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The Stacks project

Lemma 53.20.13. Let f : X \to S be a morphism of schemes. Assume

  1. f is proper,

  2. f is at-worst-nodal of relative dimension 1, and

  3. the geometric fibres of f are connected.

Then (a) f_*\mathcal{O}_ X = \mathcal{O}_ S and this holds after any base change, (b) R^1f_*\mathcal{O}_ X is a finite locally free \mathcal{O}_ S-module whose formation commutes with any base change, and (c) R^ qf_*\mathcal{O}_ X = 0 for q \geq 2.

Proof. Part (a) follows from Derived Categories of Schemes, Lemma 36.32.6. By Derived Categories of Schemes, Lemma 36.32.5 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 Schemes, Lemma 36.30.4). 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 conclude. \square


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