Monomial sequences of linear type

Abstract

Let R be a Noetherian commutative ring, 〈a₁, …, a_n〉 a sequence of elements of R, I=(a₁, …, a_n) the ideal generated by the elements a_i and I_i=(a₁, …, a_i), i=0, 1, …, n, the ideal generated by the first i elements of the sequence. A c-sequence is a sequence 〈a₁, …, a_n〉 which satisfies the condition

[I_i−1I^k : a_i]∩I^k=I_i−1I^k−1

for every i∈{1, …, n} and every k≥1. It generates an ideal of linear type. We characterize c-sequences in terms of the corresponding sequences in the Rees algebra of the ideal generated by the elements of the sequence. We then characterize monomial c-sequences of three terms.