Toroidalization of generating sequences in dimension two function fields of positive characteristic

We give a characteristic free proof of the main result of [L. Ghezzi, H.T. Hà, O. Kashcheyeva, Toroidalization of generating sequences in dimension two function fields, J. Algebra 301 (2) (2006) 838–866. ArXiv:math.AC/0509697.] concerning toroidalization of generating sequences of valuations in dimension two function fields. We show that when an extension of two-dimensional algebraic regular local rings Rsubset ofS satisfies the conclusions of the Strong Monomialization theorem of Cutkosky and Piltant, the map between generating sequences in R and S has a toroidal structure.