Zhongming Tang, Guifen Zhuang
Let $Δ$ be a stable simplicial complex on $n$ vertexes. Over an arbitrary base field $K$, the symmetric algebraic shifted complex $Δ^s$ of $Δ$ is defined. It is proved that the Betti numbers of the Stanley-Reisner ideals in the polynomial ring $K[x_1,x_2,...,x_n]$ of the symmetric algebraic shifted, exterior algebraic shifted and combinatorial shifted complexes of $Δ$ are equal.
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We provide a simple method to compute the Betti numbers if the Stanley-Reisner ideal of a simplicial tree and its Alexander dual.
We prove that multigraded Betti numbers of a simplicial forest are always either 0 or 1. Moreover a nonzero multidegree appears exactly at one homological degree in the resolution. Our work generalizes work of Bouchat on edge ideals of graph forests.
We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij--Soederberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure diagrams be totally ordered, we are able to define a multiplication law for Betti diagrams that respects the decomposition and allows us to write a simple expression the decomposition of the Betti diagram of any complete intersection in terms of the degrees of its minimal generators. In the more traditional sense, the decomposition of complete in…
We find some general lower bounds of the sum of certain families of multigraded Betti numbers of any simplicial complex with a vertex coloring.
We give a sufficient condition for a monomial ideal to have a nonzero Betti number in each multidegree. In the case of facet ideals of simplicial forests, this condition becomes a necessary one and it allows us to characterize Betti numbers, projective dimension and regularity of such ideals combinatorially. Our condition is expressed in terms of minimal facet covers of simplicial complexes.
This paper concerns the study of a class of clutters called simplicial subclutters. Given a clutter $\mathcal{C}$ and its simplicial subclutter $\mathcal{D}$, we compare some algebraic properties and invariants of the ideals $I, J$ associated to these two clutters, respectively. We give a formula for computing the (multi)graded Betti numbers of $J$ in terms of those of $I$ and some combinatorial data about $\mathcal{D}$. As a result, we see that if $\mathcal{C}$ admits a simplicial subclutter, then there exists a monomial $u \notin I$ such that the (multi)graded Betti numbers of $I+(u)$ can be…
Using the existence of a good leaf in every simplicial tree, we order the facets of a simplicial tree in order to find combinatorial information about the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire splitting of the ideal, as well as a refinement of a recursive formula of Hà and Van Tuyl for computing the graded Betti numbers of simplicial trees.
Let $G$ be a chordal graph and $I(G)$ its edge ideal. Let $β(I(G)) = (β_0, β_1, ..., β_p)$ denote the Betti sequence of $I(G)$, where $β_i$ stands for the $i$th total Betti number of $I(G)$ and where $p$ is the projective dimension of $I(G)$. It will be shown that there exists a simplicial complex $Δ$ of dimension $p$ whose $f$-vector $f (Δ) = (f_0, f_1, ..., f_p)$ coincides with $β(I(G))$.
We prove the subadditivity property for the maximal degrees of the syzygies of facet ideals simplicial forests. For such an ideal $I$, if the $i$-th Betti number is nonzero and $i=a+b$, we show that there are monomials in the lcm lattice of $I$ that are complements in part of the lattice, each supporting a nonvanishing $a$-th and $b$-th Betti numbers. The subadditivity formula follows from this observation.
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…
We study the regularity and the projective dimension of the Stanley-Reisner ring of a $k$-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of $k$-decomposable monomial ideals which is the dual concept for $k$-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a chordal clutter $\mathcal{H}$, an upper bound for $reg(I(\mathcal{H}))$ is given in terms of the regularities of edge ideals of some chordal clutters which are minors of $\mathcal{H}$.
In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex $Δ$ is vertex decomposable if and only if $I_{Δ^{\vee}}$ is a vertex splittable ideal. Moreover, the properties of vertex splittable ideals are studied. As the main result, it is proved that any vertex splittable ideal has a Betti splitting and the graded Betti numbers of such ideals are explained with a recursive formula. As a corollary, recursive formulas for the regularity and projective dimension of $R/I_Δ$, when $Δ$ is…
Well ordered covers of square-free monomial ideals are subsets of the minimal generating set ordered in a certain way that give rise to a Lyubeznik resolution for the ideal, and have guaranteed nonvanishing Betti numbers in certain degrees. This paper is about square-free monomial ideals which have a well ordered cover. We consider the question of subadditivity of syzygies of square-free monomial ideals via complements in the lcm lattice of the ideal, and examine how lattice complementation breaks well ordered covers of the ideal into (well ordered) covers of subideals. We also introduce a fam…
An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of Artinian Gorenstein algebras with the Weak Lefschetz Property, a property shared by most Artinian Gorenstein algebras. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration $G$ of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the Weak Lefschetz Property. Furthermore, we s…
On Stanley-Reisner Rings with Minimal Betti Numbers
研究具有给定顶点数的简单复形，其斯坦利-雷伊斯纳环具有最小可能的贝蒂数，发现其具有特殊的组合和拓扑结构，如贝蒂数由二项式系数给出，其全子复形同调等价于点或球面。
We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.
