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Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals
Abstract
We give a generalization of Hochster's formula for local cohomologies of
square-free monomial ideals to monomial ideals, which are not necessarily
square-free. Using this formula, we give combinatorial characterizations of
generalized Cohen-Macaulay monomial ideals. We also give other applications of
the generalized Hochster's formula.
... In Theorem 3.3, we provide an alternative proof for the above inequality. While the proof in [14] is based on constructing a splittable map between distinct symbolic powers of I, our proof is based on a formula due to Takayama [19]. Next, we use this inequality to reprove that the sequence {depth(S/I (k) )} ∞ k=1 is convergent and min k depth(S/I (k) ) = lim k→∞ depth(S/I (k) ). ...
... Our proof is based on a formula due to Takayama [19]. Hence, we first recall this formula. ...
... x αn n . Takayama [19,Theorem 2.2] proves that for any vector α ∈ Z n and for every integer i, we have ...
Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $I\subset S$ is a squarefree monomial ideal. For every integer $k\geq 1$, we denote the $k$-th symbolic power of $I$ by $I^{(k)}$. Recently, Monta\~no and N\'u\~nez-Betancourt \cite{mn} proved that for every pair of integers $m, k\geq 1$,$${\rm depth}(S/I^{(m)})\leq {\rm depth}(S/I^{(\lceil\frac{m}{k}\rceil)}).$$We provide an alternative proof for this inequality. Moreover, we reprove the known results that the sequence $\{{\rm depth}(S/I^{(k)})\}_{k=1}^{\infty}$ is convergent and$$\min_k{\rm depth}(S/I^{(k)})=\lim_{k\rightarrow \infty}{\rm depth}(S/I^{(k)})=n-\ell_s(I),$$where $\ell_s(I)$ denotes the symbolic analytic spread of $I$. We also determine an upper bound for the index of depth stability of symbolic powers of $I$. Next, we consider the Stanley depth of symbolic powers and prove that the sequences $\{{\rm sdepth}(S/I^{(k)})\}_{k=1}^{\infty}$ and $\{{\rm sdepth}(I^{(k)})\}_{k=1}^{\infty}$ are convergent and the limit of each sequence is equal to its minimum. Furthermore, we determine an upper bound for the indices of sdepth stability of symbolic powers.
... In Theorem 4.5, we give a positive answer to this question and even more, we show that one can choose s to be µ(I ℓ(I)−1 )!, where for every monomial ideal J, we denote by µ(J) the number of minimal monomial generators of J. The proof of Theorem 4.5 is based on a formula due to Takayama [25,Theorem 2.2] which is a generalization of the so-called Hochster's formula and relates the local cohomology modules of a (nonsquarefree) monomial ideal to reduced homologies of particular simplicial complexes. ...
... In Theorem 4.5, we show this is the case. Our proof is base on a formula due to Takayama [25], which is presented as follows. ...
... x αn n . Takayama [25,Theorem 2.2] proves that for every vector α ∈ Z n and for every integer i, we have ...
Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $G$ is a graph with edge ideal $I(G)$. We prove that the modules $S/\overline{I(G)^k}$ and $\overline{I(G)^k}/\overline{I(G)^{k+1}}$ satisfy Stanley's inequality for every integer $k\gg 0$. If $G$ is a non-bipartite graph, we show that the ideals $\overline{I(G)^k}$ satisfy Stanley's inequality for all $k\gg 0$. For every connected bipartite graph $G$ (with at least one edge), we prove that ${\rm sdepth}(I(G)^k)\geq 2$, for any positive integer $k\leq {\rm girth}(G)/2+1$. This result partially answers a question asked in [20]. For any proper monomial ideal $I$ of $S$, it is shown that the sequence $\{{\rm depth}(\overline{I^k}/\overline{I^{k+1}})\}_{k=0}^{\infty}$ is convergent and $\lim_{k\rightarrow\infty}{\rm depth}(\overline{I^k}/\overline{I^{k+1}})=n-\ell(I)$, where $\ell(I)$ denotes the analytic spread of $I$. Furthermore, it is proved that for any monomial ideal $I$, there exists an integer $s$ such that $${\rm depth} (S/I^{sm}) \leq {\rm depth} (S/\overline{I}),$$for every integer $m\geq 1$. We also determine a value $s$ for which the above inequality holds. If $I$ is an integrally closed ideal, we show that ${\rm depth}(S/I^m)\leq {\rm depth}(S/I)$, for every integer $m\geq 1$. As a consequence, we obtain that for any integrally closed monomial ideal $I$ and any integer $m\geq 1$, we have ${\rm Ass}(S/I)\subseteq {\rm Ass}(S/I^m)$. \end{abstract}
... For an arbitrary monomial ideal, Hochster's formula can not be employed. Takayama's formula given in [57] can perform a similar task as Hochster's formula for this class of ideals. ...
