Lemma 53.20.13 (0GKA)—The Stacks project

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

$f$ is proper,
$f$ is at-worst-nodal of relative dimension $1$ , and
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. $\square$

The Stacks project

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