Let $f: X \to S$ be a proper flat morphism of schemes. Suppose that the geometric fibers of $f$ are connected and reduced. Does the condition $f_\ast(\mathcal O_X) = \mathcal O_S$ necessarily hold? (When $S = \mathop{\mathrm{Spec}}(k)$ is the spectrum of some field $k$, this statement is true by 0BUG. Conversely, if $f_\ast(\mathcal O_X) = \mathcal O_S$, then Stein factorization tells us the geometric fibers of $f$ are connected.)

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    $\begingroup$ Yes, this follows from EGA III, Cor. 7.8.8. $\endgroup$
    – abx
    Nov 9, 2020 at 4:57


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