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Local-global principles for tori over arithmetic curves. (English) Zbl 07262980
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
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