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Some remarks on Borel type ideals
[article]

2007
*
arXiv
*
pre-print

We give new equivalent

arXiv:math/0703757v3
fatcat:qug24b2k2nbqtcohujedbfk5qa
*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. ...##
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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. ...##
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Regularity of ideals of Borel type is linearly bounded
[article]

2007
*
arXiv
*
pre-print

We show that the regularity

arXiv:math/0610537v2
fatcat:dhpz5o3qlfgxrb53srhrkcfh24
*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. ...##
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Quasi-Stable ideals and Borel-fixed ideals with a given Hilbert Polynomial
[article]

2015
*
arXiv
*
pre-print

The present paper investigates properties

arXiv:1409.5569v2
fatcat:5joajrt2bbgvdjnr4acxjm3dau
*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. ...##
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Partial regularities and a^*-invariants of Borel type ideals
[article]

2014
*
arXiv
*
pre-print

We express the Partial regularities

arXiv:1412.3962v1
fatcat:2c62fzcngjctti2dsfcj7nofby
*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*. ...##
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An intriguing ring structure on the set of d-forms
[article]

2013
*
arXiv
*
pre-print

in degrees \leq n

arXiv:1308.6237v1
fatcat:umtlxofeffgw7bhjz6gypfw66q
*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 ...##
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Splitting families and forcing

2007
*
Annals of Pure and Applied Logic
*

Kurilić, Cohen-

doi:10.1016/j.apal.2006.08.002
fatcat:ey5urj6hvza7jlih7scxwtiqca
*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). ...##
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Monomial Ideals under Ideal Operations

2015
*
Communications in Algebra
*

We also show that the k^th symbolic power I^(k)

doi:10.1080/00927872.2014.952012
fatcat:ybt3wginujcwjjg3m2uw77ydua
*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 ...##
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Borel-fixed ideals and reduction number

2003
*
Journal of Algebra
*

The aim

doi:10.1016/s0021-8693(03)00367-3
fatcat:4z5u3uf3evaltkyiwmqyhx7dbu
*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. ...##
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A note on minimal resolutions of vector-spread Borel ideals
[article]

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*. ...

##
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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). ...

##
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Some properties of Borel ideals

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. ...

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

2017
*
Mathematics
*

Given

doi:10.3390/math5040056
fatcat:j4hsycj2gvajbfk246dm7zu454
*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. ...##
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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». ...##
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The range of all regularities for polynomial ideals with a given Hilbert function
[article]

2019
*
arXiv
*
pre-print

Precisely, I give

arXiv:1901.10974v1
fatcat:3jkdqrgehncldd2mi76sifl7xe
*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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