On the Ext-modules of ideals of Borel type

Some Remarks on Borel Type Ideals

“…We mention that this concept appears also in [3,Definition 1.3] as the so called weakly stable ideal. Herzog, Popescu and Vladoiu proved in [8] that I is of Borel type, if and only if for any monomial u ∈ I and for any 1 ≤ j < i ≤ n with x i |u, there exists an integer t > 0 such that x t j u/x ν i (u) i ∈ I, where ν i (u) is the exponent of x i in u. This allows us to prove that the property of an ideal to be of Borel type is preserved for several operations, like sum, intersection, product, colon, see Proposition 1.1.…”

“…Herzog, Popescu and Vladoiu [8] define a monomial ideal I to be of Borel type if it satisfies ( * ). We mention that this concept appears also in [3,Definition 1.3] as the so called weakly stable ideal.…”

Finite filtrations of modules and shellable multicomplexes

“…. , n. In [6], any monomial ideal satisfying condition (2) Proof. Let P ∈ Ass(S/I), and let j be the largest integer such that x j ∈ P .…”

A Stable Property of Borel Type Ideals

“…We mention that this concept appears also in [3,Definition 1.3] as the so called weakly stable ideal. Herzog, Popescu and Vladoiu proved in [7] that I is of Borel type, if and only if for any monomial u ∈ I and for any 1 ≤ j < i ≤ n and q > 0 with x q i |u, there exists an integer t > 0 such that x t j u/x q i ∈ I. As a consequence, it became obvious that the sum of ideals of Borel type is an ideal of Borel type.…”