Some Remarks on Borel Type Ideals

Abstract

We give new equivalent characterizations for ideals of Borel type. Also, we prove that the regularity of a product of ideals of Borel type is bounded by the sum of the regularities of those ideals.

Bounds for the regularity of product of edge ideals

“…In [6], Chardin, Minh and Trung proved that if I and J are monomial complete intersections, then reg(IJ) ≤ reg(I) + reg(J). Cimpoeaş proved that for two monomial ideals of Borel type I, J, reg(IJ) ≤ reg(I) + reg(J), [7]. Caviglia in [5] and Eisenbud, Huneke and Ulrich in [9] studied the more general problem of regularity of tensor products and various Tor modules of R/I and R/J.…”

On the Stanley depth of a special class of Borel type ideals

“…, x j ) ∞ ), (∀)1 ≤ j ≤ n. The Mumford-Castelnuovo regularity of I is the number reg(I) = max{j − i : β ij (I) = 0}, where β ij 's are the graded Betti numbers. The regularity of the ideals of Borel type was extensively studied, see for instance [12], [1] and [5]. In the first section, we study the invariant sdepth(I), for an ideal of Borel type.…”

“…Proposition 5.2] or[5, Corollary 1.2], it follows that I is an ideal of Borel type. As a direct consequence of Lemma 1.3 and Proposition 1.4, we get the following corollary.…”

Monomial Ideals under Ideal Operations

“…In [4,Proposition 1], Mircea Cimpoeas observed that the afore mentioned property is preserved under several operations, such as sum, intersection, product, colon. For a monomial ideal I of Borel type, note that I : m ∞ = I : m r holds for r >> 0, thus the saturation I : m ∞ is a monomial ideal of Borel type.…”