Embeddings and intersections of adelic groups

TL;DR

This paper investigates the structure and intersections of adelic groups on algebraic schemes, establishing new equalities and cohomological properties for various types of schemes and sheaves.

Contribution

It proves new intersection equalities for adelic groups on schemes of special type and computes cohomology of adelic complexes, advancing understanding of adelic structures.

Findings

Proved intersection equality for adelic groups on normal schemes of special type.

Established limit of global sections equals global sections of the sheaf on Cohen-Macaulay schemes.

Computed cohomology groups of adelic complexes on three-dimensional varieties.

Abstract

We prove embeddings of adelic groups on an excellent scheme of special type and a flat quasicoherent sheaf on it. For a normal excellent scheme of special type we establish the equality $\mathbb{A}_I(X,\mathcal{F})\cap\mathbb{A}_J(X,\mathcal{F})=\mathbb{A}_{I\setminus0}(X,\mathcal{F})$ in the case $I\cap J=I\setminus0$. 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 equals the group of global sections of this sheaf. Using this result, we deduce a theorem on intersections of adelic groups for normal projective surfaces. We also compute cohomology groups of a curtailed adelic complex and, as a consequence, show that on a three-dimensional regular projective variety over a countable field the intersection $\mathbb{A}_I(X,\mathcal{F})\cap\mathbb{A}_J(X,\mathcal{F})$ equals $\mathbb{A}_{I\cap J}(X,\mathcal{F})$ for any $I,J\subset\{0,1,2,3\}$ and any locally free sheaf $\mathcal{F}$ on $X$.