In this paper, we give the necessary and sufficient conditions for the Cohen-Macaulayness of the associated graded ring of a simplicial affine semigroups using Gröbner basis. We generalize the concept of homogeneous numerical semigroup for the simplicial affine semigroup and show that the Betti numbers of the corresponding semigroup ring matches with the Betti numbers of the associated graded ring. We also define the nice extension for simplicial affine semigroups, motivated by the notion of a nice extension of the numerical semigroups.
We show that the ideal generated by the $(n-2)$ minors of a general symmetric $n$ by $n$ matrix has an initial ideal that is the Stanley-Reisner ideal of the boundary complex of a simplicial polytope and has the same Betti numbers.
The Graded Betti Numbers of the Skeletons of Simplicial Complexes
本文研究了一类平方自由单项理想，探讨其同调性质，并引入度数解析概念，推导出单纯复形与其骨架的分层Betti数关系。
We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squ…
Chordal clutters in the sense of [14] and [3] are defined via simplicial orders. Their circuit ideal has a linear resolution, independent of the characteristic of the base field. We show that any Betti sequence of an ideal with linear resolution appears as the Betti sequence of the circuit ideal of such a chordal clutter. Associated with any simplicial order is a sequence of integers which we call the $λ$-sequence of the chordal clutter. All possible $λ$-sequences are characterized. They are intimately related to the Hilbert function of a suitable standard graded $K$-algebra attached to the ch…
We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. We also generalize $α$-rectangular numerical semigroups to the context of simplicial affine semigroups and study their isolated factorizations. As a consequence of our results, we characterize those complete intersection simplicial affine semigroups with only one Betti minimal element in several ways. Moreover, we define Betti sorted and Betti divisible simpl…
Given a square-free monomial ideal $I$, we define a simplicial complex labeled by the generators of $I^2$ which supports a free resolution of $I^2$. As a consequence, we obtain (sharp) upper bounds on the Betti numbers of the second power of any square-free monomial ideal.
Multigraded Betti numbers of Veronese embeddings
本文研究非常面嵌入的多格Betti数，利用Hochster公式和离散Morse理论推导其消失与非消失结果。
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.
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.
We investigate resolutions of letterplace ideals of posets. We develop topological results to compute their multigraded Betti numbers, and to give structural results on these Betti numbers. If the poset is a union of no more than $c$ chains, we show that the Betti numbers may be computed from simplicial complexes of no more than $c$ vertices. We also give a recursive procedure to compute the Betti diagrams when the Hasse diagram of $P$ has tree structure.
A natural extension of bipartite graphs are $d$-partite clutters, where $d \geq 2$ is an integer. For a poset $P$, Ene, Herzog and Mohammadi introduced the $d$-partite clutter $\mathcal{C}_{P,d}$ of multichains of length $d$ in $P$, showing that it is Cohen-Macaulay. We prove that the cover ideal of $\mathcal{C}_{P,d}$ admits an $x_i$-splitting, determining a recursive formula for its Betti numbers and generalizing a result of Francisco, Hà and Van Tuyl on the cover ideal of Cohen-Macaulay bipartite graphs. Moreover we prove a Betti splitting result for the Alexander dual of a Cohen-Macaulay s…
Let $S = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $°x_i = 1$. Let $Δ$ be a simplicial complex on $[n] = \{1, ..., n \}$ and $I_Δ\subset S$ its Stanley--Reisner ideal. We write $Δ^e$ for the exterior algebraic shifted complex of $Δ$ and $Δ^c$ for a combinatorial shifted complex of $Δ$. Let $β_{ii+j}(I_Δ) = \dim_K \Tor_i(K, I_Δ)_{i+j}$ denote the graded Betti numbers of $I_Δ$. In the present paper it will be proved that (i) $β_{ii+j}(I_{Δ^e}) \leq β_{ii+j}(I_{Δ^c})$ for all $i$ and $j$, where the base field is infinite, and (ii) $β_{ii+j}(I_Δ) \leq…
We define a set of invariants of a homogeneous ideal $I$ in a polynomial ring called the symmetric iterated Betti numbers of $I$. For $I_Γ$, the Stanley-Reisner ideal of a simplicial complex $Γ$, these numbers are the symmetric counterparts of the exterior iterated Betti numbers of $Γ$ introduced by Duval and Rose. We show that the symmetric iterated Betti numbers of an ideal $I$ coincide with those of a particular reverse lexicographic generic initial ideal $\Gin(I)$ of $I$, and interpret these invariants in terms of the associated primes and standard pairs of $\Gin(I)$. We verify that for an…
In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary submodule of a free S-module are preserved when taking the generic initial module. We relate extremal multigraded Betti numbers in the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the Alexander dual simplicial complex and generalize theorems of Eagon-Reiner and Terai. As an application we give easy (alte…
We give a bijective proof that the Betti numbers of a minimal free resolution of the Stanley-Reisner ring of a cycle graph (viewed as a one-dimensional simplicial complex) are given by the number of standard Young tableaux of a given shape.
