Monomial ideals arising from distributive lattices

Monomial ideals arising from flag complexes whose generic initial ideals do not depend on term orders

We will study monomial ideals $I$ in the exterior algebra as well as in the polynomial ring whose generic initial ideal is constant for all term orders up to permutations of variables. First, in the exterior algebra, we determine all graphs and all flag complexes whose exterior face ideal satisfies the above condition. Second, in the polynomial ring, it will be shown that the generic initial ideal $\gin_σ(I(G))$ of the edge ideal $I(G)$ of a graph $G$ is constant for all term orders $σ$ up to permutations of variables if and only if $G$ is a complete bipartite graph.

Duality, a-invariants and canonical modules of rings arising from linear optimization problems

The aim of this paper is to study integer rounding properties of various systems of linear inequalities to gain insight about the algebraic properties of Rees algebras of monomial ideals and monomial subrings. We study the normality and Gorenstein property--as well as the canonical module and the a-invariant--of Rees algebras and subrings arising from systems with the integer rounding property. We relate the algebraic properties of Rees algebras and monomial subrings with integer rounding properties and present a duality theorem.

Face posets of tropical polyhedra and monomial ideals

We exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relations. We focus on monomial tropical polyhedra, and deduce how their geometry reflects properties of monomial ideals. Their vertex-facet lattice is homotopy equivalent to a sphere and encodes the Betti numbers of an associated monomial ideal.

The monomial ideal of independent sets associated to a graph

Independent sets play a key role into the study of graphs and important problems arising in graph theory reduce to them. We define the monomial ideal of independent sets associated to a finite simple graph and describe its homological and algebraic invariants in terms of the combinatorics of the graph. We compute the minimal primary decomposition and characterize the Cohen--Macaulay ideals. Moreover, we provide a formula for computing the Betti numbers, which depends only on the coefficients of the independence polynomial of the graph.

Finite lattices and Gröbner bases

Gröbner bases of binomial ideals arising from finite lattices will be studied. In terms of Gröbner bases and initial ideals, a characterization of finite distributive lattices as well as planar distributive lattices will be given.

Toric rings arising from vertex cover ideals

We extend the sortability concept to monomial ideals which are not necessarily generated in one degree and as an application we obtain normal Cohen-Macaulay toric rings attached to vertex cover ideals of graphs. Moreover, we consider a construction on a graph called a clique multi-whiskering which always produces vertex cover ideals with componentwise linear powers.

Cellular resolutions of powers of monomial ideals

There are many connections between the invariants of the different powers of an ideal. We investigate how to construct minimal resolutions for all powers at once using methods from algebraic and polyhedral topology with a focus on ideals arising from combinatorics. In one construction, we obtain cellular resolutions for all powers of edge ideals of bipartite graphs on n vertices, supported by (n-2)-dimensional complexes. Our main result is an explicit minimal cellular resolution for all powers of edge ideals of paths. These cell complexes are constructed by first subdividing polyhedral complex…

Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals

There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J(H) be its cover ideal in a polynomial ring R. We give an explicit description of all associated primes of R/J^s, for any power J^s of J, in terms of the coloring properties of hypergraphs arising from H. We also give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J. Our work yields two new purely al…

源于顶点可分解单纯复形的Rees代数

Normal Rees algebras arising from vertex decomposable simplicial complexes

本文证明了顶点可分解单纯复形的Rees代数是正规Cohen-Macaulay整环，并据此推导出无平方弱多拟阵理想及相关图覆盖理想和超图边理想的正规性。

Divisors on graphs, binomial and monomial ideals, and cellular resolutions

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their $\mathbb{Z}$-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other r…

五个变量的单项式理想的贝蒂数的特征独立性

Characteristic Independence of Betti Numbers of Monomial Ideals in Five Variables

研究五个变量的单项式理想贝蒂数与基域特征的关系，通过将其简化为无平方因子孪生理想并应用霍赫斯特公式，证明其贝蒂数与特征无关，给出相关计算结果，且表明佩娃六变量例子的界限是精确的。

关于 licci 无平方单项式理想

On licci squarefree monomial ideals

本文研究了图及相关组合结构中无平方单项式理想 licci 属性，确定了 licci bi-Cohen-Macaulay 无平方单项式理想、互补边理想等，并完整列出了树的 licci 路径理想，扩展了 licci 边理想分类至更广路径理想家族。

