is surely a global thing?

The intuition for $\widehat{\mathcal O_{X,x}}$ is that it contains "Taylor expansions" of functions around $x$. If you know what the normal cone is, then you can also convince yourself that if $x$ is a closed point, then $\widehat{\mathcal{O_{X,x}}}$ contains essentially the same information as the normal cone of $x$ in $X$. You can also verify that both of those objects are isomorphic in the following two cases:

$x$ is the node of the nodal curve and

$x$ is the origin in $\text{Spec}k[x,y]/(xy)$.

And by looking at pictures, you can immediately see that the two points are geometrically locally similar.

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  • $\begingroup$ Ok, this is clear. And what about the spectrum of $\widehat{\mathcal O_{X,x}}$? $\endgroup$ Commented Mar 11, 2016 at 14:08
  • $\begingroup$ I don't know. I don't think a lot is known in general. For example, look at this MO question, which occurs for a regular point on a surface: mathoverflow.net/questions/24082/…. But if X is a curve, there is the correspondence that you hinted at of minimal prime ideals to formal branches through x, which are basically just the tangent directions of X at x. Maybe someone more knowledgeable than me can chime in.. $\endgroup$
    – user00000
    Commented Mar 11, 2016 at 20:38

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