On real local-global principles
Let \(F\) be a formally real field. For every ordering \(P\) of \(F\) let \(F_ P\) denote a real closure of \(F\) at \(P\). Let \(W_ T\) be the torsion ideal and \(I\) the augmentation ideal of the Witt ring of \(F\). Let \(k_{n_ T} F\) be the subgroup of the Milnor functor \(k_ n F\) generated by all \(\ell (s)\ell (a_ 2)...\ell (a_ n)\) where \(s\) is a sum of squares. \textit{M. Marshall} [Can. J. Math. 29, 606--614 (1977; Zbl 0332.12101)] stated the following two conjectures: (1) The kernel of \(k_ n F\to \prod k_ n F_ P \) is \(k_{n_ T} F\), (2) \(I^ n\cap W_ T=I^{n-1} W_ T\). The main results of this paper are: If the stability index of \(F\) is less than or equal to \(k\) then (1) and (2) hold for \(n\geq k\). If (1) holds for all finitely generated extensions of \(\mathbb Q\) then (1) and (2) hold in general.
scientific article; zbMATH DE number 4132291