The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.
Betti numbers of skeletons of thick trees
研究厚树骨架的Betti数及其环的性质，确定其项目维度、深度和正则性，并探讨Cohen-Macaulay条件。
Buchstaber invariant is a numerical characteristic of a simplicial complex, arising from torus actions on moment-angle complexes. In the paper we study the relation between Buchstaber invariants and classical invariants of simplicial complexes such as bigraded Betti numbers and chromatic invariants. The following two statements are proved. (1) There exists a simplicial complex U with different real and ordinary Buchstaber invariants. (2) There exist two simplicial complexes with equal bigraded Betti numbers and chromatic numbers, but different Buchstaber invariants. To prove the first theorem …
We use the results by Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the Multiplicity Conjecture of Herzog, Huneke and Srinivasan for modules that are not necessarily Cohen-Macaulay. We give a combinatorial proof of the convexity of the simplicial fan spanned by the pure diagrams.
In this paper we study some algebraic properties of hypergraphs, in particula their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge, are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.
Let $R=\Bbbk[x_1,...,x_m]$ be the polynomial ring over a field $\Bbbk$ with the standard $\mathbb Z^m$-grading (multigrading), let $L$ be a Noetherian multigraded $R$-module, let $β_{i,α}(L)$ the $i$th (multigraded) Betti number of $L$ of multidegree $\a$. We introduce the notion of a generic (relative to $L$) multidegree, and the notion of multigraded module of generic type. When the multidegree $\a$ is generic (relative to $L$) we provide a Hochster-type formula for $β_{i,α}(L)$ as the dimension of the reduced homology of a certain simplicial complex associated with $L$. This allows us to sh…
The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel's conjecture for the maximum value of the Euler characteristic of a $2k$-dimensional simplicial manifold on $n$ vertices as well as Kalai's conjecture providing a lower bound on the number of edges of a simplicial ma…
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 $Δ$ 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 for the components of the $h$--vector of the clique complex of an arbitrary chordal graph.
We give a bound for the Betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex by applying unprojection theory. From this we derive a bound for the Betti numbers of iterated stellar subdivisions of the boundary complex of a simplex. The bound depends only on the number of subdivisions, and we construct examples which prove that it is sharp.
For a simplicial complex $Δ$, the graded Betti number $β_{i,j}(k[Δ])$ of the Stanley-Reisner ring $k[Δ]$ over a field $k$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $Δ$ is the boundary complex of a $d$-dimensional stacked polytope with $n$ vertices for $d\geq3$, then $β_{k-1,k}(k[Δ])=(k-1)\binom{n-d}{k}$. We prove this combinatorially.
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…
Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we charac…
Simplicial Resolutions of Powers of Square-free Monomial Ideals
本文构造支撑任意无平方单项式理想幂自由分解的单纯复形，显著缩小 Taylor 分解，并通过极值理想给出更优 Betti 数上界。
Weighted simplicial complexes (WSCs) are powerful tools for describing weighted cloud data or networks with weighted nodes. In this paper, we propose a novel approach to study WSCs via the concept of polarization. Polarization of a WSC allows one to construct a new (unweighted) simplicial complex which coincides with an object called the mixed wreath product. This new construction preserves several properties and invariants of the underlying simplicial complex of a WSC. Our main focus is to analyze WSCs through their underlying simplicial complexes and mixed wreath products. Combinatorially, w…
Frattini Geometry of Non-Generating Complexes: Symmetry, Homology and Betti Numbers
本文通过Frattini拓扑约化将有限群的非生成复形化为初等阿贝尔向量空间的非张成复形，从而确定其同构、自同构群、同调及Betti数，并给出完整Cohen-Macaulay分类。
Line graphs of simplicial complexes
本文研究纯单纯复形的线图，证明可基于其线图的组合结构计算面理想的第二阶分次Betti数，并刻画了线图为完全（二部）图的纯单纯复形及线图应满足的条件。
Inspired by recent results of Ein, Lazarsfeld, Erman and Zhou on the non-vanishing of Betti numbers of high Veronese subrings, we describe the behaviour of the Betti numbers of Stanley-Reisner rings associated with iterated barycentric or edgewise subdivisions of a given simplicial complex. Our results show that for a simplicial complex $Δ$ of dimension $d-1$ and for $1\leq j\leq d-1$ the number of $0$'s the j-th linear strand of the minimal free resolution of the r-th barycentric or edgewise subdivision is bounded above only in terms of $d$ and $j$ (and independently of $r$).
