On a conjecture of Vasconcelos via Sylvester forms
TL;DR: It is shown that the Rees algebra has a natural quasi-homogeneous structure and its presentation ideal is generated by explicit Sylvester forms, thus providing an affirmative partial answer to a conjecture of W. Vasconcelos.
About: This article is published in Journal of Symbolic Computation.The article was published on 2016-11-01 and is currently open access. It has received 6 citations till now. The article focuses on the topics: Monomial ideal & Rees algebra.
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TL;DR: The Sally module of a Rees algebra relative to one of its Rees subalgebras is a construct that can be used as a mediator for the trade-off of cohomological (e.g. depth) information between $\BB$ and the corresponding associated graded ring for several types of filtrations as discussed by the authors.
Abstract: The Sally module of a Rees algebra $\BB$ relative to one of its Rees subalgebras $\AA$ is a construct that can be used as a mediator for the trade-off of cohomological (e.g. depth) information between $\BB$ and the corresponding associated graded ring for several types of filtrations. While originally devised to deal with filtrations of finite colength, here we treat aspects of these developments for filtrations in higher dimensions as well.
4 citations
TL;DR: In this article, it was shown that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen-Macaulay, which is a conjecture of Vasconcelos.
Abstract: In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen–Macaulay.
3 citations
TL;DR: In this paper , a generalization of the plane de Jonquières transformation to arbitrary dimension is studied, with an eye for the ideal theoretic side, and some results on the homological behavior of the graph of the transformation are given.
Abstract: A generalization of the plane de Jonquières transformation to arbitrary dimension is studied, with an eye for the ideal theoretic side. In particular, one considers structural properties of the corresponding base ideal and of its defining relations. Useful throughout is the idea of downgraded sequences of forms, a tool considered in many sources for the rounding-up of ideals of defining relations. The emphasis here is on the case where the supporting Cremona transformation of the de Jonquières transformation is the identity map. In this case we give some results on the homological behavior of the graph of the transformation.
2 citations
01 Jan 2014
TL;DR: In this article, a minimal bigraded resolution of the Rees Algebra associated to a proper rational parametrization of a monomial plane curve is presented, and the maps of the resolution in terms of a generalized version of the Euclidean algorithm are described explicitly.
Abstract: We compute a minimal bigraded resolution of the Rees Algebra associated to a proper rational parametrization of a monomial plane curve We describe explicitly both the bigraded Betti numbers and the maps of the resolution in terms of a generalized version of the Euclidean Algorithm We also explore the relation between pencils of adjoints of the monomial plane curve and elements in a suitable piece of the defining ideal of the Rees Algebra
1 citations
Posted Content•
TL;DR: In this article, a generalization of the plane de Jonquieres transformation to arbitrary dimension is studied, with an eye for the ideal theoretic side, where structural properties of the corresponding base ideal and of its defining relations are considered.
Abstract: A generalization of the plane de Jonquieres transformation to arbitrary dimension is studied, with an eye for the ideal theoretic side. In particular, one considers structural properties of the corresponding base ideal and of its defining relations. Useful throughout is the idea of downgraded sequences of forms, a tool considered in many sources for the rounding-up of ideals of defining relations. The emphasis here is on the case where the supporting Cremona transformation of the de Jonquieres transformation is the identity map. In this case we establish aspects of the homological behavior of the graph of the transformation.
1 citations
References
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Book•
30 Mar 1995
TL;DR: In this article, the authors define basic constructions and dimension theory, and apply them to the problem of homological methods for combinatorial problem solving in the context of homology.
Abstract: Introduction.- Elementary Definitions.- I Basic Constructions.- II Dimension Theory.- III Homological Methods.- Appendices.- Hints and Solutions for Selected Exercises.- References.- Index of Notation.- Index.
5,674 citations
"On a conjecture of Vasconcelos via ..." refers background in this paper
...It is a corollary to the standard Taylor resolution that the syzygies of I are generated by the Koszul relations of the pure powers xa1, . . . , x a n and by the reduced relations of (x1 · · · xn)b with each one of the pure powers (see for example Eisenbud, 1995, Exercise 17.11)....
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11 Jan 1994
TL;DR: In this article, the structure of ideals generated by binomials and the schemes and varieties associated to them were investigated, and structural results were given for affine algebraic sets that can be defined by binomial ideals.
Abstract: We investigate the structure of ideals generated by binomials (polynomials
with at most two terms) and the schemes and varieties associated to them The
class of binomial ideals contains many classical examples from algebraic
geometry, and it has numerous applications within and beyond pure mathematics
The ideals defining toric varieties are precisely the binomial prime ideals
Our main results concern primary decomposition: If $I$ is a binomial ideal then
the radical, associated primes, and isolated primary components of $I$ are
again binomial, and $I$ admits primary decompositions in terms of binomial
primary ideals A geometric characterization is given for the affine algebraic
sets that can be defined by binomials Our structural results yield
sparsity-preserving algorithms for finding the radical and primary
decomposition of a binomial ideal
341 citations
87 citations
"On a conjecture of Vasconcelos via ..." refers background or methods in this paper
...They have been largely used in many sources, such as (Busé, 2009; Cortadellas and D’Andrea, 2014, 2015; Cox, 2008; Cox et al., 2008; Hassanzadeh and Simis, 2014; Jouanolou, 1997; Hong et al., 2008, 2012, 2013; Simis and Tohǎneanu, 2015)....
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...As a consequence, the Rees algebra RR(I) will be almost Cohen–Macaulay, thus answering affirmatively in this case a conjecture of Vasconcelos stated in (Hong et al., 2013, Conjecture 4.15)....
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...All rights reserved. monomials is almost Cohen–Macaulay (cf. Hong et al., 2013, Conjecture 4.15)....
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...In particular, for a ≤ 2b, the ideal J is a minimal reduction of I with reduction number 1, and it is well-known that in this case RR (I) is Cohen–Macaulay (see, e.g. Hong et al., 2013, Theorem 3.3 (i))....
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70 citations
"On a conjecture of Vasconcelos via ..." refers background in this paper
...The appearance of Sylvester forms goes back at least to the late sixties in a paper of Wiebe (Wiebe, 1969; see also Cox et al., 2008)....
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...Wiebe, H., 1969....
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