Valuations, equimultiplicity and normal flatness

Let RR be a regular noetherian local ring of dimension n≥2n≥2 and (Ri)≡R=R0⊂R1⊂R2⊂⋯⊂Ri⊂⋯(Ri)≡R=R0⊂R1⊂R2⊂⋯⊂Ri⊂⋯ be a sequence of successive quadratic transforms along a regular prime ideal pp of RR (i.e if pipi is the strict transform of pp in RiRi, then pi≠Ripi≠Ri, i≥0i≥0). We say that pp is maximal for (Ri)(Ri) if for every non-negative integer j≥0j≥0 and for every prime ideal qjqj of RjRj such that (Ri)(Ri) is a quadratic sequence along qjqj with pj⊂qjpj⊂qj, we have pj=qjpj=qj. We show that pp is maximal for (Ri)(Ri) if and only if V=∪i≥0Ri/piV=∪i≥0Ri/pi is a valuation ring of dimension one. In this case, the equimultiple locus at pp is the set of elements of the maximal ideal of RR for which the multiplicity is stable along the sequence (Ri)(Ri), provided that the series of real numbers given by the multiplicity sequence associated with VV diverges. Furthermore, if we consider an ideal JJ of RR, we also show that Spec(R/J) is normally flat along Spec(R/p) at the closed point if and only if the Hironaka’s character ν∗(J,R)ν∗(J,R) is stable along the sequence (Ri)(Ri). This generalizes well known results for the case where pp 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]).