The behavior of quadratic and differential forms under function field extensions in characteristic two

Abstract

Let F be a field of characteristic 2. Let ΩFn be the F-space of absolute differential forms over F. There is a homomorphism ℘: ΩFn → ΩFn/dΩFn-1 given by ℘ (x dx1/x1 ∧ ⋯ ∧ dxn/xn) = (x2-x)dx1/x1 ∧ ⋯ ∧ dxn/xn mod dΩFn-1. Let Hn+1 (F) = Coker (℘). We study the behavior of Hn+1 (F) under the function field F(Φ)/F, where Φ = «b1,..., bn» is an n-fold Pfister form and F(Φ) is the function field of the quadric Φ = 0 over F. We show that ker(Hn+1(F) → Hn+1 (F(Φ))) = F · db1/b1 ∧ ⋯ ∧ dbn/bn. Using Kato's isomorphism of Hn+1 (F) with the quotient InW(inf)q((/inf)F)/In+1W(inf)q(/inf)(F), where W(inf)q(/inf)(F) is the Witt group of quadratic forms over F and I ⊂ W(F) is the maximal ideal of even-dimensional bilinear forms over F, we deduce from the above result the analogue in characteristic 2 of Knebusch's degree conjecture, i.e. InW(inf)q(/inf) (F) is the set of all classes q̄ with deg (q) ≥ n. © 2003 Elsevier Science (USA). All rights reserved.