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Let $\hat{X} = \text{Spf} \hat{A}$ be obtained as the formal completion of an affine scheme $X = \text{Spec} A$ where $A$ is an adic noetherian ring. Given a coherent sheaf $\mathfrak{F}$ on $\hat{X}$, is it always possible to find a coherent sheaf $\mathcal{F}$ on $X$ such that $\hat{\mathcal{F}} = \mathfrak{F}$?

flag	asked Mar 17 2011 at 15:03 lorca 71●1

Let me just make the orthogonal comment that this is true if $X$ is projective, by formal GAGA. Algebraicity in the affine case is pretty rare as you can see from ts3uji's example below. – Keerthi Madapusi Pera Mar 18 2011 at 0:59

t3suji · Answer 1 · 2011-03-17 19:48:33Z UTC

Take $A=k[x,y]$, $\hat A=k[[x,y]]$, and suppose coherent sheaf corresponds to the $\hat A$-module $N=\hat A/(f)$, where $f\in k[[x,y]]$ is not "algebraic": say, $f=y-exp(x)$, assuming $k$ has characteristic zero. It is clear that $N$ does not come from completion of any f.generated $A$-module $M$. (Proof: Suppose $N=\hat M$. Since $N$ has no $A$-torsion, any torsion of $M$ maps to zero, so we can replace $M$ with $M/(torsion)$ and assume $M$ is torsion-free. Then $M$ embeds in a locally free module (its second dual), but that would make $N$ embed in the free $\hat A$-module, which is false.)

Torsten Wedhorn · Answer 2 · 2011-03-17 17:29:47Z UTC

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According to EGA I (Springer edition), Theoreme (10.10.2), there is an equivalence of categories between the category of finitely generated $A$-modules and the category of coherent $O_{{\rm Spf}(A)}$-modules. In particular, the answer to your question is Yes.

answered Mar 17 2011 at 17:29

Torsten Wedhorn
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But does this answer the original question? You show there is a f.g. $\hat A$-module, but is it induced by a f.g. $A$-module. – t3suji Mar 17 2011 at 19:49

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I think this is a problem of definitions. Usually an adic ring $A$ is already defined to be complete and thus $A = \hat A$. Then coherent modules over $O_{{\rm Spec}(A)}$ and $O_{{\rm Spf}(A)}$ are defined in a different way but EGAI tells you that there is not really a difference. If you define it to be only a topological ring endowed with some $I$-adic topology for some ideal $I$ - and thus $A \ne \hat A$ in general, then the answer is No as you explained. – Torsten Wedhorn Mar 18 2011 at 8:51

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@Wedhorn: I completely agree that there is a potential confusion. My interpretation was based on the fact that the OP uses two different letters $A$ and $\hat A$ for the rings. – t3suji Mar 18 2011 at 11:41

coherent sheaves on affine formal schemes

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