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…
We study face numbers of simplicial complexes that triangulate manifolds (or even normal pseudomanifolds) with boundary. Specifically, we establish a sharp lower bound on the number of interior edges of a simplicial normal pseudomanifold with boundary in terms of the number of interior vertices and relative Betti numbers. Moreover, for triangulations of manifolds with boundary all of whose vertex links have the weak Lefschetz property, we extend this result to sharp lower bounds on the number of higher-dimensional interior faces. Along the way we develop a version of Bagchi and Datta's $σ$- an…
Using the concept of $d$-collapsibility from combinatorial topology, we define chordal simplicial complexes and show that their Stanley-Reisner ideals are componentwise linear. Our construction is inspired by and an extension of "chordal clutters'' which was defined by Bigdeli, Yazdan Pour and Zaare-Nahandi in 2017, and characterizes Betti tables of all ideals with linear resolution in a polynomial ring. We show $d$-collapsible and $d$-representable complexes produce componentwise linear ideals for appropriate $d$. Along the way, we prove that there are generators that when added to the ideal,…
In this paper we study the resolution of a facet ideal associated with a special class of simplicial complexes introduced by S. Faridi. These simplicial complexes are called trees, and are a generalization (to higher dimensions) of the concept of a tree in graph theory. We show that the Koszul homology of the facet ideal I of a tree is generated by the homology classes of monomial cycles, determine the projective dimension and the regularity of I if the tree is 1-dimensional, show that the graded Betti numbers of I satisfy an alternating sum property if the tree is connected in codimension 1, …
Given any finite simplicial complex Δ, we show how to construct a new simplicial complex Δ_χ that is balanced and vertex decomposable. Moreover, we show that the h-vector of the simplicial complex Δ_χ is precisely the f-vector, denoted f(Δ), of the original complex Δ. We deduce this result by relating f(Δ) with the graded Betti numbers of the Alexander dual of Δ_χ. Our construction generalizes the "whiskering" construction of Villarreal, and Cook and Nagel. As a corollary of our work, we add a new equivalent statement to a theorem of Björner, Frankl, and Stanley that classifies the f-vectors o…
For a simplicial complex $Δ$, the affect of the expansion functor on combinatorial properties of $Δ$ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal $I(Δ)$ and its Alexander dual which we denote by $J_Δ$ to see how the expansion functor alter the algebraic properties of these ideals. It is shown that for any expansion $Δ^α$ the ideals $J_Δ$ and $J_{Δ^α}$ have the same total Betti numbers and their Cohen-Macaulayness are equivalent, which implies that the regularities of the ideals $I(Δ)$ and $I(Δ^α)$ are …
We define the uniform face ideal of a simplicial complex with respect to an ordered proper vertex colouring of the complex. This ideal is a monomial ideal which is generally not squarefree. We show that such a monomial ideal has a linear resolution, as do all of its powers, if and only if the colouring satisfies a certain nesting property. In the case when the colouring is nested, we give a minimal cellular resolution supported on a cubical complex. From this, we give the graded Betti numbers in terms of the face-vector of the underlying simplicial complex. Moreover, we explicitly describe the…
Let $k$ be a field and $R=k[x_1,\ldots,x_n]/I=S/I$ a graded ring. Then $R$ has a $t$-linear resolution if $I$ is generated by homogeneous elements of degree $t$, and all higher syzygies are linear. Thus $R$ has a $t$-linear resolution if ${\rm Tor}^S_{i,j}(S/I,k)=0$ if $j\ne i+t-1$. For a simplicial complex $Δ$ on $[{\bf n}]=\{1,\ldots,n\}$ and a field $k$, the Stanley-Reisner ring $k[Δ]$ is $k[x_1,\ldots,x_n]/I$, where $I$ is generated by those squarefree monomials $x_{i_1}\cdots x_{i_k}$ for which $\{ i_1,\ldots,i_k\}$ does not belong to $Δ$. In \cite{Fr} the Stanley-Reisner rings with 2-lin…
Simplicial Resolutions of the Quadratic Power of Monomial Ideals
本研究定义了支撑单项式理想二次幂消解的单纯复形及其子复形，得到投射维数与Betti数的更紧界，引入置换理想并证明其Betti数上界性质及对应单纯复形为Scarf复形。
We extend the construction of moment-angle complexes to simplicial posets by associating a certain T^m-space Z_S to an arbitrary simplicial poset S on m vertices. Face rings Z[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen--Macaulay rings. Our primary motivation is to study the face rings Z[S] by topological methods. The space Z_S has many important topological properties of the original moment-angle complex Z_K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z_S is iso…
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