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This theorem is: let f:X--->Y be a proper morphism of noetherian schemes,and the induced morphism of sheaves f^#:O_Y---->f_*O_X is isomorphic.Then for any point y belongs to Y,f^-1(y) is nonempty and connected. I have seen a proof by use of formal schemes.My question is：Is there any proof without this trick？

flag	asked Feb 7 2010 at 7:47 Haoyu Hu 23●2

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The proof in EGA III.4.3 only uses the theorem of formal functions for $Rf^0_*$. If you accept formal completion of modules, there is no need of formal schemes. – Qing Liu Feb 7 2010 at 13:28

Though, I think there is a reasonable argument that if you have accepted formal completion of modules, you have accepted formal schemes. – Ben Webster♦ Feb 7 2010 at 16:15

Emerton · Accepted Answer · 2010-02-08 04:18:38Z UTC

A typical context in which one has the condition $f_{*}\mathcal O_X = \mathcal O_Y$ in that in which $f$ is a birational morphism and $Y$ is normal. In this context, the proof that the fibres are connected is due to Zariski (I believe that it's the original version of his ``main theorem'') and certainly predates EGA methods. One can find the paper in his collected works. (It's been a long time since I looked at it, but I would guess that his main technical tool is valuation theory; I might check this when I get a chance.)

However, it is worth bearing in mind the evolution of Zariski's work on this kind of question: his investiations of this sort of connectedness theorem culminated in his proof of his connectedness theorem, to the effect that a specialization of connected varieties is again connected. His was the first purely algebraic proof of this result, I think. To give the proof, he invented his theory of formally holomorphic functions, which I believe was regarded at the time as being the most difficult part of the algebraic theory of algebraic geometry developed by Weil, Chevalley, and Zariski. This theory served as one of the inspirations for the theory of formal schemes, and is the precursor to Grothendieck's theorem on formal functions (the proof via formal schemes mentioned in the question).

If so great a geometer as Zariski was led to introduce these kinds of formal methods to study connectedness problems, it is probably reasonable to regard them as somewhat intrinsic to the problem.

the proof of “theorem of connectedness”

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