964 research outputs found
Products of Borel fixed ideals of maximal minors
We study a large family of products of Borel fixed ideals of maximal minors.
We compute their initial ideals and primary decompositions, and show that they
have linear free resolutions. The main tools are an extension of straightening
law and a very surprising primary decomposition formula. We study also the
homological properties of associated multi-Rees algebra which are shown to be
Cohen-Macaulay, Koszul and defined by a Gr\"obner basis of quadrics
Generalizing the Borel property
We introduce the notion of Q-Borel ideals: ideals which are closed under the
Borel moves arising from a poset Q. We study decompositions and homological
properties of these ideals, and offer evidence that they interpolate between
Borel ideals and arbitrary monomial ideals.Comment: 19 pages, 1 figur
Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs
In this paper we study irreducible representations and symbolic Rees algebras
of monomial ideals. Then we examine edge ideals associated to vertex-weighted
oriented graphs. These are digraphs having no oriented cycles of length two
with weights on the vertices. For a monomial ideal with no embedded primes we
classify the normality of its symbolic Rees algebra in terms of its primary
components. If the primary components of a monomial ideal are normal, we
present a simple procedure to compute its symbolic Rees algebra using Hilbert
bases, and give necessary and sufficient conditions for the equality between
its ordinary and symbolic powers. We give an effective characterization of the
Cohen--Macaulay vertex-weighted oriented forests. For edge ideals of transitive
weighted oriented graphs we show that Alexander duality holds. It is shown that
edge ideals of weighted acyclic tournaments are Cohen--Macaulay and satisfy
Alexander dualityComment: Special volume dedicated to Professor Antonio Campillo, Springer, to
appea
Symbolic Powers of Monomial Ideals
We investigate symbolic and regular powers of monomial ideals. For a
square-free monomial ideal in we show
is a subset of for all
positive integers , and , where is the big-height of and . This captures two conjectures ( and ): one of
Harbourne-Huneke and one of Bocci-Cooper-Harbourne. We also introduce the
symbolic polyhedron of a monomial ideal and use this to explore symbolic powers
of non-square-free monomial ideals.Comment: 15 pages. Fixed typ
Linear resolutions of powers and products
The goal of this paper is to present examples of families of homogeneous
ideals in the polynomial ring over a field that satisfy the following
condition: every product of ideals of the family has a linear free resolution.
As we will see, this condition is strongly correlated to good primary
decompositions of the products and good homological and arithmetical properties
of the associated multi-Rees algebras. The following families will be discussed
in detail: polymatroidal ideals, ideals generated by linear forms and Borel
fixed ideals of maximal minors. The main tools are Gr\"obner bases and Sagbi
deformation
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