On splitting the Knebusch-Milnor exact sequence (Q855711)

Let \(K\) be a formally real algebraic function field of one variable with real closed field of constants \(k\) and let \(\gamma(K)\) be the set of all points of \(K\), whose residue field is equal to \(k\). The set \(\gamma(K)\) may be viewed as a real algebraic curve. That curve is a disjoint union of finitely semi-algebraically connected components. The ring \(R\) of regular functions on \(\gamma(K)\) is a Dedekind domain, so its Witt ring \(WR\) can be embedded into the Witt ring \(WK\) of field \(K\). The main result of the paper states that the Witt ring \(WR\) is a direct summand of the group \(WK\). As a consequence of this theorem the Author shows that, in this case, the Knebusch-Milnor exact sequence slices into and is patched by two split exact sequences.