The behavior of quadratic and differential forms under function field extensions in characteristic two
- 作者：Aravire ; R. ; Baeza ; R.
- 关键词：Quadratic forms ; Differential forms ; Bilinear forms ; Witt-groups ; Function fields ; Generic splitting fields of quadratic forms ; Degree of quadratic forms
- 刊名：Journal of Algebra
- 出版年：2003
- 期刊代码：69_00218693
- 类别：mt
- 出版时间：January 15, 2003
- 卷：259
- 期：2
- 页码：361-414
- 文件大小：412 K
摘要
Let F be a field of characteristic 2. Let ΩnF be the F-space of absolute differential forms over F. There is a homomorphism 
:ΩnF→ΩnF/dΩn−1F given by (xdx1/x1
dxn/xn)=(x2−x)dx1/x1dxn/xnmoddΩFn−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)))=. Using Kato's isomorphism of Hn+1(F) with the quotient InWq(F)/In+1Wq(F), where Wq(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. InWq(F) is the set of all classes with deg(q)≥n.
:ΩnF→ΩnF/dΩn−1F given by (xdx1/x1dxn/xn)=(x2−x)dx1/x1dxn/xnmoddΩFn−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)))=. Using Kato's isomorphism of Hn+1(F) with the quotient InWq(F)/In+1Wq(F), where Wq(F) is the Witt group of quadratic forms over F and IW(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. InWq(F) is the set of all classes with deg(q)≥n.