Lemma Tag 055J: Let R be a dvr. Let X be flat over Spec(R), with reduced special fibre, and connected total space. Then the generic fibre of the structure morphism f : X —> Spec(R) is connected.
You can find a version of this lemma as EGA IV, Lemma 15.5.6 where the hypotheses are that f is locally of finite type and open instead of flat. But in the proof of the lemma in EGA it is remarked that the hypotheses “loc. fin. type + open over dvr” imply “flat”, hence the lemma above implies the lemma in EGA. I urge you to try to prove the lemma above before looking it up, because it is fun when you find it!
Why is flatness necessary? If X is not flat over R, then a counter example is X = Spec(R[x]/(px(x-1))) where p ∈ R is a uniformizer. In words: X is a union of two copies of Spec(R) glued at 0, 1 of the affine line over the residue field of R.
Why is a reduced special fibre necessary? If not then a counter example is X = Spec(R[x]/(x(x – p))) where p is a uniformizer in R. In words: X is a union of two copies of Spec(R) glued at their special points.
Wie het kleine niet eert is het grote niet weerd!