A numerical characterization is given of the so-called h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result characterizes the number of faces of various dimensions and codimensions in such a complex, generalizing the classical Macaulay-Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree at most d and shifted pure (d-1)-dimensional simplicial complexes.
A monomial ideal $I$ admits a Betti splitting $I=J+K$ if the Betti numbers of $I$ can be determined in terms of the Betti numbers of the ideals $J,K$ and $J \cap K$. Given a monomial ideal $I$, we prove that $I=J+K$ is a Betti splitting of $I$, provided $J$ and $K$ are componentwise linear, generalizing a result of Francisco, Hà and Van Tuyl. If $I$ has a linear resolution, the converse also holds. We apply this result recursively to the Alexander dual of vertex-decomposable, shellable and constructible simplicial complexes and to determine the graded Betti numbers of the defining ideal of thr…
We give a necessary and sufficient condition for a simplicial complex to be approximately Cohen-Macaulay. Namely it is approximately Cohen-Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear and generated in two consecutive degrees. This completes the result of Herzog and Hibi who proved that a simplicial complex is sequentially Cohen-Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear.
This paper uses dualities between facet ideal theory and Stanley-Reisner theory to show that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay. The proof involves showing that the Alexander dual (or the cover dual, as we call it here) of a simplicial tree is a componentwise linear ideal. We conclude with additional combinatorial properties of simplicial trees.
In 1999 Herzog and Hibi introduced componentwise linear ideals. A homogeneous ideal $I$ is componentwise linear if for all non-negative integers $d$, the ideal generated by the homogeneous elements of degree $d$ in $I$ has a linear resolution. For square-free monomial ideals, componentwise linearity is related via Alexander duality to the property of being sequentially Cohen-Macaulay for the corresponding simplicial complexes. In general, the property of being componentwise linear is not preserved by taking powers. In 2011, Herzog, Hibi, and Ohsugi conjectured that if $I$ is the cover ideal of…
In this work, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of squarefree divisor complexes. We characterize when these rings are Cohen-Macaulay and we the study the shape of the Betti tables for the pinched Veronese in the two variables. As a byproduct we obtain information on the linearity of such rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.
We study the dependence of graded Betti numbers of monomial ideals on the characteristic of the base field. The examples we describe include bipartite ideals, Stanley--Reisner ideals of vertex-decomposable complexes and ideals with componentwise linear resolutions. We give a description of bipartite graphs and, using discrete Morse theory, provide a way of looking at the homology of arbitrary simplicial complexes through bipartite ideals. We also prove that the Betti table of a monomial ideal over the field of rational numbers can be obtained from the Betti table over any field by a sequence o…
Let G be a simple undirected graph on n vertices, and let I(G) \subseteq R = k[x_1,...,x_n] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some ex…
Let S be a polynomial ring and R=S/I where I is a graded ideal of S. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan which was recently proved using the Boij-Soederberg theory states that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S as well as bounded below by a function of the minimal shifts if R is Cohen-Macaulay. In this paper we study the related problem to show that the total Betti-numbers of R are also bounded above by a function of the shifts in the minimal graded free resolution of R as well a…
We prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings $S/I$, where $S$ is a polynomial ring and $I\subseteq S$ is an homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they on…
Let $Δ$ be simplicial complex and let $k[Δ]$ denote the Stanley--Reisner ring corresponding to $Δ$. Suppose that $k[Δ]$ has a pure free resolution. Then we describe the Betti numbers and the Hilbert--Samuel multiplicity of $k[Δ]$ in terms of the $h$--vector of $Δ$. As an application, we derive a linear equation system and some inequalities for the components of the $h$--vector of the clique complex of an arbitrary chordal graph. As an other application, we derive a linear equation system and some inequalities for the components of the $h$--vector of Cohen--Macaulay simplicial complexes.
Let R=k[x_1,...,x_n] be a polynomial ring over a field k. Let J={j_1,...,j_t} be a subset of [n]={1,...,n}, and let m_J denote the ideal (x_{j_1},...,x_{j_t}) of R. Given subsets J_1,...,J_s of [n] and positive integers a_1,...,a_s, we study ideals of the form I=m_{J_1}^{a_1} \cap ... \cap m_{J_s}^{a_s}. These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that I is always componentwise linea…
We study algebraic and homological properties of facet ideals of order complexes of posets which we call ideals of maximal flags of posets or simply flag ideals. We characterize the unmixed and Cohen-Macaulay flag ideals of graded posets. We also give structural results for the multigraded Betti numbers of flag ideals of such posets. The structural results on multigraded Betti numbers of flag ideals are used to characterize the class of flag ideals with linear resolutions.
We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent.
