Initially regular sequences and depths of ideals

Abstract

For an arbitrary ideal I in a polynomial ring R we define the notion of initially regular sequences on R/I. These sequences share properties with regular sequences. In particular, the length of an initially regular sequence provides a lower bound for the depth of R/I. Using combinatorial information from the initial ideal of I we construct sequences of linear polynomials that form initially regular sequences on R/I. We identify situations where initially regular sequences are also regular sequences, and we show that our results can be combined with polarization to improve known depth bounds for general monomial ideals.2010 Mathematics Subject Classification. 13C15, 13D05, 05E40, 13F20, 13P10.

Regular Sequences On Squares Of Monomial Ideals

“…Let R be a polynomial ring over an arbitrary field k and let I ⊆ R be a homogeneous ideal. In [6], we introduced a new notion of an initially regular sequence with respect to I, whose length gives a lower bound for the depth of R/I. We further constructed sequences of linear forms that are initially regular and discussed special situations in which these constructed sequences are also regular.…”

“…We further constructed sequences of linear forms that are initially regular and discussed special situations in which these constructed sequences are also regular. In this paper, we extend the study in [6] a step further and examine when the constructed sequences of linear forms are initially regular or regular sequences on R/I t with t ≥ 2. This leads to a criterion for when depth R/I 2 > 1, partially answering a question raised by Terai and Trung in [15].…”

“…To be more precise, let us first recall the notion of initially regular sequences from [6]. Given a term order >, a sequence of nonconstant polynomials f 1 , .…”

“…, s, f i is a regular element on R/I i , where I 1 = in > (I) and I i = in > (I i−1 , f i ) for i > 1. We focus on the setting where each linear form in the sequence associated to I described in [6] is a sum of the form f = b 0 +b 1 +• • •+b t , where b 0 > b 1 > . .…”

“…[1,5,7,9,10,11,13,16]). This motivates the following natural question: when does a linear form or a sequence of linear forms constructed in [6] remain regular or initially regular with respect to powers of I? We shall address this question; particularly, we shall investigate when a linear form f = b 0 + • • • + b t satisfying (⋆), or a sequence of such forms, is regular or initially regular with respect to powers of I.…”