On commutative diagrams consisting of low term exact sequences

TL;DR

This paper constructs and analyzes commutative diagrams of low term exact sequences related to Grothendieck spectral sequences, extending previous work and aiding in local-global principles for rational points.

Contribution

It introduces new commutative diagrams of low term exact sequences linked to Grothendieck spectral sequences, enhancing tools for studying rational points and obstructions.

Findings

Extended existing diagrams to include more cases

Provided new tools for local-global principles

Connected diagrams to previous literature and obstructions

Abstract

We establish several useful commutative diagrams consisting of low term exact sequences attached to {\Grot} spectral sequences, which extends and integrates the previous ones appeared in literature such as Alexei~N. Skorobogatov [Beyond the {M}anin obstruction, Invent. Math. (1999)], and [On the elementary obstruction to the existence of rational points, Mathematical Notes (2007)]. Parts of the diagrams was frequently used in local-global principle to rational points.