... The simplicial complex ∆ a (I) is called the degree complex of I. The construction of ∆ a (I) was first given in [57] and then simplified in [47]. We recall the construction from [47]. ...
... Theorem 3.2 (Takayama's Formula [57]). Let I ⊆ S be a monomial ideal, and let ∆(I) denote the simplicial complex corresponding to √ I. Then ...
We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{ reg } I(G)$ and the asymptotic linear function $\text{ reg } I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.
... , x n ] and I a monomial ideal in R. Then R/I has a natural N n -graded structure inherited from that of R. Therefore, the local cohomology module H i m (R/I ) has a Z n -graded structure. Takayama's formula [33,Theorem 1] describes the dimension of the Z n -graded component H i m (R/I ) a , for a ∈ Z n , in terms of a simplicial complexes a (I ). We shall recall the construction of a (I ), as given in [29], which is simpler than the original construction of [33]. ...
... Takayama's formula [33,Theorem 1] describes the dimension of the Z n -graded component H i m (R/I ) a , for a ∈ Z n , in terms of a simplicial complexes a (I ). We shall recall the construction of a (I ), as given in [29], which is simpler than the original construction of [33]. ...
... It contains additional conditions on a for H i m (R/I ) a = 0. However, the proof in [33] shows that we may drop these conditions, which is more convenient for our investigation. From Takayama's formula, we immediately obtain the following characterizations of depth and regularity of monomial ideals in terms of the degree complexes. ...
We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of hypegraphs and of powers of ideals.
... Then, R/I has a natural N n -graded structure. Takayama [17] showed that the Z n -graded components of the local cohomology modules of R/I can be described in terms of certain complexes, which are defined as follows. ...
... Then, a (I ) is a simplicial complex, which we call a degree complex of I because the a-graded component of the local cohomology modules of R/I depends on the reduced cohomology of a (I ). The above definition of a (I ) is due to [13, Lemma 1.2], which is simpler than the original definition in [17]. ...
Let I be the edge ideal of a connected non-bipartite graph and R the base polynomial ring. Then, \({\text {depth}}R/I \ge 1\) and \({\text {depth}}R/I^t = 0\) for \(t \gg 1\). This paper studies the problem when \({\text {depth}}R/I^t = 1\) for some \(t \ge 1\) and whether the depth function is non-increasing thereafter. Furthermore, we are able to give a simple combinatorial criterion for \({\text {depth}}R/I^{(t)} = 1\) for \(t \gg 1\) and show that the condition \({\text {depth}}R/I^{(t)} = 1\) is persistent, where \(I^{(t)}\) denotes the t-th symbolic powers of I.
... For a monomial ideal in , Takayama in [14] found a combinatorial formula for dim ( ∕ ) for all ∈ ℤ in terms of certain simplicial complexes which are called degree complexes. For every = ( 1 , … , ) ∈ ℤ we set = { | < 0} and write = Π =1 . ...
... Let be a monomial ideal in and be a vector in ℕ . In the proof of [14,Theorem 1], the author showed that if there exists ∈ [ ] such that ≥ = max{deg ( ) | is a minimal monomial generator of } then Δ ( ) is either a cone over or the void complex. Definition 2.10. ...
https://onlinelibrary.wiley.com/share/author/CWHCBFB7QSVCTZCCAJ5R?target=10.1002/mana.202100485
... Then R/I has a natural N n -graded structure. Takayama [16] showed that the Z n -graded components of the local cohomology modules of R/I can be described in terms of certain complexes, which are defined as follows. ...
... Then ∆ a (I) is a simplicial complex, which we call a degree complex of I because the a-graded component of the local cohomology modules of R/I depends on the reduced cohomology of ∆ a (I). The above definition of ∆ a (I) is due to [12, Lemma 1.2], which is simpler than the original definition in [16]. ...
Let I be the edge ideal of a connected non-bipartite graph and R the base polynomial ring. Then depth R/I \ge 1 and depth R/I^t = 0 for t \gg 1. We give conditions for depth R/I^t = 1 for some t in between and show that the depth function is non-increasing thereafter. Especially, the depth function quickly decreases to 0 after reaching 1. We show that if depth R/I = 1 then depth R/I^2 = 0 and if depth R/I^2 = 1 then depth R/I^5 = 0. Other similar cases suggest that if depth R/I^t = 1 then depth R/I^{t+3} = 0. Furthermore, we give a combinatorial criterion for depth R/I^(t) = 1 for some t \ge 1 and show that the condition depth R/I^(t) = 1 is persistent.