Betti numbers for cochordal zero-divisor graphs of commutative rings
School of Mathematics and Statistics, Shandong University of Technology（山东理工大学数学与统计学院）；School of Mechanical Engineering, Shandong University of Technology（山东理工大学机械工程学院）
The face numbers of simplicial complexes without missing faces of dimension larger than $i$ are studied. It is shown that among all such $(d-1)$-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal $f$-vector; and moreover, among all such 2-Cohen--Macaulay (2-CM) complexes, the same sphere has the componentwise minimal $h$-vector. It is also verified that the $l$-skeleton of a flag $(d-1)$-dimensional 2-CM complex is $2(d-l)$-CM while the $l$-skeleton of a flag PL $(d-1)$-sphere is $2(d-l)$-homotopy CM. In addition, tight lower bounds …
A certain squarefree monomial ideal $H_P$ arising from a finite partially ordered set $P$ will be studied from viewpoints of both commutative algebra and combinatorics. First, it is proved that the defining ideal of the Rees algebra of $H_P$ possesses a quadratic Gröbner basis. Thus in particular all powers of $H_P$ have linear resolutions. Second, the minimal free graded resolution of $H_P$ will be constructed explicitly and a combinatorial formula to compute the Betti numbers of $H_P$ will be presented. Third, by using the fact that the Alexander dual of the simplicial complex $Δ$ whose Stan…
We introduce sequentially $S_r$ modules over a commutative graded ring and sequentially $S_r$ simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre's condition $S_r$. In analogy with the sequentially Cohen-Macaulay property, we show that a simplicial complex is sequentially $S_r$ if and only if its pure $i$-skeleton is $S_r$ for all $i$. For $r=2$, we provide a more relaxed characterization. As an algebraic criterion, we prove that a simplicial complex is sequentially $S_r$ if and only if the minimal …
Let M in k[x,y] be a monomial ideal M=(m_1,m_2,...,m_r), where the m_i are a minimal generating set of M. We construct an explicit free resolution of k over S=k[x,y]/M for all monomial ideals M, and provide recursive formulas for the Betti numbers. In particular, if M is any monomial ideal (excepting five degenerate cases,) the total Betti numbers β_i^S(k) are given by β_0^S(k)=1, β_1^S(k)=2, and β_i^S(k)=β_{i-1}(k)+(r-1)β_{i-2}^S(k), where r is the number of minimal generators of M. This specializes to the classic example S=k[x,y]/(x^2,xy), which has β_i^S(k)=F_{i+1}, where F_{i+1} is the (i+…
Given a linear space L in affine space A^n, we study its closure L' in the product of projective lines (P^1)^n. We show that the degree, multigraded Betti numbers, defining equations, and universal Grobner basis of its defining ideal I(L') are all combinatorially determined by the matroid M of L. We also prove I(L') and all of its initial ideals are Cohen-Macaulay with the same Betti numbers. In so doing, we prove that the initial ideals of I(L') are the Stanley-Reisner ideals of an interesting family of simplicial complexes related to the basis activities of M. We also describe the state poly…
We study the irreducibility of resonance varieties of graded rings over an exterior algebra E with particular attention to Orlik-Solomon algebras. We prove that for a stable monomial ideal in E the first resonance variety is irreducible. If J is an Orlik- Solomon ideal of an essential central hyperplane arrangement, then we show that its first resonance variety is irreducible if and only if the subideal of J generated by all degree 2 elements has a 2-linear resolution. As an application we characterize those hyperplane arrangements of rank less than or equal to 3 where J is componentwise linea…
Inspired by the study of random graphs and simplicial complexes, and motivated by the need to understand average behavior of ideals, we propose and study probabilistic models of random monomial ideals. We prove theorems about the probability distributions, expectations and thresholds for events involving monomial ideals with given Hilbert function, Krull dimension, first graded Betti numbers, and present several experimentally-backed conjectures about regularity, projective dimension, strong genericity, and Cohen-Macaulayness of random monomial ideals.
Fix a field $k$. When $Δ$ is a simplicial complex on $n$ vertices with Stanley-Reisner ideal $I_Δ$, we define and study an invariant called the $\textit{type defect}$ of $Δ$. Except when $Δ$ is of a single simplex, the type defect of $Δ$, $\textrm{td}(Δ)$, is the difference $ \dim_k \textrm{Tor}_c^S(S/ I_Δ,k) - c$, where $c$ is the codimension of $Δ$ and $S = k[x_1, \ldots x_n]$. We show that this invariant admits surprisingly nice properties. For example, it is well-behaved when one glues two complexes together along a face. Furthermore, $Δ$ is Cohen-Macaulay if $\textrm{td}(Δ) \leq 0$. On th…
Georgia Institute of Technology（佐治亚理工学院）；University of California, Berkeley（加州大学伯克利分校）；Microsoft Research AI（微软研究院人工智能部门）
Function classes are collections of Boolean functions on a finite set, which are fundamental objects of study in theoretical computer science. We study algebraic properties of ideals associated to function classes previously defined by the third author. We consider the broad family of intersection-closed function classes, and describe cellular free resolutions of their ideals by order complexes of the associated posets. For function classes arising from matroids, polyhedral cell complexes, and more generally interval Cohen-Macaulay posets, we show that the multigraded Betti numbers are pure, a…
