Embeddings and intersections of adelic groups
Dmitry Badulin
TL;DR
The paper advances higher-dimensional adelic theory by establishing embeddings and precise intersection formulas for Parshin–Beilinson adeles on a broad class of schemes. It first proves local-to-global embedding results for adelic groups on thickenings, then derives a fundamental intersection identity on normal excellent schemes with flat sheaves, answering Parshin’s question in broad settings. By passing to limits over power and symbolic thickenings, it connects adelic data to global cohomology through curtailed adelic complexes, yielding cohomology and intersection equalities on normal projective surfaces and on regular threefolds. The results generalize known surface cases, provide new tools for computing cohomology via adelic limits, and solidify the higher-dimensional Krichever–Parshin framework with concrete intersection theorems and cohomological applications.
Abstract
We prove the embeddings of adelic groups $\mathbb{A}_I(X, \mathcal{F})\hookrightarrow \mathbb{A}_J(X, \mathcal{F})$ on an excellent scheme $X$ of special type, where $I\subset J$ and $\mathcal{F}$ is a flat quasicoherent sheaf on $X$. We prove the equality $\mathbb{A}_I(X, \mathcal{F})\cap \mathbb{A}_J(X, \mathcal{F}) = \mathbb{A}_{I\setminus 0}(X, \mathcal{F})$ for a normal excellent scheme of special type $X$ and a flat quasicoherent sheaf $\mathcal{F}$ on it, where $I\cap J = I\setminus 0$. We show that the limit of restrictions of global sections of a locally free sheaf on a Cohen-Macaulay projective scheme to power thickenings of integral subschemes is equal to the group of global sections of this sheaf. Using this result, we prove the equality $\mathbb{A}_{\dim X - 1}(X, \mathcal{F})\cap\mathbb{A}_{\dim X}(X, \mathcal{F}) = H^0(X, \mathcal{F})$ for a Cohen-Macaulay projective variety $X$ and a locally free sheaf $\mathcal{F}$ on it. Hence we deduce a theorem on intersections of adelic groups for the case of a normal projective surface. We also compute cohomology groups for a particular case of a curtailed adelic complex. Hence we show that on a three-dimensional regular projective variety over a countable field $X$ there is an equality $\mathbb{A}_I(X, \mathcal{F})\cap \mathbb{A}_J(X, \mathcal{F}) = \mathbb{A}_{I\cap J}(X, \mathcal{F})$ for any $I, J\subset \{0, 1, 2, 3\}$ and any locally free sheaf $\mathcal{F}$ on $X$.