5
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 ? $\endgroup$ – Captain Lama Mar 11 '16 at 12:53
  • $\begingroup$ Yes of course, but I'm looking at the geometric picture for varieties over fields. $\endgroup$ – ginevracoal Mar 11 '16 at 13:00
4
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Ok, this is clear. And what about the spectrum of $\widehat{\mathcal O_{X,x}}$? $\endgroup$ – ginevracoal Mar 11 '16 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 Mar 11 '16 at 20:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.