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

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

Linkage of fields in characteristic

Baeza R. Instituto de Matemática y Física,Universidad de Talca,Casilla 747,Talca,Chile

Annihilators of quadratic and bilinear forms over fields of characteristic two

AbstractLet F be a field with 2=0, W(F) the Witt ring of symmetric bilinear forms over F and Wq(F) the W(F)-module of quadratic forms over F. Let IF⊂W(F) be the maximal ideal. We compute explicitly in IFm and ImWq(F) the annihilators of n-fold bilinear and quadratic Pfister forms, thereby answering positively, in the case 2=0, certain conjectures stated by Krüskemper in [M. Krüskemper, On annihilators in graded Witt rings and in Milnor's K-theory, in: B. Jacob et al. (Eds.), Recent Advances in Real Algebraic Geometry and Quadratic Forms, in: Contemp. Math., vol. 155, 1994, pp. 307–320]

characteristic

The behavior of quadratic and differential forms under function field extensions i

On some invariants of fields of characteristic p>0

Baeza, R. Instituto de Matemáticas, Universidad de Talca, Casilla 721, Talca, Chil