A Variant of a Theorem by Springer

The theorem in question gives a sufficient condition for a quadratic space over a local ring R to contain a hyperbolic plane over R. Our main aim in this paper is to prove the following result, which is a variant of the Springer theorem [La] for quadratic spaces over local rings. Theorem. Let R be a local Noetherian domain that has an infinite residue field of characteristic different from 2. Let S = R[T ]/(F (T )) be an integral étale extension over R. Let (V, q) be a quadratic space over R such that the S-quadratic space (V ⊗R S, q ⊗R S) contains a hyperbolic plane HS. If the degree of the polynomial F (T ) is odd, then the space (V, q) already contains a hyperbolic plane over R. This theorem is a main ingredient in the proof of the following result established in [P]. 0.1. Theorem. 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)⊗RK is isotropic over K, then (V, φ) is isotropic over R; i.e., there exists a unimodular vector v ∈ V with φ(v) = 0. It is well known that any finite étale extension S of R has the form S = R[T ]/(F (T )), where F (T ) is a monic separable polynomial. If A is a semilocal ring and (W,φ) is a quadratic space over A, then W contains a hyperbolic plane if and only if W contains a unimodular isotropic vector w. A vector w is said to be unimodular if w can be taken as the first vector w1 of a free A-base w1, . . . , wn of the A-module W . §1. Preliminaries In this section we formulate two results to be used in the proof of the main theorem. These two results will be proved in §3. We need to fix some notation: k is an infinite field (char(k) = 2); f(t) is a monic separable polynomial of degree n over k; l = k[t]/(f(t)) is a separable k-algebra; θ = t mod f(t) is an element of l; (W,φ) is a quadratic space over k of rank ≥ 3; (Wl, φl) is the quadratic space (W ⊗k l, φ⊗k l) over l; W [t] = W · 1⊕W · t⊕ · · · ⊕W · tm−1 ⊂ W [t] = W ⊗k k[t]; k[t] = k · 1⊕ k · t⊕ · · · ⊕ k · tm−1 ⊂ k[t]; 2000 Mathematics Subject Classification. Primary 53A04; Secondary 52A40, 52A10.