Higher Direct Images of the Structure Sheaf Over a Dedekind Domain

Abstract

We prove that for Noetherian, smooth, separated, integral, finite type schemes $X$ and $Y$ over an excellent Dedekind domain $R$, that are properly birational over $R$, we have $R^if_{*}\mathcal{O}_X \cong R^ig_{*} \mathcal{O}_Y$ and $R^i f_{*}\Omega_{X/S}^{d} \cong R^ig_{*} \Omega_{Y/S}^d$, where $d$ is the relative dimension of $X$ and $Y$ over $S= Spec(R)$, and $f$ and $g$ are the structure maps of $X$ and $Y$, respectively, as $S$-schemes. As a corollary we obtain the vanishing of higher direct images of the structure sheaf for proper birational morphisms beteween such schemes. These results extend those obtained by Chatzistamatiou--R\"ulling over perfect fields of positive characteristic and we obtain them by extending their method of algebraic correspondences. We furthermore obtain as a corollary that if $K$ is a number field and $\mathcal{O}_K$ its ring of integers and if $X$ is a smooth and proper $K$-scheme with $\mathcal{X}$ and $\mathcal{Y}$ two smooth proper models of $X$ over some dense open subscheme $U \subseteq S = Spec(\mathcal{O}_K)$, that if $H^j(\mathcal{X},\mathcal{O}_{\mathcal{X}})$ is $\mathcal{O}_S(U)$-torsion-free we have $H^j(\mathcal{X}_t,\mathcal{O}_{\mathcal{X}_t}) = H^j(\mathcal{Y}_t,\mathcal{O}_{\mathcal{Y}_t}),$ for all closed points $t \in U$.