$F$-Invariants of Stanley-Reisner Rings
Wágner Badilla-Céspedes
该研究聚焦素特征下衡量奇点的不变量的有理性问题，证明了若干情形下Stanley-Reisner环的F-不变量为有理数。
素特征下存在可衡量奇点的重要不变量。部分情形下已知它们为有理数。本文针对若干情形证明了Stanley-Reisner环的该性质。
In prime characteristic there are important invariants that allow us to measure singularities. For certain cases, it is known that they are rational numbers. In this article, we show this property for Stanley-Reisner rings in several cases.
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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 certain class of Stanley--Reisner rings having sufficiently large multiplicities are Cohen--Macaulay using Alexander duality.
When a cone is added to a simplicial complex $Δ$ over one of its faces, we investigate the relation between the arithmetical ranks of the Stanley-Reisner ideals of the original simplicial complex and the new simplicial complex $Δ'$. In particular, we show that the arithmetical rank of the Stanley-Reisner ideal of $Δ'$ equals the projective dimension of the Stanley-Reisner ring of $Δ'$ if the corresponding equality holds for $Δ$.
We show that the Stanley-Reisner ideal of the one-dimensional simplicial complex whose diagram is an $n$-gon is always a set-theoretic complete intersection in any positive characteristic.
Stanley-Reisner Ideals with Pure Resolutions
研究具有纯分解的Stanley-Reisner理想，描述两个与高度对称复形相关的无穷族，并证明Boij-Söderberg第一猜想的部分类比，给出构造任意形状纯Betti图的算法。
We present some examples of squarefree monomial ideals whose arithmetical rank can be computed using linear algebraic considerations.
The v-numbers of Stanley-Reisner ideals from the viewpoint of Alexander dual complexes
本研究从Alexander对偶复形视角探究Stanley-Reisner理想的v-数，推导其与合冲模初始次数、Serre深度等的关联，还得到2-纯单纯复形下v-数与Stanley-Reisner环维数的等式关系。
We give a purely combinatorial characterization of complete Stanley-Reisner rings having principally generated (equivalently, finitely generated) Cartier algebras.
In this note, we characterize the Hilbert regularity of the Stanley-Reisner ring $K[\bigtriangleup]$ in terms of the $f$-vector and the $h$-vector of a simplicial complex $\bigtriangleup$. We also compute the Hilbert regularity of a Gorenstein algebra.
We study the symbolic powers of the Stanley-Reisner ideal $I_{B_n}$ of a bipyramid $B_n$ over a $n-$gon $Q_n$. Using a combinatorial approach, based on analysis of subtrees in $Q_n$ we compute the Waldschmidt constant of $I_{B_n}$.
Local cohomology modules, even over a Noetherian ring $R$, are typically unwieldly. As such, it is of interest whether or not they have finitely many associated primes. We prove the affirmative in the case where $R$ is a Stanley-Reisner ring over a field, whose associated simplicial complex is a $T$-space.
We prove that for m > 2, the m-th symbolic power of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex is a matroid. Similarly, the m-th ordinary power is Cohen-Macaulay for some m > 2 if and only if the complex is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.
Let $Δ$ be a one-dimensional simplicial complex. Let $I_Δ$ be the Stanley-Reisner ideal of $Δ$. We prove that for all $s \ge 1$ and all intermediate ideals $J$ generated by $I_Δ^s$ and some minimal generators of $I_Δ^{(s)}$, we have $${\rm reg} J = {\rm reg} I_Δ^s = {\rm reg} I_Δ^{(s)}.$$
We consider ideals $I$ in a Stanley-Reisner ring $k[Δ]$ over the simplical complex $Δ$, such that the tight closure of $I$, $I^*$, is equal to $\mathfrak{m}$, the standard graded maximal ideal of $k[Δ]$. We determine the minimal number of generators of $I$ to be the $\dim Δ+1$ and note the important role this value plays in bounding the intersection of all such ideals $I$. We make mention of this intersection in special cases of Stanley-Reisner rings. We conclude with a description of how this work relates to integral closure.
