Higher Direct Images of the Structure Sheaf Over a Dedekind Domain

TL;DR

This paper proves the invariance of higher direct images of the structure sheaf and differentials under proper birational maps over Dedekind domains, extending previous results to a broader algebraic setting.

Contribution

It extends the known invariance results of higher direct images to schemes over Dedekind domains using algebraic correspondences, generalizing prior work over perfect fields.

Findings

Higher direct images of the structure sheaf are invariant under proper birational maps.

Vanishing of higher direct images for proper birational morphisms over Dedekind domains.

Equality of certain cohomology groups for different models over number fields.

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 $t \in U$.