The Stacks project

Lemma 109.11.2. We have $\mathcal{C}\! \mathit{urves}^{grc, 1} \subset \mathcal{C}\! \mathit{urves}^{h0, 1}$ as open substacks of $\mathcal{C}\! \mathit{urves}$. In particular, given a family of curves $f : X \to S$ whose geometric fibres are reduced, connected and of dimension $1$, then $R^1f_*\mathcal{O}_ X$ is a finite locally free $\mathcal{O}_ S$-module whose formation commutes with arbitrary base change.

Proof. This follows from Varieties, Lemma 33.9.3 and Lemmas 109.9.1 and 109.11.1. The final statement follows from Lemma 109.9.3. $\square$

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