Consider the scheme $\mathbb{P}^1$, and the point $0 \in \mathbb{P}^1$. What is the formal neighbourhood of $0$ in $\mathbb{P}^1$?
Or if you know a good reference, that would be helpful.
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1My first guess would be that for a ring $R$, the formal neighborhood of $0$ in $\Bbb{P}^1_R$ is $R[[X]]$, the ring of formal power series in the variable $X$ with coefficients in $R$. Does that make any sense in your context? – Nils Matthes Jul 04 '13 at 12:50
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1One reference for a general exposition is Hartshorne II.9. – Andrew Jul 04 '13 at 19:28
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2@NilsMatthes I would probably say it is $Spf (R[[X]])$ rather than the ring itself. – Matt Jul 04 '13 at 22:29
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pg 190 of Hartshorne (II.9) seems helpful. – Puraṭci Vinnani Jul 05 '13 at 03:25
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The formal neighbourhood of a closed subscheme $X$ of $Y$ refers to the formal scheme obtained by completing $Y$ along $X$; see the discussion of formal schemes in Hartshorne for (unfortunately not that many) more details.
This is certainly not an open subscheme of $Y$; indeed, as Marci notes, the underlying topological space is simply $X$ itself. However, the ringed space structure has been "thickened up" to incorporate the normal diretions to $X$ in $Y$.
One thing you could look at is a paper of Beauville and Laszlo where they explain how to make certain constructions, e.g. bundles, by gluing data on the formal n.h. of $X$ to data on the complement of $X$. (A discussion of this paper and of related earlier work can be found in the comments here and also in this answer.)
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One thing to add. If you are interested in the formal neighborhood F of X in Y, then the underlying topological space of the formal neighborhood is X.
Marci
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