Higher Direct Images of the Structure Sheaf Over a Dedekind Domain

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 $R^i{f}_{\ast }{\mathcal{O}}_X\cong R^i{g}_{\ast }{\mathcal{O}}_Y$ and $R^i{f}_{\ast }{\mathrm{\Omega }}_{X\mathrm{/}S}^d\cong R^i{g}_{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}S}^d$, 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–Rülling 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 $t\in U$.