Valuations, equimultiplicity and normal flatness

Abstract: Let be a regular noetherian local ring of dimension and be a sequence of successive quadratic transforms along a regular prime ideal of (i.e if is the strict transform of in , then , ). We say that is maximal for if for every non-negative integer and for every prime ideal of such that is a quadratic sequence along with , we have . We show that is maximal for if and only if is a valuation ring of dimension one. In this case, the equimultiple locus at is the set of elements of the maximal ideal of for which the multiplicity is stable along the sequence , provided that the series of real numbers given by the multiplicity sequence associated with diverges. Furthermore, if we consider an ideal of , we also show that is normally flat along at the closed point if and only if the Hironaka’s character is stable along the sequence . This generalizes well known results for the case where has height one (see [B.M. Bennett, On the characteristic functions of a local ring, Ann. of Math. Second Series 91 (1) (1970) 25–87]). [Copyright &y& Elsevier]