The canonical trace of Stanley-Reisner rings that are Gorenstein on the punctured spectrum
本文证明维数至少3的近乎Gorenstein Stanley-Reisner环必为Gorenstein，并给出其在穿孔谱上为Gorenstein的充要条件，即近乎Gorenstein或典范迹为无关极大理想平方，后者对应非可定向同调流形。
A triangulation of a polygon has an associated Stanley-Reisner ideal. We obtain a full algebraic and combinatorial understanding of these ideals, and describe their separated models. More generally we do this for stacked simplicial complexes, in particular for stacked polytopes.
This paper studies properties of simplicial complexes for which the m-th symbolic power of the Stanley-Reisner ideal equals to the m-th ordinary power for a given m > 1. The main results are combinatorial characterizations of such complexes in the two-dimensional case. It turns out that there exist only a finite number of complexes with this property and that these complexes can be described completely. As a consequence we are able to determine all complexes for which the m-th ordinary power of the Stanley-Reisner ideal is Cohen-Macaulay for a given m > 1.
The second vanishing theorem in Stanley-Reisner ring with topological interpretation
本文在Stanley-Reisner环上推广了局部上同调的第二消没定理，并从组合拓扑视角给出了解释。
Stanley-Reisner Theory in Mixed Characteristic
本文定义了混合特征下类似Stanley-Reisner环的环类，并推广了Hochster公式与Terai判据，利用细胞层上同调作为工具。
The Frobenius complexity of a local ring $R$ measures asymptotically the abundance of Frobenius operators of order $e$ on the injective hull of the residue field of $R$. It is known that, for Stanley-Reisner rings, the Frobenius complexity is either $-\infty$ or $0$. This invariant is determined by the complexity sequence $\{c_ e\}_e $ of the ring of Frobenius operators on the injective hull of the residue field. We will show that $\{c_ e\}_e $ is constant for $e\geq 2,$ generalizing work of Alvarez Montaner, Boix and Zarzuela. Our result settles an open question mentioned by Alvarez Montaner …
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 $\mathcal{H}$.
Canonical traces of graded fiber products: applications to disconnected Stanley--Reisner rings
本文研究了不连通Stanley-Reisner环的规范迹，通过建立graded纤维积的显式公式并应用于不连通simplicial复形，最终给出了不连通复形的分类结果。
In this note we define a Stanley-Reisner ring for quasi-arithmetic matroids and more general structures. To this end, we define two types of CW complexes associated with a quasi-arithmetic matroid that generalize independence complexes of matroids. Then we use Stanley's construction of Stanley-Reisner rings for simplicial posets.
In this paper, we prove that the Stanley--Reisner ideal of any connected simplicial complex of dimension $\ge 2$ that is locally complete intersection is a complete intersection ideal. As an application, we show that the Stanley--Reisner ideal whose powers are Buchsbaum is a complete intersection ideal.
Let R be a Stanley-Reisner ring (that is, a reduced monomial ring) with coefficients in a domain k, and K its associated simplicial complex. Also let D_k(R) be the ring of k-linear differential operators on R. We give two different descriptions of the two-sided ideal structure of D_k(R) as being in bijection with certain well-known subcomplexes of K; one based on explicit computation in the Weyl algebra, valid in any characteristic, and one valid in characteristic p based on the Frobenius splitting of R. A result of Traves [Tra99] on the D_k(R)-module structure of R is also given a new proof a…
On Stanley-Reisner rings with linear resolution
本文在已有图复形$Δ_k^t(G)$拓扑性质研究的基础上，进一步探究该复形及其Alexander对偶的Stanley-Reisner环的Cohen-Macaulay性、Betti数、线性消解等代数性质。
We prove that the Frobenius algebra of the injective hull of a complete Stanley-Reisner ring as well as its Matlis dual notion of Cartier algebra can be only principally generated or infinitely generated. As a consequence we are able to show that the set of F-jumping numbers of generalized test ideals associated to complete Stanley-Reisner rings form a discrete set.
