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

We give a characteristic free proof of the main result of our previous paper (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 $R\subset S$ 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

Algebraic series and valuation rings over nonclosed fields

Suppose that $k$ is an arbitrary field. Consider the field $k((x_1,...,x_n))$, which is the quotient field of the ring $k[[x_1,...,x_n]]$ of formal power series in the variables $x_1,...,x_n$, with coefficients in $k$. Suppose that $\sigma$ is a formal power series in $x_1,...,x_n$ with coefficints in the algebraic closure of $k$. We give a very simple necessary and sufficient condition for $\sigma$ to be algebraic over $k((x_1,...,x_n))$. As an application of our methods, we give a characterization of valuation rings $V$ which dominate an excellent, Noetherian local domain $R$ of dimension two, and such that the rank increases after passing to the completion of a birational extension of $R$.Comment: 17 pages; final version to appear in JPA

Toroidalization of generating sequences in dimension two function fields

Abstract. We give a characteristic free proof of the main result of [5] 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 R ⊂ S 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. 1

TOROIDALIZATION OF GENERATING SEQUENCES IN DIMENSION TWO FUNCTION FIELDS

Abstract. Let k be an algebraically closed field of characteristic 0, and let K ∗ /K be a finite extension of algebraic function fields of transcendence degree 2 over k. Let ν ∗ be a k-valuation of K ∗ with valuation ring V ∗ , and let ν be the restriction of ν ∗ to K. Suppose that R → S is an extension of algebraic regular local rings with quotient fields K and K ∗ respectively, such that V ∗ dominates S and S dominates R. We prove that there exist sequences of quadratic transforms R → ¯ R and S → ¯ S along ν ∗ such that ¯ S dominates ¯ R and the map between generating sequences of ν and ν ∗ has a toroidal structure. Our result extends the Strong Monomialization theorem of Cutkosky and Piltant. 1