Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local-global principle for over semiglobal fields

Philippe Gille, Raman Parimala

Abstract

We compare different local-global principles for torsors under a reductive group G defined over a semiglobal field F. In particular if the F-group G s a retract rational F-variety, we prove that the local global principle holds for the completions with respect to divisorial valuations of F.

Local-global principle for over semiglobal fields

Abstract

We compare different local-global principles for torsors under a reductive group G defined over a semiglobal field F. In particular if the F-group G s a retract rational F-variety, we prove that the local global principle holds for the completions with respect to divisorial valuations of F.

Paper Structure

This paper contains 16 sections, 23 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Pairs of a group scheme together with a torsor
  3. Refinements on the reduction of torsors to finite subgroups
  4. Patching techniques
  5. Setting
  6. Torsors and models
  7. Other Tate-Schafarevich sets
  8. Various sets of valuations
  9. The constant case.
  10. Variant with $0$-cycles of degree one
  11. Serre's injectivity question
  12. A local variant
  13. The $0$-cycle variant
  14. Applications
  15. Torsion primes
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $G$ be an $S$--sheaf in groups and let $E$ be a $G$--torsor over $S$. We denote by $^EG$ the twisted $S$--sheaf of $G$ by $E$ by inner automorphisms. (1) We have a natural exact sequence of fppf $S$--sheaves in groups which is locally split for the flat topology. (2) If $E=G$ the sequence above splits and the $S$--functor $\mathop{\mathrm{\underline{Aut}}}\nolimits_{\fam\eufmfam P}\bigl( G,G\

Theorems & Definitions (44)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • proof
  • Theorem 3.4
  • proof
  • ...and 34 more