The Stacks project

Lemma 36.32.6. Let $f : X \to S$ be a morphism of schemes. Assume

  1. $f$ is proper, flat, and of finite presentation, and

  2. the geometric fibres of $f$ are reduced and connected.

Then $f_*\mathcal{O}_ X = \mathcal{O}_ S$ and this holds after any base change.

Proof. By Lemma 36.32.5 it suffices to show that $\kappa (s) = H^0(X_ s, \mathcal{O}_{X_ s})$ for all $s \in S$. This follows from Varieties, Lemma 33.9.3 and the fact that $X_ s$ is geometrically connected and geometrically reduced. $\square$

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