Related papers: On Borel fixed ideals generated in one degree

Let g be a complex simple Lie algebra and b a fixed Borel subalgebra of g. We shall describe the abelian ideals of b in a uniform way, that is, independent of the classification of complex simple Lie algebras.

Some affirmative answers are given to Huneke's problems. The calculation of local cohomology modules with respect to an arbitrary pair of ideals $I,J$ can be reduced to calculation of local cohomology modules with respect to a pair of…

We prove sharp estimates on the quadratic strand of the resolution of any homogeneous prime ideal in a standard graded polynomial ring over an arbitrary field. Our bounds only depend on the height of the prime ideal, and they are optimal…

This paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a Borel subalgebra. Our main result is a nice uniform formula for the dimension of maximal ideals in these…

The support poset of a monomial ideal $I\subseteq\mathbf{k}[x_1,\dots,x_n]$ encodes the relation between the variables $x_1,\dots,x_n$ and the minimal monomial generators of $I$. It is known that not every poset is realizable as the support…

Stillman posed a question as to whether the projective dimension of a homogeneous ideal I in a polynomial ring over a field can be bounded by some formula depending only on the number and degrees of the minimal generators of I. More…

A celebrated theorem of Fr\"oberg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem to higher degree square-free monomial ideals is an active…

Let A be a finite dimensional symmetric cllular algebras. We construct a nilpotent ideal in A. The ideal connects the radicals of cell modules with the radical of the algebra. It also reveals some information on the dimensions of simple…

We construct H(curl) and H(div) conforming finite elements on convex polygons and polyhedra with minimal possible degrees of freedom, i.e., the number of degrees of freedom is equal to the number of edges or faces of the polygon/polyhedron.…

We determine, in a polynomial ring over a field, the arithmetical rank of certain ideals generated by a set of monomials and one binomial.

Let I be the toric ideal defined by a 2 x n matrix of integers, A = ((1 1 ... 1)(a_1 a_2 ... a_n)) with a_1<a_2<...<a_n. We give a combinatorial proof that I is generated by elements of degree at most the sum of the two largest differences…

Let $\Bbbk$ be a field and let $I$ be a monomial ideal in the polynomial ring $Q=\Bbbk[x_1,\ldots,x_n]$. In her thesis, Taylor introduced a complex which provides a finite free resolution for $Q/I$ as a $Q$-module. Later, Gemeda constructed…

We focus on the structure of a homogeneous Gorenstein ideal $I$ of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. For such an ideal $I$ this…

Let $\mm=(m_0,m_1,m_2,n)$ be an almost arithmetic sequence, i.e., a sequence of positive integers with ${\rm gcd}(m_0,m_1,m_2,n) = 1$, such that $m_0<m_1<m_2$ form an arithmetic progression, $n$ is arbitrary and they minimally generate the…

Our focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal…

While sporadic examples of virtual resolutions with homology have been constructed, their occurrence is not well understood or controlled. Our results build a new set of tools for studying virtual resolutions of monomial ideals as arising…

We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic square-free monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with…

Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra over a field generated in degree $\geq d$, resp. $\geq d+1$ . Suppose that $I$ is either generated by three monomials of degrees $d$ and a set of monomials of…

Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the…

In classical and real algebraic geometry there are several notions of the radical of an ideal I. There is the vanishing radical defined as the set of all real polynomials vanishing on the real zero set of I, and the real radical defined as…