Standard monomial theory and toric degenerations of Richardson varieties in the Grassmannian

Richardson varieties are obtained as intersections of Schubert and opposite Schubert varieties. We provide a new family of toric degenerations of Richardson varieties inside Grassmannians by studying Gröbner degenerations of their corresponding ideals. These degenerations are parametrised by block diagonal matching fields in the sense of Sturmfels-Zelevinsky. We associate a weight vector to each block diagonal matching field and study its corresponding initial ideal. In particular, we characterise when such ideals are toric, hence providing a family of toric degenerations for Richardson variet…

Spheres arising from multicomplexes

In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex $Δ$ on the vertex set $V$ with $Δ\ne 2^V$, the deleted join of $Δ$ with its Alexander dual $Δ^\vee$ is a combinatorial sphere. In this paper, we extend Bier's construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable, which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that thes…

相对Hochster—Takayama公式与Cohen—Macaulay单项理想商理想

Relative Hochster--Takayama formula and Cohen--Macaulay monomial ideal quotients

本文提出相对Hochster—Takayama公式，用于研究单项理想商理想，给出Cohen-Macaulay性判别准则及符号幂滤相关模的性质。

Non-very ample configurations arising from contingency tables

In this paper, it is proved that, if a toric ideal possesses a fundamental binomial none of whose monomials is squarefree, then the corresponding semigroup ring is not very ample. Moreover, very ample semigroup rings of Lawrence type are discussed. As an application, we study very ampleness of configurations arising from contingency tables.

Total of each of the 1-th and 2-th Betti numbers of the join-meet ideal of Special Crystal Lattice

The join-meet ideal was introduced by Takayuki Hibi in 1987. It is binomial ideals that are defined by finite lattices. We study the join-meet ideal of non-distributive finite lattices that do not always satisfy modular. In particular, we work on the case of Crystal lattice which is one of them. In this paper, we give the computational result of a total of each of the 1-th and 2-th of grade Betti numbers of the join-meet ideal of a special Crystal lattice. The important point about this result is that it is characterized by the number of combinations of special generators. Moreover, we can con…

Syzygies of Oriented Matroids

We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of independent sets, and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of Stanley's formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by Bayer, Popescu and Sturmfels. We resolve the combinatorial problems posed in their paper by computing Mobius in…

Minimal primes of ideals arising from conditional independence statements

We consider ideals arising in the context of conditional independence models that generalize the class of ideals considered by Fink [7] in a way distinct from the generalizations of Herzog-Hibi-Hreinsdottir-Kahle-Rauh [13] and Ay-Rauh [1]. We introduce switchable sets to give a combinatorial description of the minimal prime ideals, and for some classes we describe the minimal components. We discuss many possible interpretations of the ideals we study, including as 2 \times 2 minors of generic hypermatrices. We also introduce a definition of diagonal monomial orders on generic hypermatrices and…

Contracted ideals and the Groebner fan of the rational normal curve

The paper has two goals: the study the associated graded ring of contracted homogeneous ideals in $K[x,y]$ and the study of the Groebner fan of the ideal $P$ of the rational normal curve in ${\bf P}^d$. These two problems are, quite surprisingly, very tightly related. We completely classify the contracted ideals with a Cohen-Macaulay associated graded rings in terms of the numerical invariants arising from Zariski's factorization. We determine explicitly all the initial ideals (monomial or not) of $P$ that are Cohen-Macaulay.

On minimal free resolutions of sub-permanents and other ideals arising in complexity theory

We compute the linear strand of the minimal free resolution of the ideal generated by k x k sub-permanents of an n x n generic matrix and of the ideal generated by square-free monomials of degree k. The latter calculation gives the full minimal free resolution by work of Biagioli-Faridi-Rosas. Our motivation is to lay groundwork for the use of commutative algebra in algebraic complexity theory. We also compute several Hilbert functions relevant for complexity theory.

Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields

Let K be a finite field with q elements and let X be a subset of a projective space P^{s-1}, over the field K, which is parameterized by Laurent monomials. Let I(X) be the vanishing ideal of X. Some of the main contributions of this paper are in determining the structure of I(X) and some of their invariants. It is shown that I(X) is a lattice ideal. We introduce the notion of a parameterized code arising from X and present algebraic methods to compute and study its dimension, length and minimum distance. For a parameterized code arising from a connected graph we are able to compute its length …