Colliot-Thélène, Jean-Louis; Harbater, David; Hartmann, Julia; Krashen, Daniel; Parimala, Raman; Suresh, Venapally Local-global principles for tori over arithmetic curves. (English) Zbl 07262980 Algebr. Geom. 7, No. 5, 607-633 (2020). Summary: 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. MSC: 11E72 Galois cohomology of linear algebraic groups 12G05 Galois cohomology 14G05 Rational points 14H25 Arithmetic ground fields for curves 20G15 Linear algebraic groups over arbitrary fields 14G27 Other nonalgebraically closed ground fields in algebraic geometry Keywords:linear algebraic groups; torsors; tori; local-global principles; Galois cohomology; semi-global fields; patching; flasque resolutions PDF BibTeX XML Cite \textit{J.-L. Colliot-Thélène} et al., Algebr. Geom. 7, No. 5, 607--633 (2020; Zbl 07262980) Full Text: DOI