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

AbstractLet 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/x1∧⋯∧dxn/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)))=F·db1/b1∧⋯∧dbn/bn. 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 q with deg(q)⩾n

Open Problems on Central Simple Algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered, compared to v

A note on generic splitting of quadratic forms

Let F be a field of any characteristic. For n ≥ 0, let J(n) = {q̄ ∈ Wq(F)| deg(q) ≥ n}. The degree conjecture asserts that for each n ≥ 0 (DC) J(n) = InWq(F) Let p be any n-fold quadratic Pfister form over F and F(p) the function field of p. Then the function field conjecture asserts (FFC) ker [InWq(F)/In+1Wq(F) → InWq(F(p))/In+1Wq(F(p))] = {0, p̄} We prove that (DC) is equivalent to (FFC)

On the Witt-equivalence of fields of characteristic 2

Let F be a field with 2 = 0 and ϕ = ≪ a1,..., an ≫ an n-fold anisotropic bilinear Pfister form over F with function field F (ϕ). In this paper we compute ker[I n F /In+1 F → In F (ϕ) /In+1 F (ϕ) ] where IF ⊂ W (F) is the maximal ideal in the Witt ring W (F) of F. We use this computation to prove a n-linkage property of the subfields F 2 (a1,..., an). In this paper F will denote throughout a field with 2 = 0. Let Ω ∗ F = � ∞ n=0 ΩnF be the F-algebra of differential forms over F and let d: Ωn F → Ωn+

Quadratic and differential forms over function fields of Pfister quadrics in characteristic two

Let F be a field of characteristic 2. Let Ω n F be the F-space of differ-ential forms over F. There is a homomorphism ℘ : Ω n F − → Ω n F/dΩ n−1 ş ť F d x1 d xn given by ℘ x ∧ · · · ∧ = (x x1 xn 2 d x1 d xn −x) ∧ · · · ∧ mod dΩn−1 x1 xn F. Let H n+1 (F) = coker(℘). If p = ≪ a1,..., an; b]] is an anisotropic quadratic Pfister form over F and F(p) the function field of the Pfister quadric {p = 0}, we compute the kernel H n+1 (F(p)/F) = ker č H n+1 (F) → H n+1 (F(p)) ď for all m. Using Kato’s correspondence between differential and quadratic forms we compute the kernels I m Wq(F(p)/F) = ker[I m Wq(F) → I m Wq(F(p))], where Wq(F) denotes the Witt group of quadratic forms over F and IF is the maximal ideal of the Witt ring W(F) of symmetric bilinear forms over F