Year: 2020
Type: article
Abstract: In this paper, we study local-global principles for tori over semi-global fields, which are one-variable function fields over complete discretely valued fields.In particular, we show that for principal homogeneous spaces for tori over the underlying discrete valuation ring, the obstruction to a local-global principle with respect to discrete valuations can be computed using methods coming from patching.We give a sufficient condition for the vanishing of the obstruction, as well as examples where the obstruction is nontrivial or even infinite.A major tool is the notion of a flasque resolution of a torus.
Source: Algebraic geometry
DOI: 10.14231/ag-2020-022
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Funder: National Science Foundation
Awards: FRG: Collaborative Research: Model Theory of Differential and Difference Equations with Applications, FRG: Collaborative Research: Lifting Problems and Galois Theory, Algebraic Structures and the Arithmetic of Fields, RTG: Algebra, Algebraic Geometry, and Number Theory, Arithmetic of Homogeneous Spaces under Linear Algebraic Groups, FRG: Obstructions to Local-Global Principles and Applications to Algebraic Structures, Galois Cohomology of function fields of curves, FRG: Obstructions to Local-Global Principles and Applications to Algebraic Structures, FRG: Collab: Obstructions to Local-Global Principles and Applications to Algebraic Structures, Rational points on homogeneous spaces, quadractic forms and Brauer groups
OpenAlex ID: W2952209592