R. Aravire, R. Baeza: The behavior of quadratic and differential forms under function field extensions in characteristic two


Submission: 2001, Nov 6

Let $F$ be a field of characteristic 2. Let $\Omega^n_F$ be the $F$-space of absolute differential forms over $F$. There is a homomorphism $\wp:\Omega^n_F \rightarrow \Omega^n_F/d\Omega^{n-1}_F$ given by \begin{equation*} {\ \wp }(x\frac{dx_{1}}{x_{1}}\wedge \cdots \wedge \frac{dx_{n}}{x_{n}} )=(x^{2}-x)\frac{dx_{1}}{x_{1}}\wedge \cdots \wedge \frac{dx_{n}}{x_{n}} \;\;\;\; \mbox{ mod }d\Omega _{F}^{n-1} \end{equation*} Let $H^{n+1}(F)=\mbox{ Coker}(\wp)$. We study the behavior of $H^{n+1}(F)$ under the function field $F(\phi)/F$, where $\phi=\ll b_1,\ldots,b_n\gg$ is a $n$-fold Pfister form and $F(\phi)$ is the function field of the quadric $\phi=0$ over $F$. We show that \begin{equation} \ker(H^{n+1} (F) \rightarrow H^{n+1} (F(\phi)) = \overline{F \cdot\frac{ db_{1}}{b_{1}} \wedge\cdots\wedge\frac{db_{n}}{b_{n}}}\nonumber \end{equation} Using Kato's isomorphism of $H^{n+1}(F)$ with the quotient $I^nW_q(F)/$ $I^{n+1}W_q(F)$, where $W_q(F)$ is the Witt group of quadratic forms over $F$ and $I\subset W(F)$ 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. $I^nW_q(F)$ is the set of all classes $\overline{q}$ with $\deg(q)\geq n$.

2000 Mathematics Subject Classification: 11E04, 11E81, 12E05, 12F20, 15A63, 15A75.

Keywords and Phrases: quadratic forms, differential forms, bilinear forms, Witt-groups, function fields, generic splitting fields of quadratic forms, degree of quadratic forms.

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