Ramification of valuations (Q597135)

A theory of ramification of general valuations in algebraic function fields of arbitrary dimension in characteristic \(0\) is developed generalizing the classical ramification theory of local Dedekind domains. In positive characteristic similar results are obtained in dimension two. Conditions are given for the existence of local monomialization and simultaneous resolution. Pathological examples are given. In characteristic \(p>0\) the following is proved. Let \(K^\ast| K\) be a finite, separable extension of two--dimensional algebraic function fields over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(V^\ast\) be the valuation ring of \(K^\ast\) and \(V=V^\ast\cap K\). Let \(\Gamma^\ast\) be the value group of \(V^\ast\), \(\Gamma\) be the value group of \(V\). Strong monomialization holds whenever \(\Gamma^\ast\) is finitely generated. Simultaneous resolution is true whenever the nondiscrete rational group \(\Gamma\) is not \(p\)--divisible. The defect of \(V^\ast\) over \(V\) is considered as a new invariant in characteristic \(p\). It is proved that \(V^\ast \text{ over } V\) is defectless if \(\Gamma^\ast\) is finitely generated. Strong monomialization holds whenever \(V^\ast\text{ over }V\) is defectless. An example is given for \(V^\ast\text{ over }V\) having positive defect and strong monomialization does not hold.