Lemma 109.9.3. 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 109.9.1). Then
f_*\mathcal{O}_ X = \mathcal{O}_ S and this holds universally,
R^1f_*\mathcal{O}_ X is a finite locally free \mathcal{O}_ S-module,
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 75.26.7. 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 75.25.4). 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 15.75.7. Since everything is compatible with base change we also see that (3) holds.
\square
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