Generalized Hamming weights and symbolic powers of Stanley-Reisner ideals of matroids
本文研究了matroid的广义Hamming权重与Stanley-Reisner理想符号幂的关系，揭示了其子可加性及在码理论中的应用。
We give a purely algebraic proof of the hypersurface case of Toric Residue Mirror Conjecture recently proposed by Batyrev and Materov.
We introduce a class of Stanley-Reisner ideals called generalized complete intersection, which is characterized by the property that all the residue class rings of powers of the ideal have FLC. We also give a combinatorial characterization of such ideals.
We study the properties of the rook complex $\mathcal{R}$ of a polyomino $\mathcal{P}$ seen as independence complex of a graph $G$, and the associated Stanley--Reisner ideal $I_\mathcal{R}$. In particular, we characterize the polyominoes $\mathcal{P}$ having a pure rook complex, and the ones whose Stanley--Reisner ideal has linear resolution. Furthermore, we prove that for a class of polyominoes the Castelnuovo-Mumford regularity of $I_\mathcal{R}$ coincides with the induced matching number of $G$.
Unprojection theory aims to analyze and construct complicated commutative rings in terms of simpler ones. Our main result is that, on the algebraic level of Stanley-Reisner rings, stellar subdivisions of non-acyclic Gorenstein simplicial complexes correspond to unprojections of type Kustin-Miller. As an application, we inductively calculate the minimal graded free resolutions of Stanley-Reisner rings associated to stacked polytopes, recovering results of Terai, Hibi, Herzog and Li Marzi.
Steenrod problem and some graded Stanley-Reisner rings
本文针对Steenrod提出的环的空间奇异上同调表示问题，研究分次Stanley-Reisner环的可实现性，在一定假设下给出了其可实现的充要条件。
Recently, I defined a squarefree module over a polynomial ring $S = k[x_1, >..., x_n]$ generalizing the Stanley-Reisner ring $k[Δ] = S/I_Δ$ of a simplicial complex $Δ\subset 2^{1, ..., n}$. In this paper, from a squarefree module $M$, we construct the $k$-sheaf $M^+$ on an $(n-1)$ simplex $B$ which is the geometric realization of $2^{1, ..., n}$. For example, $k[Δ]^+$ is (the direct image to $B$ of) the constant sheaf on the geometric realization $|Δ| \subset B$. We have $H^i(B, M^+) = [H^{i+1}_m(M)]_0$ for all $i > 0$. The Poincare-Verdier duality for sheaves $M^+$ on $B$ corresponds to the l…
A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley--Reisner ring has a linear resolution. It turns out that the Stanley--Reisner ring of the sphere which is the boundary complex of a linear ball satisfies the multiplicity conjecture. A class of shellable spheres arising naturally from commutative algebra whose Stanley--Reisner rings satisfy the multiplicity conjecture will be presented.
A geometrization of Stanley--Reisner theory
本文提出Stanley--Reisner对应的几何解释，将其推广至概形，并用一个元素的域进行诠释。
In this paper, we study several properties of the second power $I_Δ^2$ of a Stanley-Reisner ideal $I_Δ$ of any dimension. As the main result, we prove that $S/I_Δ$ is Gorenstein whenever $S/I_Δ^2$ is Cohen-Macaulay over any field $K$. Moreover, we give a criterion for the second symbolic power of $I_Δ$ to satisfy $(S_2)$ and to coincide with the ordinary power, respectively. Finally, we provide new examples of Stanley-Reisner ideals whose second powers are Cohen-Macaulay.
We study the structure of Stanley-Reisner rings associated to cyclic polytopes, using ideas from unprojection theory. Consider the boundary simplicial complex Delta(d,m) of the d-dimensional cyclic polytope with m vertices. We show how to express the Stanley-Reisner ring of Delta(d,m+1) in terms of the Stanley-Reisner rings of Delta(d,m) and Delta(d-2,m-1). As an application, we use the Kustin-Miller complex construction to identify the minimal graded free resolutions of these rings. In particular, we recover results of Schenzel, Terai and Hibi about their graded Betti numbers.
