When I study formal completion and formal schemes, on p.194 of Hartshorne's "Algebraic Geometry", he said "However, one should note that the local rings of $\hat{X}$ are in general not complete."

But I can't see the reason for that the local rings of $\hat{X}$ are in general not complete.

Could someone explains this for me, thanks.

  • 7
    $\begingroup$ Let $A=k[[x,y]]$. This is complete with respect to $m=(x,y)$.Now,localize at theprime $p=(x)$. Note that $x/y \in m_p$,but$\sum_{n=1}^{\infty} (x/y)^n \notin A_p$ $\endgroup$ – the L Feb 23 '11 at 17:18
  • 1
    $\begingroup$ For which formal scheme $\hat X$ and which point $x\in\hat X$ is this $A_p$ the local ring? $\endgroup$ – user2035 Feb 25 '11 at 10:05
  • $\begingroup$ I see your point, the ideal $p$ is not open... Thanks! $\endgroup$ – the L Feb 25 '11 at 10:57
  • $\begingroup$ Even for open $p$, the local ring of $\mathrm{Spf}(A)$ at $p$ is not $A_p$, but rather the direct limit of the completions of $A_f$ for $f\notin p$. $\endgroup$ – user2035 Feb 25 '11 at 12:35

Consider the completion of $\mathrm{Spec}(\mathbb Q[U,V])$ along $U=0$, i.e., $\hat X=\mathrm{Spf}(\mathbb Q[[U]]\{V\})$, and let $x$ be the point $V=0$. Then $U^k/(V-k)$ for $k=1,2,\dots$ is a sequence in $\mathcal O_{\hat X,x}$ converging $U$-adically to $0$, but the corresponding series does not converge. Each element in $\mathcal O_{\hat X,x}$ comes from an open subset of $\hat X$, so is regular outside a finite subset.


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.