11 research outputs found
The primary components of positive critical binomial ideals
A natural candidate for a generating set of the (necessarily prime) defining
ideal of an -dimensional monomial curve, when the ideal is an almost
complete intersection, is a full set of critical binomials. In a somewhat
modified and more tractable context, we prove that, when the exponents are all
positive, critical binomial ideals in our sense are not even unmixed for , whereas for they are unmixed. We further give a complete
description of their isolated primary components as the defining ideals of
monomial curves with coefficients. This answers an open question on the number
of primary components of Herzog-Northcott ideals, which comprise the case
. Moreover, we find an explicit, concrete description of the irredundant
embedded component (for ) and characterize when the hull of the ideal,
i.e., the intersection of its isolated primary components, is prime. Note that
these last results are independent of the characteristic of the ground field.
Our techniques involve the Eisenbud-Sturmfels theory of binomial ideals and
Laurent polynomial rings, together with theory of Smith Normal Form and of
Fitting ideals. This gives a more transparent and completely general approach,
replacing the theory of multiplicities used previously to treat the particular
case .Comment: 21 page
Binomial ideals in quantum tori and quantum affine spaces
The article targets binomial ideals in quantum tori and quantum affine
spaces. First, noncommutative analogs of known results for commutative
(Laurent) polynomial rings are obtained, including the following: Under the
assumption of an algebraically closed base field, it is proved that primitive
ideals are binomial, as are radicals of binomial ideals and prime ideals
minimal over binomial ideals. In the case of a quantum torus
, the results are strongest: In this situation, the
binomial ideals are parametrized by characters on sublattices of the free
abelian group whose group algebra is the center of ; the
sublattice-character pairs corresponding to primitive ideals as well as to
radicals and minimal primes of binomial ideals are determined. As for
occurrences of binomial ideals in quantum algebras: It is shown that
cocycle-twisted group algebras of finitely generated abelian groups are
quotients of quantum tori modulo binomial ideals. Another appearance is as
follows: Cocycle-twisted semigroup algebras of finitely generated commutative
monoids, as well as quantum affine toric varieties, are quotients of quantum
affine spaces modulo certain types of binomial ideals
Primary Components of Binomial Ideals
Binomials are polynomials with at most two terms. A binomial ideal is an ideal generated by binomials. Primary components and associated primes of a binomial ideal are still binomial over algebraically closed fields. Primary components of general binomial ideals over algebraically closed fields with characteristic zero can be described combinatorially by translating the operations on binomial ideals to operations on exponent vectors. In this dissertation, we obtain more explicit descriptions for primary components of special binomial ideals. A feature of this work is that our
results are independent of the characteristic of the field.
First of all, we analyze the primary decomposition of a special class of binomial ideals, lattice ideals, in which every variable is a nonzerodivisor modulo the ideal. Then we provide a description for primary decomposition of lattice ideals in fields with positive characteristic.
In addition, we study the codimension two lattice basis ideals and we compute their primary components explicitly.
An ideal I β k[x_(1),β¦.x_(n) ] is cellular if every variable is either a nonzerodivisor modulo I or is nilpotent modulo I. We characterize the minimal primary components of cellular binomial ideals explicitly. Another significant result is a computation of the Hull of a cellular binomial ideal, that is the intersection of all of its minimal primary components.
Lastly, we focus on commutative monoids and their congruences. We study properties of monoids that have counterparts in the study of binomial ideals. We provide a characterization of primary ideals in positive characteristic, in terms of the congruences they induce