In this paper firstly, we generalize the concept of codismantlable graphs to hypergraphs and show that some special vertex decomposable hypergraphs are codismantlable. Then we generalize the concept of bouquet in graphs to hypergraphs to extend some combinatorial invariants of graphs about disjointness of a set of bouquets. We use these invariants to characterize the projective dimension of Stanley-Reisner ring of special hypergraphs in some sense.
Let $Δ$ be a pure simplicial complex and $I_Δ$ its Stanley-Reisner ideal in a polynomial ring $S$. We show that $Δ$ is a matroid (complete intersection) if and only if $S/I_Δ^{(m)}$ ($S/I_Δ^m$) is clean for all $m\in\mathbb{N}$. If $\dim(Δ)=1$, we also prove that $S/I_Δ^{(2)}$ ($S/I_Δ^2$) is clean if and only if $S/I_Δ^{(2)}$ ($S/I_Δ^2$) is Cohen-Macaulay.
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.
In the present note we study Waldschmidt constants of Stanley-Reisner ideals of a hypergraph and a graph with vertices forming a bipyramid over a planar n-gon. The case of the hypergraph has been studied by Bocci and Franci. We reprove their main result. The case of the graph is new. Interestingly, both cases provide series of ideals with Waldschmidt constants descending to 1. It would be interesting to known if there are bounded ascending sequences of Waldschmidt constants.
Let $Δ$ be a simplicial complex. We study the expansions of $Δ$ mainly to see how the algebraic and combinatorial properties of $Δ$ and its expansions are related to each other. It is shown that $Δ$ is Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum or $k$-decomposable, if and only if an arbitrary expansion of $Δ$ has the same property. Moreover, some homological invariants like the regularity and the projective dimension of the Stanley-Reisner ideals of $Δ$ and those of their expansions are compared.
In this paper, we study non-Cohen--Macaulay Buchsbaum Stanley--Reisner rings with linear free resolution. In particular, for given integers $c$, $d$, $q$ with $c \ge 1$, $2 \le q \le d$, we give an upper bound $h_{c,d,q}$ on the dimension of the unique non-vanishing homology $\widetilde{H}_{q-2}(Δ;k)$ of a $d$-dimensional Buchsbaum ring $k[Δ]$ with $q$-linear resolution and codimension $c$. Also, we discuss about existence for such Buchsbaum rings with $\dim_k \widetilde{H}_{q-2}(Δ;k) = h$ for any $h$ with $0 \le h \le h_{c,d,q}$, and prove an existence theorem in the case of $q=d=3$ using the…
In a paper from 2002, Bruns and Gubeladze conjectured that graded algebra retracts of polytopal algebras over a field $k$ are again polytopal algebras. Motivated by this conjecture, we prove that graded algebra retracts of Stanley-Reisner rings over a field $k$ are again Stanley-Reisner rings. Extending this result further, we give partial evidence for a conjecture saying that monomial quotients of standard graded polynomial rings over $k$ descend along graded algebra retracts.
The goal of the present paper is the study of some algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension $d - 1$. We prove that the inequality $d \leq \mathrm{reg}(Δ) \cdot \mathrm{type}(Δ)$ holds for any $(d-1)$-dimensional Cohen-Macaulay simplicial complex $Δ$ satisfying $Δ=\mathrm{core}(Δ)$, where $\mathrm{reg}(Δ)$ (resp. $\mathrm{type}(Δ)$) denotes the Castelnuovo-Mumford regularity (resp. Cohen-Macaulay type) of the Stanley-Reisner ring $\Bbbk[Δ]$. Moreover, for any given integers $d,r,t$ satisfying $r,t \geq 2$ and $r \leq d \leq rt$, we constr…
We consider simplicial complexes admitting a free action by an abelian group. Specifically, we establish a refinement of the classic result of Hochster describing the local cohomology modules of the associated Stanley--Reisner ring, demonstrating that the topological structure of the free action extends to the algebraic setting. If the complex in question is also Buchsbaum, this new description allows for a specialization of Schenzel's calculation of the Hilbert series of some of the ring's Artinian reductions. In further application, we generalize to the Buchsbaum case the results of Stanley …
We show that the Golod property of a Stanley-Reisner ring can depend on the characteristic of the base field. More precisely, for every finite set $T$ of prime numbers we construct simplicial complexes $Δ$ and $Γ$, such that $\mathbb{K}[Δ]$ is Golod exactly in the characteristics in $T$ and $\mathbb{K}[Γ]$ is Golod exactly in the characteristics not in $T$. Along the way, we show that a one-dimensional simplicial complex is Golod if and only if it is chordal.
