Quadratic forms over rational function fields in characteristic 2
The paper provides a generalization of the authors' earlier result on the characteristic 2 version of the Milnor exact sequence for the Witt group of a rational function field (a localization exact sequence for a projective line) [cf. Contemp. Math. 344, 1--17 (2004; Zbl 1092.11022)]. The four-term exact sequence terms except one are the Witt groups of quadratic forms \(W_q F\). The sequence starts with a canonical monomorphism of the Witt group \(W_q F\) of a base field \(F\) into \(W_q F(x)\). The third term is a direct sum over points \(p\) of a projective line (over all discrete valuations of \(F(x)\) trivial on \(F\)) of natural targets of second residue maps, namely groups \(W_1 F(x)_p = \text{coker} (W_q F[x]/p \rightarrow W_q F(x)_p )\), where the map is induced by a Teichmüller lifting. Next the sum of traces is an epimorphism onto Witt group \(W_q F\) of the base field. The authors have built such an exact sequence assuming the base field \(F\) to be perfect. Now this assumption is released, so there is no canonical Teichmüller lifting. Thus a considerable amount of technique is involved to bootstrap all relevant maps. The authors are preparing a study of Witt groups of fields of functions on algebraic curves in characteristic 2 as an application of this result.
scientific article; zbMATH DE number 5190385