Algebraic resolution of formal ideals along a valuation
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Let X be a possibly singular complete algebraic variety, defined over a field [kappa] of characteristic zero. X is nonsingular at [rho] [element of] X if OX,[rho] is a regular local ring. The problem of resolution of singularities is to show that there exists a nonsingular complete variety X, which birationally dominates X. Resolution of singularities (in characteristic zero) was proven by Hironaka in 1964. The valuation theoretic analogue to resolution of singularities is local uniformization. Let [logical or] be a valuation of the function field of X, [logical or] dominates a unique point [rho], on any complete variety [upsilon] , which birationally dominates X. The problem of local uniformization is to show that, given a valuation [logical or] of the function field of X, there exists a complete variety [upsilon] , which birationally dominates X such that the center of [logical or] on [upsilon], is a regular local ring. Zariski proved local uniformization (in characteristic zero) in 1944. His proof gives a very detailed analysis of rank 1 valuations, and produces a resolution which reflects invariants of the valuation. We extend Zariski's methods to higher rank to give a proof of local uniformization which reflects important properties of the valuation. We simultaneously resolve the centers of all the composite valuations, and resolve certain formal ideals associated to the valuation.