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.
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 $r$).
In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex $Δ$. Indeed, the differentials in the linear part are simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of $Δ$. Along the way, we also show that if a monomial ideal has at least one generator of degree $2$, then the linear strand of its minimal free resolution can be written using only $\pm 1$ coefficients.
Let $R$ be a Noetherian commutative ring of positive dimension. The Hochster-Huneke graph of $R$ (sometimes called the dual graph of Spec $R$ and denoted by $\mathcal{G} (R)$) is defined as follows: the vertices are the minimal prime ideals of $R$, and the edges are the pairs of prime ideals $(P_1,P_2)$ with height $(P_1 + P_2) = 1$. If $R$ satisfies Serre's property $(S_2)$, then $\mathcal{G} (R)$ is connected. In this note, we provide lower and upper bounds for the maximum diameter of Hochster-Huneke graphs of Stanley-Reisner rings satisfying $(S_2)$. These bounds depend on the number of var…
Asymptotic Resurgence of Facet and Stanley-Reisner ideals of Matroids
研究拟阵的斯坦利 - 赖斯纳理想的渐近复苏，通过与面理想渐近复苏的等价性，利用拟阵收缩的面理想的瓦尔施密特常数证明公式，得出渐近复苏与弱序的关系及相关上界，还证明了许多类拟阵面理想渐近复苏的下界等式并计算部分结果。
For a simplicial complex $Δ$, we introduce a simplicial complex attached to $Δ$, called the expansion of $Δ$, which is a natural generalization of the notion of expansion in graph theory. We are interested in knowing how the properties of a simplicial complex and its Stanley-Reisner ring relate to those of its expansions. It is shown that taking expansion preserves vertex decomposable and shellable properties and in some cases Cohen-Macaulayness. Also it is proved that some homological invariants of Stanley-Reisner ring of a simplicial complex relate to those invariants in the Stanley-Reisner …
In this paper we settle long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed faces of Minkowski sums. This has a wide range of applications and generalizes the classical the Upper Bound Theorems of McMullen and Stanley. Our main tool is relative Stanley--Reisner theory, a powerful generalization of the algebraic theory of simplicial complexes inaugurated by Hochster, Reis…
It is known that the Frobenius algebra of the injective hull of the residue field of a complete Stanley--Reisner ring (i.e. a formal power series ring modulo a squarefree monomial ideal) can be only principally generated or infinitely generated as algebra over its degree zero piece, and that this fact can be read off in the corresponding simplicial complex; in the infinite case, we exhibit a 1--1 correspondence between potential new generators appearing on each graded piece and certain pairs of faces of such a simplicial complex, and we use it to provide an alternative proof of the fact that t…
We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property.
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 develop the theory of integrable representations for an arbitrary maximal parabolic subalgebra of an affine Lie algebra. We see that such subalgebras can be thought of as arising in a natural way from a Borel--de Siebenthal pair of semisimple Lie algebras. We see that although there are similarities with the represenation thery of the standard maximal parabolic subalgebra there are also very interesting and non--trivial differences; including the fact that there are examples of non--trivial global Weyl modules which are irreducible and finite--dimensional. We also give a presentation of the…
We review a class of problems on the borders of topology of torus actions, commutative homological algebra and combinatorial geometry, which is currently being investigated by Victor Buchstaber and the author. The text builds on the lectures delivered on the transformation group courses in Osaka City University and Universitat Autonoma de Barcelona. We start with discussing several well-known results and problems on combinatorial geometry of polytopes and simplicial complexes, and then move gradually towards investigating the combinatorial structures associated with spaces acted on by the toru…
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