The Leading Ideal of a Complete Intersection of Height Two in a 2-dimensional Regular Local Ring
نویسندگان
چکیده
Let (S,n) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I∗ be the leading ideal of I in the associated graded ring grn(S), and set R = S/I and m = n/I. In [GHK2], we prove that if μG(I ∗) = n, then I∗ contains a homogeneous system {ξi}1≤i≤n of generators such that deg ξi + 2 ≤ deg ξi+1 for 2 ≤ i ≤ n−1, and htG(ξ1, ξ2, · · · , ξn−1) = 1, and we describe precisely the Hilbert series H(grm(R), λ) in terms of the degrees ci of the ξi and the integers di, where di is the degree of Di = GCD(ξ1, . . . , ξi). To the complete intersection ideal I = (f, g)S we associate a positive integer n with 2 ≤ n ≤ c1 + 1, an ascending sequence of positive integers (c1, c2, . . . , cn), and a descending sequence of integers (d1 = c1, d2, . . . , dn = 0) such that ci+1 − ci > di−1 − di > 0 for each i with 2 ≤ i ≤ n − 1. We establish here that this necessary condition is also sufficient for there to exist a complete intersection ideal I = (f, g) whose leading ideal has these invariants. We give several examples to illustrate our theorems.
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