Higher Direct Images of the Structure Sheaf Over a Dedekind Domain
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912906080157696 |
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| author | Amazeen, Grétar |
| author_facet | Amazeen, Grétar |
| contents | 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 $R^i f_{*}Ω_{X/S}^{d} \cong R^ig_{*} Ω_{Y/S}^d$, 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--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 $t \in U$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13881 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Higher Direct Images of the Structure Sheaf Over a Dedekind Domain Amazeen, Grétar Algebraic Geometry 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 $R^i f_{*}Ω_{X/S}^{d} \cong R^ig_{*} Ω_{Y/S}^d$, 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--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 $t \in U$. |
| title | Higher Direct Images of the Structure Sheaf Over a Dedekind Domain |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2602.13881 |