J. London Math. Soc. (2) 90 (2014) 725–740 C2014 London Mathematical Society doi:10.1112/jlms/jdu045 The analogue of Izumi’s Theorem for Abhyankar valuations

Shreeram Abhyankar and David Rees A well-known theorem of Izumi, strengthened by Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring (R,m) are linearly comparable to each other. This is equivalent to saying that any divisorial valuation ν centered in R is linearly comparable to the m-adic order. In the present paper, we generalize this theorem to the case of Abhyankar valuations ν with archimedian value semigroup Φ. Indeed, we prove that in a certain sense linear equivalence of topologies characterizes Abhyankar valuations with archimedian semigroups, centered in analytically irreducible local noetherian rings. In other words, saying that R is analytically irreducible, ν is Abhyankar and Φ is archimedian is equivalent to linear equivalence of topologies plus another condition called weak noetherianity of the graded algebra grνR. We give some applications of Izumi’s Theorem and of Lemma 2.8, which is a crucial step in our proof of the main theorem. We show that some of the classical results on equivalence of topologies in noetherian rings can be strengthened to include linear equivalence of topologies. We also prove a new comparison result between the m-adic topology and the topology defined by the symbolic powers of an arbitrary ideal. 1