The Stacks project

Lemma 107.9.3. Let $f : X \to S$ be a family of curves such that $\kappa (s) = H^0(X_ s, \mathcal{O}_{X_ s})$ for all $s \in S$, i.e., the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{h0, 1}$ (Lemma 107.9.1). Then

  1. $f_*\mathcal{O}_ X = \mathcal{O}_ S$ and this holds universally,

  2. $R^1f_*\mathcal{O}_ X$ is a finite locally free $\mathcal{O}_ S$-module,

  3. for any morphism $h : S' \to S$ if $f' : X' \to S'$ is the base change, then $h^*(R^1f_*\mathcal{O}_ X) = R^1f'_*\mathcal{O}_{X'}$.

Proof. We apply Derived Categories of Spaces, Lemma 73.26.7. This proves part (1). It also implies that locally on $S$ we can write $Rf_*\mathcal{O}_ X = \mathcal{O}_ S \oplus P$ where $P$ is perfect of tor amplitude in $[1, \infty )$. Recall that formation of $Rf_*\mathcal{O}_ X$ commutes with arbitrary base change (Derived Categories of Spaces, Lemma 73.25.4). Thus for $s \in S$ we have

\[ H^ i(P \otimes _{\mathcal{O}_ S}^\mathbf {L} \kappa (s)) = H^ i(X_ s, \mathcal{O}_{X_ s}) \text{ for }i \geq 1 \]

This is zero unless $i = 1$ since $X_ s$ is a $1$-dimensional Noetherian scheme, see Cohomology, Proposition 20.20.7. Then $P = H^1(P)[-1]$ and $H^1(P)$ is finite locally free for example by More on Algebra, Lemma 15.74.6. Since everything is compatible with base change we also see that (3) holds. $\square$

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