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 field of characteristic zero. If (V, φ)⊗R K is isotropic over K, then (V, φ) is isotropic over R, that is there exists a unimordular vector v ∈ V with φ(v) = 0. This theorem is trivial for all discrete valuation rings. The case of any two dimensional regular local ring is proved in [O]. To prove the Theorem we need certain auxiliary preliminaries. 1.0.2 Lemma. Let U = Spec(R) and let X p −→ U be a smooth projective morphism. Let u ∈ U be the closed point of U and let X = p(u) be the closed fibre of p. Let f : Y → X be a projective morphism of an essentially k-smooth scheme Y with dim(Y ) = dim(U). Suppose that f is transversal to the closed imbedding i : X →֒ X. Then the morphism q = p ◦ f : Y → U is finite etale. In fact, since f is transversal to i the scheme f(X) is a k-smooth scheme of dimension zero. Since f(X) = q(u) the morphism q is a quasi-finite. Since q is projective it is finite and surjective. The schemes Y and U are both regular. Therefore q is flat. Since the scheme q(u) is a k-smooth scheme of dimension zero it is etale over Spec(k) and hence it is etale over the point u. Since q(u) is the closed fibre of the flat morphism q the morphism q is etale. The next lemma is a variant of the lemma [LM, Lemma 7.1]. 1.0.3 Lemma. Suppose that k is an infinite field admitting a resolution of singularities. Let W be a k-smooth scheme and let i : Z →֒ W be a smooth closed subscheme. Then CH∗(W ) is generated by the elements of the form f∗(1) where f : Y → W is a projective morphism transverse to i and f∗ : CH∗(Y ) → CH∗(W ) is the push-forward.