The Stacks project

Lemma 37.72.1. Let $f : X \to Y$ be a proper morphism of schemes. Let $y \in Y$ be a point with $\dim (X_ y) \leq 1$. If

  1. $R^1f_*\mathcal{O}_ X = 0$, or more generally

  2. there is a morphism $g : Y' \to Y$ such that $y$ is in the image of $g$ and such that $R'f'_*\mathcal{O}_{X'} = 0$ where $f' : X' \to Y'$ is the base change of $f$ by $g$.

Then $H^1(X_ y, \mathcal{O}_{X_ y}) = 0$.

Proof. To prove the lemma we may replace $Y$ by an open neighbourhood of $y$. Thus we may assume $Y$ is affine and that all fibres of $f$ have dimension $\leq 1$, see Morphisms, Lemma 29.28.4. In this case $R^1f_*\mathcal{O}_ X$ is a quasi-coherent $\mathcal{O}_ Y$-module of finite type and its formation commutes with arbitrary base change, see Limits, Lemmas 32.19.3 and 32.19.2. The lemma follows immediately. $\square$

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