Arason's filtration of the Witt group of dyadic valued fields (Q1628502)

Let $F$ be a field with a valuation $v:\Gamma: \rightarrow \{\infty\}$, where $\Gamma$ is a totally ordered abelian group (which may be assumed to be divisable). Let $\Gamma_F$ be the value group of $(F,v)$ and let $\overline{F}$ be the residue field. \par The well-known result of Springer states that if $(F,v)$ is complete and discretely valued, and the characteristic of $\overline{F}$ is not $2$, the Witt group of quadratic forms over $F$, $W_q(F)$, is isomorphic to $W_q(\overline{F})\oplus W_q(\overline{F})$. In particular, any quadratic form over such a field $F$ can be decomposed into a sum of quadratic forms $q$ and $\pi\cdot q'$, where $q$ and $q'$ have diagaonlisations whose entries are units and $\pi$ is a uniformiser with respect to the valuations. \par In general, this result fails when the characteristic of $\overline{F}$ is $ 2$, in part due to quadratic forms not being diagonalisable in characteristic $2$. For example, for $F=k((t))$, the field of Laurent series over $k$ a field of characteristic $2$, it is relatively easy to show that the quadratic form given by $(x,y)\mapsto x^2 +xy + t^{-1}y^2$ does not have a Springer-like decomposition. \par The obstruction to a Springer-like result for the case of residue characteristic $2$ has been investigated partially before by various authors. This paper uses more intrinsic methods to give a more complete and consistent explanation of the difficulties of this case. The main result of the paper is as follows. \par It is well known that there that there exists a subgroup $W_q(F)_{\mathrm{tame}}$ of $W_q(F)$ and an epimorphism \[ W_q(F)_{\mathrm{tame}} \rightarrow \bigoplus_{\Gamma_F/\Gamma_F} W_q(F), \] which is an isomorphism when $F$ is Henselian (Springer's theorem). If the characteristic of $\overline{F}$ is not $2$, then $W_q(F)_{\mathrm{tame}}=W_q(F)$. For the case where the characteristic of $\overline{F}$ is $2$, the author shows the existence of and precisely describes a filtration \[ (W_q(F)_\varepsilon)_{\varepsilon\in F}, \quad E= \left\{ \varepsilon \in \frac{1}{2}\Gamma_F \mid 0 \leq \varepsilon \leq v(2) \right\} \] of $W_q(F)$ by subgroups $W_q(F)_\varepsilon\subset W_q(F)$ satisfying \[ W_q(F)_0 = W_q(F)_{\mathrm{tame}} \quad \text{and} \quad \bigcup_{\varepsilon\in E} W_q(F)_\varepsilon =W_q(F). \] The introduction also gives several illustrative examples showing this filtration in use and the various types of situation that can occur.