Monomial sequences of linear type

Abstract

Let $R$ be a Noetherian commutative ring, $\langle a_1,\dots, a_n\rangle$ a sequence of elements of $R$, $I=(a_1,\dots,a_n)$ the ideal generated by the elements $a_i$ and $I_i=(a_1,\dots,a_i)$, $i=0,1,\dots, n$, the ideal generated by the first $i$ elements of the sequence. A c-sequence is a sequence $\langle a_1,\dots, a_n\rangle$ which satisfies the condition \[ [I_{i-1}I^k : a_i] \cap I^k=I_{i-1}I^{k-1} \] for every $i \in\{1, \dots, n\}$ and every $k\geq1$. 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.