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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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