Holmstrom Obstructions

arXiv:1002.1423 The Étale Homotopy Type and Obstructions to the Local-Global Principle from arXiv Front: math.AT by Yonatan Harpaz, Tomer M. Schlank In 1969 Artin and Mazur defined the étale homotopy type of an algebraic variety [AMa69]. In this paper we define various obstructions to the local-global principle on a variety $X$ over a global field using the étale homotopy type of $X$ and the concept of homotopy fixed points. We investigate relations between those “homotopy obstructions” and connect them to various known obstructions such as the Brauer -Manin obstruction, the étale-Brauer obstruction and finite descent obstructions. This gives a reinterpretation of known arithmetic obstructions in terms of homotopy theory.

nLab page on Obstructions?