... By focusing on monomial ideals, whose symbolic powers are then also monomial ideals, one can invoke a formula of Takayama [31], which relates local cohomology modules of a monomial ideal with the reduced homology groups of certain simplicial complexes. Since depth can be characterized by the vanishing of the local cohomology modules, the study of symbolic depth functions can be reduced to the investigation of combinatorial properties of monomial ideals. ...
... For a ∈ Z n , let H i m (R/K) a denote the degree a component of H i m (R/K). Takayama [31] gave a formula to relate the dimension and vanishing of H i m (R/K) a to that of the reduced homology groups of certain simplicial complex ∆ a (K), which depends on the primary component of K. The simplicial complex ∆ a (K) is a subcomplex of the Stanley-Reisner simplicial complex ∆(K) of the squarefree monomial ideal √ K. Particularly, the facets of ∆ a (K) are facets of ∆(K) if a ∈ N n . ...
We survey recent studies and results on the following problem: for which function \(f: {\mathbb N} \rightarrow {\mathbb Z}_{\ge 0}\) does there exist a homogeneous ideal Q in a polynomial ring S such that (a) depthS∕Qt = f(t) for all t ≥ 1, or (b) depthS∕Q(t) = f(t) for all t ≥ 1?KeywordsDepthProjective dimensionHomogeneous idealMonomial idealPowerSymbolic powerBertini-type theorem1991 Mathematics Subject ClassificationPrimary 13C1513D0214B05
... By focusing on monomial ideals, whose symbolic powers are then also monomial ideals, one can invoke a formula of Takayama [31], which relates local cohomology modules of a monomial ideal with the reduced homology groups of certain simplicial complexes. Since depth can be characterized by the vanishing of the local cohomology modules, the study of symbolic depth functions can be reduced to the investigation of combinatorial properties of monomial ideals. ...
... For a ∈ Z n , let H i m (R/K) a denote the degree a component of H i m (R/K). Takayama [31] gave a formula to relate the dimension and vanishing of H i m (R/K) a to that of the reduced homology groups of certain simplicial complex ∆ a (K), which depends on the primary component of K. The simplicial complex ∆ a (K) is a subcomplex of the Stanley-Reisner simplicial complex ∆(K) of the squarefree monomial ideal √ K. Particularly, the facets of ∆ a (K) are facets of ∆(K) if a ∈ N n . ...
We survey recent studies and results on the following problem: which numerical functions can be the depth function of powers and symbolic powers of homogeneous ideals.
... In order to do that, together with studying reg(R/I n ) we also study the so-called a i -invariants a i (R/I n ), which can be regarded as partial Castelnuovo-Mumford regularities, see (3.1). Using a technique of computing local cohomology modules H i m (R/I) of a monomial ideal given in [32] and developed further in a series of papers [15,19,21,25,33], one can translate the problem of computing H i m (R/I n ) into studying the sets of integer points in some rational polyhedra. Now, dealing with partial Castelnuovo-Mumford regularities, for each n, the computation of a i (R/I n ) can be formulated as a finite set of integer programs, see Theorem 3.8 and Corollary 3.5. ...
... Further, we need a formula for computing local cohomology modules H i m (R/I) given by Takayama in [32]. Note that H i m (R/I) admits an Z r -grading over R. For every degree α ∈ Z r we denote by H i m (R/I) α the α-component of H i m (R/I). ...
The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integer coefficients depending linearly on $n$. An integer $N_*$ is determined such that the optima of these integer programs are a quasi-linear function of $n$ for all $n\ge N_*$. Using results in the first part, one can bound in the second part the indices of stability of the Castelnuovo-Mumford regularities of integral closures of powers of a monomial ideal and that of symbolic powers of a square-free monomial ideal.
... In this paper, we examine the dimension of Z s -graded local cohomology modules of S/K, for K ⊂ S a monomial ideal. Due to a formula of Takayama [21], this problem can be reduced to examining the reduced homology of a related simplicial complex called the degree complex. The degree complex has been a useful tool of study for examining ideal invariants that are based on where the dimension of the local cohomology is nonzero, such as regularity and depth (see, e.g., [10], [15], [22]). ...
... The reason for studying the degree complexes of a monomial ideal is a formula due to Takayama (originally [21]; the version used here can be seen in [15]). ...
