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Let $R$ be a ring, and $\mathfrak{p}$ be a prime ideal. The stalk at $\mathfrak{p}$ with respect to the etale topology is $(R_{\mathfrak{p}})^{sh}$ (the strict henselization of $R_{\mathfrak{p}}$). The stalk at $\mathfrak{p}$ with respect to the Nisnevich topology is $(R_{\mathfrak{p}})^h$ (the henselization of $R_{\mathfrak{p}}$).

Grothendieck also spoke of formal neighborhoods, and I wonder if this fits into the pattern above. To be precise: is there some Grothendieck topology for which the stalk at $\mathfrak{p}$ would be the completion of $R_{\mathfrak{p}}$ with respect to $\mathfrak{p}$? If so, what is it?

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The stalk of a sheaf $F$ at a point $p$ is always some sort of colimit of sets of sections $F(U)$ -- this is true for any topology with enough points. The completion looks like an (inverse) limit. So the naive answer to your question is no. –  Charles Rezk Dec 19 '10 at 3:07
    
That sounds about right. I wonder how Grothendieck thought of these things. Surely it occurred to him that these notions don't work together, and yet that they are analogous... –  James D. Taylor Dec 19 '10 at 5:42
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I use to heuristically see $\mathrm{Spec}(\mathcal{O}_{X,x})$ as an "intersection of open neighbourhoods of $x$", while $\mathrm{Spec}(\widehat{\mathcal{O}}_{X,x})$ as an "increasing union of closed subsets containing $x$". –  Qfwfq Dec 23 '10 at 22:10

1 Answer 1

You may want to look at the Artin approximation theorem. Roughly, it says (in the context of varieties, say) that any phenomenon you observe at the level of completions is already achieved etale locally.

E.g. suppose that $X$ is an algebraic curve over $k$ (an algebraically closed field) and $x$ is a closed point of $X$ such that completed local ring $\hat{\mathcal O}_{X,x}$ has two irreducible components in its spectrum. Then Artin's theorem says that there is an etale n.h. of $x$ which is the union of two branches passing through $x$.

This has the practical consequence that notions such as $x$ is a node'' can be defined either by a condition on the completion $\hat{\mathcal O}_{X,x}$ or by an etale local condition. The former is more elementary, and usually easier to check; the latter is often more powerful in proofs, because it is more tightly connected to the geometry of the whole curve $X$.

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