Quadratic modules over polynomial rings

Abstract

This chapter reviews quadratic modules over polynomial rings. Let A = F[X1… |Xd] be a polynomial algebra in d variables over a field F of characteristic ≠ 2. Let S = S(X1…Xd) be a symmetric r by r matrix over A whose determinant is a nonzero constant in F. The question may then be raised: is S equivalent to a matrix with coefficients in F; more precisely, is there a U in GLr(A) such that UStU = S0, where the t denotes transpose, and S0 = S(0, …, 0) ? A theorem of Harder implies that the answer is affirmative for d = 1. A theorem of Karoubi implies that the answer is stably affirmative. However, Parimala has recently given a remarkable counterexample showing that the answer is negative for F = ℝ, d = 2, and r = 4. She raised the question of whether the answer is affirmative for F = C. It is proved that the answer is affirmative whenever F is algebraically closed and d≤ 3. The proof uses Karoubi's theorem but not Harder's, thus giving a new proof of Harder's theorem in this special case. The proof also uses a general Witt Cancellation Theorem. The effect of the Cancellation Theorem plus Karoubi's theorem is to reduce the problem to one where r is small.