The leading ideal of a complete intersection of height two, Part II

Abstract

Let ( S , n ) be a regular local ring and let I = ( f , g ) be an ideal in S generated by a regular sequence f , g of length two. Let R = S / I and m = n / I . As in [S. Goto, W. Heinzer, M.-K. Kim, The leading ideal of a complete intersection of height two, J. Algebra 298 (2006) 238–247], we examine the leading form ideal I ∗ of I in the associated graded ring G = gr n ( S ) . If gr m ( R ) is Cohen–Macaulay, we describe precisely the Hilbert series H ( gr m ( R ) , λ ) in terms of the degrees of homogeneous generators of I ∗ and of their successive GCD's. If D = GCD ( f ∗ , g ∗ ) is a prime element of gr n ( S ) that is regular on gr n ( S ) / ( f ∗ D , g ∗ D ) , we prove that I ∗ is 3-generated and a perfect ideal. If ht gr n ( S ) ( f ∗ , g ∗ , h ∗ ) = 2 , where h ∈ I is such that h ∗ is of minimal degree in I ∗ ∖ ( f ∗ , g ∗ ) gr n ( S ) , we prove I ∗ is 3-generated and a perfect ideal of gr n ( S ) , so gr m ( R ) = gr n ( S ) / I ∗ is a Cohen–Macaulay ring. We give several examples to illustrate our theorems.