# the proof of “theorem of connectedness”

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？

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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 '10 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 '10 at 16:15

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.)