Rees algebras of non-singular equimultiple prime ideals

Abstract

In a recent paper [HI2] the first named author and S. Ikeda have investigated Gorenstein properties under blowing up of height two equimultiple ideals from the arithmetical point of view. The problem is to what extent does the Gorenstein property of the Rees algebra R(a) = of an equimultiple ideal a of a local ring (A, m) describe the structure of A and a. One result is that if A is a generalized Cohen-Mac aulay ring (i.e. a ring of finite local cohomology) with dim (A) ⊇ 4 and if there is an height two equimultiple prime ideal p such that R(p) is Gorenstein, then A is Gorenstein and p is generated by a regular sequence (see [HI2], Theorem 2.6). That has led to the question whether this result is still valid for dim (∧) = 3. ∧ partial answer was given in [HI2], Proposition 2.10, where “equimultiplicity” was replaced by the stronger condition “p/p2 is flat over A/p”