All of our results are local in nature, so there is no loss in taking X to be an affine variety. In this case one can work with the coordinate ring C[X] of X in place of its structure sheaf O X .2 Swanson's theorem holds in particular on any normal variety over a field of any characteristic.
Fix a rank one valuation ν centered at a smooth point x on an algebraic variety over a field of characteristic zero. Assume that ν is Abhyankar, that is, that its rational rank plus its transcendence degree equal the dimension of the variety. Let a m denote the ideal of elements in the local ring of x whose valuations are at least m . Our main theorem is that there exists k > 0 such that a mn is contained in ( a m-k ) n for all m and n . This can be viewed as a greatly strengthened form of Izumi's Theorem for Abhyankar valuations centered on smooth complex varieties. The proof uses the theory of asymptotic multiplier ideals.
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