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### Some remarks on Borel type ideals [article]

Mircea Cimpoeas
2007 arXiv   pre-print
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.  ...  S.Ahmad and I.Anwar presented in [1] a characterization for ideals of Borel type and proved that if I ⊂ S is an ideal of Borel type, then reg(I) ≤ m(I)(deg(I) − 1) + 1, see [1, Cor 2.4 ].  ...  We recall the following result of Eisenbud-Reeves-Totaro, see [6, Proposition 12] . In the following, we will give a theorem of characterization for ideals of Borel type. (2) ⇒ (3) is obvious.  ...

### Page 119 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews
This is an important class of ideals, because the generic initial ideal of an arbitrary homogeneous ideal is Borel-fixed. There is a combinatorial characterization of Borel-fixed ideals.  ...  If k has a positive characteristic p, strongly stable ideals are Borel-fixed, but the converse is not true. The minimal free resolution of a Borel-fixed ideal in this case is still not understood.  ...

### Regularity of ideals of Borel type is linearly bounded [article]

2007 arXiv   pre-print
We show that the regularity of monomial ideals whose associated prime ideals are totally ordered by inclusion is linearly bounded.  ...  We also give a new characterization for when an ideal is of Borel type, Theorem 2.2 .  ...  Ideals of Borel Type A monomial ideal I is of Borel type if (I : x ∞ j ) = (I : (x 1 , x 2 , ..., x j ) ∞ ) for all j = 1, 2, ...n.  ...

### Quasi-Stable ideals and Borel-fixed ideals with a given Hilbert Polynomial [article]

Cristina Bertone
2015 arXiv   pre-print
The present paper investigates properties of quasi-stable ideals and of Borel-fixed ideals in a polynomial ring k[x_0,...  ...  The key tool for the proof of both algorithms is the combinatorial structure of a quasi-stable ideal, in particular we use a special set of generators for the considered ideals, the Pommaret basis.  ...  Margherita Roggero for suggesting the topic of Borel-fixed ideals in positive characteristic and her encouragement, and prof.  ...

### Partial regularities and a^*-invariants of Borel type ideals [article]

Dancheng Lu, Lizhong Chu
2014 arXiv   pre-print
We express the Partial regularities and a^*-invariants of a Borel type ideal in terms of its irredundant irreducible decomposition.  ...  In addition we consider the behaviours of those invariants under intersections and sums.  ...  The class of ideals is rather large: It includes Borel fixed ideals (i.e., strongly stable ideals if char(k) = 0 or p-Borel ideals if char(k) = p. See [2] ) and stable ideals.  ...

### An intriguing ring structure on the set of d-forms [article]

Jürgen Herzog, Leila Sharifan, Matteo Varbaro
2013 arXiv   pre-print
in degrees \leq n and block stable ideals (a class of ideals containing the Borel fixed ones) of K[x_1,\ldots,x_n] generated in degree d.  ...  As a byproduct we give a new proof of the characterization of Betti tables of ideals with linear resolution given by Murai.  ...  Actually, more generally, to characterize the possible Betti tables of a componentwise linear ideal of P is equivalent to characterize the possible Betti tables of a strongly stable monomial ideal of P  ...

### Splitting families and forcing

Miloš S. Kurilić
2007 Annals of Pure and Applied Logic
Kurilić, Cohen-stable families of subsets of the integers, J.  ...  Symbolic Logic 66 (1) (2001) 257-270], adding a Cohen real destroys a splitting family S on ω if and only if S is isomorphic to a splitting family on the set of rationals, Q, whose elements have nowhere  ...  Research supported by the MNZ ŽSRS (Project 144001: Forcing, Model Theory and Settheoretic Topology II).  ...

### Monomial Ideals under Ideal Operations

Jin Guo, Tongsuo Wu
2015 Communications in Algebra
We also show that the k^th symbolic power I^(k) of I preserves the properties of Borel type, Borel-fixed and strongly stable, and I^(k) is lexsegment if I is stably lexsegment.  ...  ,x_n] that the integral closure I is a monomial ideal of Borel type (Borel-fixed, strongly stable, lexsegment, or universal lexsegment respectively), if I has the same property.  ...  In the following, we will consider about the polarization of some class of monomial ideals with respect to <, and characterize the monomial ideals which become squarefree strongly stable with respect to  ...

