Graded Witt kernels of the compositum of multiquadratic extensions with the function fields of Pfister forms

Abstract

Let F be a field of characteristic 2 and W-q(F) be the Witt group of nonsingular quadratic forms over F. Let phi be a bilinear Pfister form over F and L be a multiquadratic extension of F of separability degree less than of equal to 2. In this paper we compute the kernel of the natural homomorphism H-2(m+1)(F) -> H-2(m+1) (L(phi)), where H-2(m+1)(F) is the cokernel of the Artin-Schreier operator p : Omega(m)(F) -> Omega(m)(F)/d Omega(m-1)(F) given by xdx1/x1 <^>...<^> dxm/xm -> (x2 - x)dx1/x1 <^>...<^> dxm/xm, where Omega(m)(F) is the space of m-differential forms over F, and F(phi) is the function field of the affine quadric given by the diagonal quadratic form associated to the bilinear form phi. As a consequence, we deduce the kernel of the natural homomorphisms <(I-q(m+1))over bar> (F) <(I-q(m+1))over bar> L(phi)) and I-q(m+1)(F) -> I-q(m+1)(L(phi)), where I-q(m+1)(F) denotes the quotient I-q(m+1)(F)/I-q(m+2)(F) such that I-q(m+1)(F) = (IF)-F-m circle times W-q(F) and (IF)-F-m is the m-th power of the fundamental ideal IF of the Witt ring of F-bilinear forms. We also include some results concerning the case where phi is replaced by a bilinear Pfister neighbor or a quadratic Pfister form. (C) 2015 Elsevier Inc. All rights reserved. Keywords. Author Keywords:Quadratic (bilinear) forms; Differential forms; Graded-Witt kernels; Function field of a quadric; Pfister forms; Pfister neighbors