On the symmetric and Rees algebra of an ideal generated by a d-sequence

Abstract

Let R be a commutative ring and I an ideal of R. In this paper, we consider the question of when the symmetric algebra of I is a domain, and hence isomorphic to the Rees algebra of I. (see Section 2 for definitions.) Several authors have studied this question (for example, [I, 4, 9, lo], or [14]). In the cases in which the symmetric algebra is a domain, other questions have been asked: Is it Cohen-Macaulay [l] ? Is it factorial [15] ? Is it integrally closed [1, 121 ? In this paper we prove the symmetric algebra of I is a domain whenever R is a domain and I is generated by a d-sequence (see [6] or [7]). A sequence of elements xi ,..., x, in R is said to be a d-sequence if (i) xi 4 (x1 ,..., xiwl , xi+r ,..., x,) for i between 1 and n and (ii) if {il ,..., ii} is a subset (possibly (6) of u,..., TZ} and K, rn~ { l,..., n}\(il ,..., ii} then((xil ,..., xi,) : xlcx,) = ((xi, ,..., xi,) : xlc). Many examples were given in [7] of d-sequences. We list some examples here.