For proper maps, stalks of higher direct images are trivial in degrees larger than the dimension of the fibre.
Lemma 69.22.9. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Let \overline{y} be a geometric point of Y. Assume
Y locally Noetherian,
f is proper, and
\dim (X_{\overline{y}}) = d.
Then for any coherent sheaf \mathcal{F} on X we have (R^ pf_*\mathcal{F})_{\overline{y}} = 0 for all p > d.
Proof. Let \kappa (\overline{y}) be the residue field of the local ring of \mathcal{O}_{Y, \overline{y}}. As in Lemma 69.22.7 we set X_{\overline{y}} = X_1 = \mathop{\mathrm{Spec}}(\kappa (\overline{y})) \times _ Y X. Moreover, the underlying topological space of each infinitesimal neighbourhood X_ n is the same as that of X_{\overline{y}}. Hence H^ p(X_ n, \mathcal{F}_ n) = 0 for all p > d by Lemma 69.10.1. Hence we see that (R^ pf_*\mathcal{F})_{\overline{y}}^\wedge = 0 by Lemma 69.22.7 for p > d. Note that R^ pf_*\mathcal{F} is coherent by Lemma 69.20.2 and hence R^ pf_*\mathcal{F}_{\overline{y}} is a finite \mathcal{O}_{Y, \overline{y}}-module. By Algebra, Lemma 10.97.1 this implies that (R^ pf_*\mathcal{F})_{\overline{y}} = 0. \square
Comments (1)
Comment #1280 by Johan Commelin on