On the powers of the fundamental ideal of a Witt ring
Let \(F\) be a field of characteristic \(\neq 2\). A recent result of \textit{J. K. Arason} and \textit{R. Elman} [``Powers of the fundamental ideal in the Witt ring``, J. Algebra 239, 150--160 (2001; Zbl 0990.11021)] describes the powers \(I^n(F)\) of the fundamental ideal \(I(F)\) of the Witt ring \(W(F)\) of \(F\) by generators and relations. The author gives several reformulations of this result. One reformulation is in terms of the Witt \(K\)-theory \(K_*^W(F)\) of \(F\), defined as follows: \(K_*^W(F)\) is the \(\mathbb Z\)-graded ring generated by symbols \(\{a\}\), \(a \in F^*\), in degree \(1\), and a symbol \(\eta\) in degree \(-1\), subject to the following relations: \[ \{ab\} = \{a\} + \{b\} - \eta.\{a\}.\{b\}, \tag{1} \] \[ \{a\}.\{1-a\} = 0 \quad \text{for } a\neq 1, \tag{2} \] \[ \eta.\{-1\} = 2, \tag{3} \] \[ \eta .\{a\} = \{a\}.\eta. \tag{4} \] Let \(I_*(F)\) denote the \(\mathbb Z\)-graded ring, which has the powers \(I^n(F)\) in degrees \(n \geq 1\) and the Witt ring \(W(F)\) in degrees \(n \leq 0\). Then the canonical homomorphism \[ K_*^W(F) \rightarrow I_*(F), \] defined via \[ \{a\} \mapsto \langle\langle a\rangle\rangle = 1-\langle a\rangle \in I(F),\quad \eta \mapsto 1 \in I_{-1}(F) = W(F) \] is an isomorphism.
scientific article; zbMATH DE number 2113817
scientific article