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When I study formal completion and formal schemes, on p.194 of Hartshorne's "Algebraic Geometry", he said "One sees easily that the stalks of the sheaf $\mathcal{O}_{\hat{X}}$ are local rings."

Notice that here $\mathcal{O}_{\hat{X}}$ is not the structure sheaf of X, there is a "hat" on the symbol $X$.

But I can't see the reason for that the stalks of the sheaf $\mathcal{O}_{\hat{X}}$ are local rings.

Could someone explains this for me, thanks.

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See 7.6.17 in Ch. 0 of EGA I (& 10.1.6 in EGA I). Discussion of formal schemes in H. is one of the low points: nothing serious happens, no indications of how to work with formal schemes or why one cares. I strongly recommend looking at discussion of formal schemes in section 10 of EGA I (and background in Ch. 0, ignoring "linear topologies" in favor of adic ones if you prefer). Everything is explained nicely in EGA on this. (Real fireworks are in EGA III, but to have a clear picture of how to think about formal schemes it is much better to read the discussion in EGA I than in Hartshorne.) –  BCnrd Jun 10 '10 at 17:43
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Illusie's paper in "FGA explained" is a very good starting point. I agree that the discussion in Hartshorne is not good. –  Angelo Jun 10 '10 at 19:09
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For what it's worth, I learnt about formal schemes first from Hartshorne, and then got a more nuanced feel from reading some papers that used them (e.g. the beautiful papers of Katz on Serre--Tate theory and related matters in LNM 868, and the paper of Deligne--Illusie on deforming K3 surfaces in the same volume). I've never had cause to carefully study the treatment in EGA. –  Emerton Jun 11 '10 at 3:46
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A cool application of formal schemes is Tate curve (Raynaud's method, in Deligne-Rap.). I don't know a way to handle its torsion properties ``over $\mathbf{Z}$'' without formal schemes, nor does Tate (I asked him). Oort said Grothendieck was especially proud of getting thm on formal functions and formal GAGA in the proper case without projective assumptions; not a routine Chow lemma thing. Buried in EGA: isom in thm on formal fns (over adic base) is topological (ker & coker inv. systems are even null). This is used in Kisin's work on pst def. rings, but left to the reader to notice that. :) –  BCnrd Jun 11 '10 at 6:58
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1 Answer

Here is a self-contained explanation (hopefully without any blunder):

Locally, $\hat{X}$ is an affine formal scheme, so each point has a neighbourhood basis admitting of open sets $U$ admitting the following description: there is a ring $A$, with ideal $I$, such that the underlying topological space is $U_0 :=$ Spec $A/I$, and the structure sheaf is the projective limit of the sheaves $\mathcal O_{U_n},$ where $U_n :=$ Spec $A/I^{n+1}$.

(Note that the underlying topological space of all the $U_n$ coincide, so as topological spaces $$U = U_0 = \cdots = U_n = \cdots .$$ The sheaves $\mathcal O_{U_n}$ are all sheaves on this same underlying topological space, which form a projective system with obvious transition maps, corresponding to the surjections of rings $A/I^{n+1} \to A/I^n.$

Now if $x$ is a point of $\hat{X}$, and if we choose a neighbourhood $U$ of $x$ as above, then the natural map on sheaves $\mathcal O_{\hat{X}} \to \mathcal O_{U_0}$ induces a natural map on stalks $\mathcal O_{\hat{X},x} \to \mathcal O_{U_0,x}$. The target is a local ring. Let $\mathfrak m_x$ denote the preimage in $\mathcal O_{\hat{X},x}$ of the maximal ideal in $\mathcal O_{U_0,x}$; I claim that it is the unique maximal ideal of $\mathcal O_{\hat{X},x}$.

To see this, suppose that $f$ is an element of $\mathcal O_{\hat{X},x}$ which does not lie in $\mathfrak m_x$. Then by definition of the stalk, $f$ extends to a section of $\mathcal O_{\hat{X}}$ over some neighbourhood of $x$, which (by shrinking $U$ as necessary) we may as well assume is our affine neighbourhood $U$. Thus we may think of $f$ as a section of the projective limit of $\mathcal O_{U_n}(U)$, i.e. the projective limit of the rings $A/I^{n+1}$.

The assumption that $f$ is not in $\mathfrak m_x$ says that its image in $\mathcal O_{U_0}(U)$ is not in the maximal ideal at $x$, and so shrinking $U$ further, if necessary, we may assume that $f$ is a unit in $\mathcal O_{U_0}(U) = A/I$.

Thus $f$ is an element of the projective limit of $A/I^{n+1}$ which is a unit in $A/I$. One easily verifies that $f$ is then a unit in every $A/I^{n+1}$, and hence in the projective limit. Thus $f$ is a unit in the ring $\mathcal O_{\hat{X}}(U)$, and so in particular in the stalk $\mathcal O_{\hat{X},x}$.

I've shown that every element of the stalk $\mathcal O_{\hat{X},x}$ not lying in $\mathfrak m_x$ is a unit, which implies that $\mathcal O_{\hat{X},x}$ is local with maximal ideal $\mathfrak m_x$.

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With a bit more work, can also show this local stalk is noetherian. The idea is to introduce an auxiliary local ring which is visibly noetherian and local over the stalk ring, as well as flat and hence f.flat over it, so the noetherian property descends to the stalk ring. This is explained in the EGA references mentioned above. –  BCnrd Jun 11 '10 at 12:21
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