Rationally trivial hermitian spaces are locally trivial

Rationally trivial quadratic spaces are locally trivial:III

It is proved the following. Let R be a regular semi-local domain containing a field such that all the residue fields are infinite. Let K be the fraction field of R. If a quadratic space (R, q : R → R) over R is isotropic over K, then there is a unimodular vector v ∈ R such that q(v) = 0. If char(R) = 2, then in the case of even n we assume that q is a non-singular space in the sense of [Kn] and...

Locally trivial W∗-bundles

The First Betti Numbers of Certain Locally Trivial Fibre Spaces

The purpose of this department is to provide early announcement of significant new results, with some indications of proof. Although ordinarily a research announcement should be a brief summary of a paper to be published in full elsewhere, papers giving complete proofs of results of exceptional interest are also solicited. Manuscripts more than eight typewritten double spaced pages long will no...

Rationally isotropic quadratic spaces are locally isotropic

Let R be a regular local ring, K its field of fractions and (V, φ) a quadratic space over R. In the case of R containing a field of characteristic zero we show that if (V, φ)⊗R K is isotropic over K, then (V, φ) is isotropic over R. 1 Characteristic zero case 1.0.1 Theorem (Main). Let R be a regular local ring, K its field of fractions and (V, φ) a quadratic space over R. Suppose R contains a f...

Locally Trivial Torsors That Are Not Weil-châtelet Divisible

For every prime p we give infinitely many examples of torsors under abelian varieties over Q that are locally trivial but not divisible by p in the Weil-Châtelet group. We also give an example of a locally trivial torsor under an elliptic curve over Q which is not divisible by 4 in the Weil-Châtelet group. This gives a negative answer to a question of Cassels.