The Witt group of a discretely valued field
Let \(K\) be a field and \(v\) a valuation on \(K\) to \(\mathbb{Z}\). Let \(\mathfrak{o}\) denote the valuation ring and \(k\) the residue class field. The relationship between the Witt group of nonsingular quadratic forms on \(K\), \(W_q(K)\), and on \(k\), \(W_q(k)\) is well-known in the case where \(k\) is of characteristic different from \(2\). We have that \(W_q(K)\) is isomorphic to \(W_q(k)\oplus W_q(k)\), and further that any quadratic form over \(K\) can be decomposed into a sum of quadratic forms \(q\) and \(\pi\cdot q'\), where \(q\) and \(q'\) have diagonalisations whose entries are units and \(\pi\) is a uniformiser with respect to the valuation. If the characteristic of \(k\) is \(2\), this relationship is more complex, in part due to quadratic forms not being diagonalisable in characteristic \(2\). For example, for \(K=F((t))\), the field of Laurent series over \(F\) 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. Let \(W_q(K)_{tame}\) be the subgroup \(W(K)W(\mathfrak{o})\) of \(W_q(K)\), where \(W(K)\) is the Witt ring of symmetric bilinear forms over \(K\). Then \(W_q(K)/W_q(K)_{tame}\) is the obstruction to a Springer-like result holding for \(K\). Indeed, this group is trivial if the characteristic of \(k\) is not \(2\). In this paper it is shown that there is a natural filtration \[ W_q(K)_0\subseteq W_q(K)_1\subseteq W_q(K)_2 \subseteq\cdots \] where \(W_q(K)_0= W_q(K)_{\text{tame}}\). This filtration is finite in the mixed characteristic case of length \(2e\), where \(e=v(2)\), and \(W_q(K)_{2e}=W_q(K)\). Conversely, the filtration is infinite when the characteristic of \(K\) is also \(2\). It is then shown how to express the quotients \(W_q(K)_n/W_q(K)_{n-1}\) in terms of \(k\). Note that this result was extended to more general valued fields in [\textit{J. Verstraete}, J. Algebra 519, 190--227 (2019; Zbl 1426.11038)].
scientific article; zbMATH DE number 6921613