This paper examines the dimension of the graded local cohomology $H_\mathfrak{m}^p(S/I)_\gamma$ for monomial ideals $I$. Due to a formula of Takayama, for $I \subset S$, the local cohomology of $S/I$ is related to the reduced homology of a simplicial complex, called the degree complex. We explicitly compute the degree complexes of ordinary and symbolic powers of sums and fiber products, as well as the degree complex of the mixed product, and then use homological techniques to discuss the cohomology of their quotient rings. In particular, this technique allows for the explicit computation of $\text{reg} ((I + J + \mathfrak{m}\mathfrak{n})^{(s)})$ in terms of the regularities of $I^{(i)}$ and $J^{(j)}$.
... Then, the main result of the paper is the following theorem. Our approach is based on a generalized Hochster's formula for computing local cohomology modules of arbitrary monomial ideals formulated by Takayama [25]. Using this formula we are able to investigate the a i -invariants of powers of monomial ideals via the integer solutions of certain systems of linear inequalities. ...
... Then, β ∈ P s . From (25) and Lemma 1.3 one has ...
Let $\mathcal H$ be a unimodular hypergraph over the vertex set $[n]$ and let $J(\mathcal H)$ be the cover ideal of $\mathcal H$ in the polynomial ring $R=K[x_1,\ldots,x_n]$. We show that ${\rm reg}\ J(\mathcal H)^s$ is a linear function in $s$ for all $s\geqslant r\left\lceil \frac{n}{2}\right\rceil+1$ where $r$ is the rank of $\mathcal H$. Moreover for every $i$, $a_i(R/J(\mathcal H)^s)$ is also a linear function in $s$ for $s \geqslant n^2$.
... Letting be the homogeneous maximal ideal of , by results of Takayama in [26] we have that 1 ( ) is supported in degree zero and finitely many positive degrees, and further that [ 1 ( )] 0 ≃ , where the Frobenius action on [ 1 ( )] 0 is simply the Frobenius map on . This implies that -depth( ) = depth( ) = 1. ...
In this paper, we address a question concerning nilpotent Frobenius actions on Rees algebras and associated graded rings. We prove a nilpotent analog of a theorem of Huneke for Cohen–Macaulay singularities. This is achieved by introducing a depth‐like invariant which captures as special cases Lyubeznik's F‐depth and the generalized F‐depth from Maddox–Miller and is related to the generalized depth with respect to an ideal. We also describe several properties of this new invariant and identify a class of regular elements for which weak F‐nilpotence deforms.
... where H i m (R/J) denotes the i-th local cohomology module of R/J with respect to the maximal homogeneous ideal m of R. Since R/J is an N n -graded ring, H i m (R/J) is Z n -graded. For a ∈ Z n , the a-components of H i m (R/J) can be expressed in terms of the reduced cohomology of a simplicial complex ∆ a (J) [31]. We call it a degree complex of J. ...
By a classical result of Brodmann, the function $\operatorname{depth} R/I^t$ is asymptotically a constant, i.e. there is a number $s$ such that $\operatorname{depth} R/I^t = \operatorname{depth} R/I^s$ for $t > s$. One calls the smallest number $s$ with this property the index of depth stability of $I$ and denotes it by $\operatorname{dstab}(I)$. This invariant remains mysterious til now. The main result of this paper gives an explicit formula for $\operatorname{dstab}(I)$ when $I$ is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize $\operatorname{dstab}(I)$ explicitly. The formula expresses $\operatorname{dstab}(I)$ in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.
... In Theorem 3.5, we provide an alternative proof for this inequality. Indeed, the proof in [9] is based on a formula due to Takayama [19,Theorem 2.2], while our proof follows from a polarization argument. ...
Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $I(G)$ and $J(G)$ denote the edge ideal and the cover ideal of $G$, respectively. We provide a combinatorial upper bound for the index of depth stability of symbolic powers of $J(G)$. As a consequence, we compute the depth of symbolic powers of cover ideals of fully clique-whiskered graphs. Meanwhile, we determine a class of graphs $G$ with the property that the Castelnuovo--Mumford regularity of $S/I(G)$ is equal to the induced matching number of $G$.
... Now we recall degree complex of a monomial ideal with respect to a ∈ Z n which was used by Takayama in [21] to give a combinatorial formula for dim K (H i m (R/I) a ), for a monomial ideal I in R. For a = (a 1 , . . . , a n ) ∈ Z n , set x a = x a 1 1 · · · x an n . ...