### Borel-fixed ideals and reduction number

Lê Tuân Hoa, Ngô Viêt Trung
2003 Journal of Algebra
The aim of this paper is to study the relationship between the reduction number and Borel-fixed ideals in all characteristics.  ...  Especially it is shown that r(R/I ) r(R/I lex ), where I lex denotes the unique lex-segment ideal whose Hilbert function is equal to that of I . This solves a recent question by Conca.  ...  Recall that a monomial ideal I is called strongly stable if whenever x A ∈ I and x A is divided by x i , then x A x j /x i ∈ I for all j i. Any strongly stable monomial ideal is Borel-fixed.  ...

### A note on minimal resolutions of vector-spread Borel ideals [article]

Marilena Crupi, Antonino Ficarra
2022 arXiv   pre-print
A characterization of the extremal Betti numbers of such a class of ideals is given. Finally, we classify all Cohen-Macaulay vector-spread Borel ideals.  ...  We consider vector-spread Borel ideals. We show that these ideals have linear quotients and thereby we determine the graded Betti numbers and the bigraded Poincar\'e series.  ...  To obtain a characterization, we only need to investigate the height of a vector-spread Borel ideal.  ...

### Page 6182 of Mathematical Reviews Vol. , Issue 2001I [page]

2001 Mathematical Reviews
Borel subalgebra of a quasi-hereditary algebra A and the notion of a strong one as well.  ...  Under the same as- sumptions as above, suppose that, in addition, R is simple, has stable rank one and does not contain minimal right ideals, and fix a nonzero element u € V(R).  ...

### Some properties of Borel ideals

Maria Grazia Marinari, Luciana Ramella
1999 Journal of Pure and Applied Algebra
Borel ideals appear as 'generic initial ideals' of ideals a ⊆ k[X1; : : : ; Xn]. We analyse them paying special attention to the three variables case.  ...  Some features of a ÿnite set X of points in P r are investigated, by studying the Borel ideals that are generic initial ideals of I(X), will respect to deg-rev-lex ordering on monomials.  ...  Mora, who proposed the idea of studying Borel ideals not distinguished by Hilbert functions and Betti numbers.  ...

### A Constructive Method for Standard Borel Fixed Submodules with Given Extremal Betti Numbers

Marilena Crupi
2017 Mathematics
Given a strongly stable submodule M of a finitely generated graded free S-module F, we propose a method for constructing a standard Borel-fixed submodule M of F so that the extremal Betti numbers of M,  ...  Such a set will come into play in the characterization of the five-degree generators of the ideals J i , for i ∈ {2, 3, 4}. Table 6 summarizes the finitely-generated Borel ideal J 1 : Table 6.  ...  Acknowledgments: The author thanks the anonymous referee for his/her useful comments that improved the quality of the paper. Conflicts of Interest: The author declares no conflict of interest.  ...

### Page 3885 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews
In zero characteristic a Borel-fixed ideal is strongly stable, so its regularity is simply the highest degree of a minimal generator [S. Eliahouand M.A. Kervaire, J.  ...  The p-Borel ideals have a quite difficult combinatorial structure. Monomials 1,..., %», € J of a p-Borel ideal J are called Borel generators of J if J is the smallest p-Borel ideal containing eee u».  ...

### The range of all regularities for polynomial ideals with a given Hilbert function [article]

Francesca Cioffi
2019 arXiv   pre-print
Precisely, I give a method to construct a strongly stable ideal defining such a scheme.  ...  Given the Hilbert function u of a closed subscheme of a projective space over an infinite field K, let m_u and M_u be, respectively, the minimum and the maximum among all the Castelnuovo-Mumford regularities  ...  Acknowledgments I am very grateful to Le Tuan Hoa for the kind suggestion of the problem that I face here and for the feedback he gave me on this work.  ...
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