In this paper, we compare the regularities of symbolic and ordinary powers of edge ideals of weighted oriented graphs. For a weighted oriented graph $D$, we give a lower bound for $\reg(I(D)^{k})$, if $V^+$ are sinks. If $D$ has an induced directed path $(x_i,x_j),(x_j,x_r) \in E(D)$ of length $2$ with $w(x_j)\geq 2$, then we show that $\reg(I(D)^{k})\leq \reg(I(D)^{k})$ for all $k\geq 2$. In particular, if $D$ is bipartite, then the above inequality holds for all $k\geq 2$. For any weighted oriented graph $D$, if $V^+$ are sink vertices, then we show that $\reg(I(D)^{k}) \leq \reg(I(D)^{k})$ with $k=2,3$. We further study when these regularities are equal. As a consequence, we give sharp upper bounds for regularity of symbolic powers of certain class of weighted oriented graphs.
... Letting m be the homogeneous maximal ideal of R, by results of Takayama in [Tak05] we have that H 1 m pRq is supported in degree zero and finitely many positive degrees, and further that rH 1 m pRqs 0 » K, where the Frobenius action on rH 1 m pRqs 0 is simply the Frobenius map on K. This implies F-depth R " depth R " 1. ...
In this article we address a question concerning nilpotent Frobenius actions on Rees algebras and associated graded rings. We prove a nilpotent analog of a theorem of Huneke for Cohen-Macaulay singularities. This is achieved by introducing a depth-like invariant which captures as special cases Lyubeznik's F-depth and the generalized F-depth from Maddox-Miller and is related to the generalized depth with respect to an ideal. We also describe several properties of this new invariant and identify a class of regular elements for which weak F-nilpotence deforms.
... For each degree α = (α 1 , . . . , α r ) ∈ Z r , in order to compute dim K H i m (R/I) α we use a formula given by Takayama [28,Theorem 2.2] which is a generalization of Hochster's formula for the case I is squarefree [22,Theorem 4.1]. ...
Let R = K[x1,...,xr] be a polynomial ring over a field K. Let G be a graph with vertex set {1,...,r} and let J be the cover ideal of G. We give a sharp bound for the stability index of symbolic depth function sdstab(J). In the case G is bipartite, it yields a sharp bound for the stability index of depth function dstab(J) and this bound is exact if G is a forest.
... Let I be a monomial ideal in S and a vector a ∈ N n . In the proof of [T,Theorem 1], the author showed that if there exists j ∈ [n] such that a j ≥ ρ j = max{deg j (u) | u is a minimal monomial generator of I} then ∆ a (I) is either a cone over {j} or the void complex. ...
Let $I(G)$ be the edge ideal of a simple graph $G$ over a field k. We prove that $${\rm reg}(\overline {I(G)^s}) = {\rm reg}(I(G)^s),$$ for all $s \le 4$. Furthermore, we provide an example of a graph $G$ such that $${\rm reg} I(G)^s = {\rm reg} \overline{I(G)^s} = {\rm reg} I(G)^{(s)} = \begin{cases} 5 + 2s & \text{ if char k} = 2 \\ 4 + 2s & \text{ if char k} \neq 2, \end{cases}$$ for all $s \ge 1.
... Let I be a monomial ideal in S and a vector a ∈ N n . In the proof of [T,Theorem 1], the author showed that if there exists j ∈ [n] such that a j ≥ ρ j = max{deg j (u) | u is a minimal monomial generator of I} then ∆ a (I) is either a cone over {j} or the void complex. ...
Let $\Delta$ be a one-dimensional simplicial complex. Let $I_\Delta$ be the Stanley-Reisner ideal of $\Delta$. We prove that for all $s \ge 1$ and all intermediate ideals $J$ generated by $I_\Delta^s$ and some minimal generators of $I_\Delta^{(s)}$, we have $${\rm reg} J = {\rm reg} I_\Delta^s = {\rm reg} I_\Delta^{(s)}.
... If I = I ∆ is the Stanley-Reisner ideal of a simplicial complex ∆, as in the proof of [T,Theorem 1],H i (∆ a (I), K) = 0 for all i for each a ∈ N n such that there is a component a j ≥ 1. Then, we only consider a j = 0 for all j (i.e. a = 0). ...
Let $G$ be a simple graph and $I$ its edge ideal. We prove that $${\rm reg}(I^{(s)}) = {\rm reg}(I^s)$$ for $s = 2,3$, where $I^{(s)}$ is the $s$-th symbolic power of $I$. As a consequence, we prove the following bounds \begin{align*} {\rm reg} I^{s} & \le {\rm reg} I + 2s - 2, \text{ for } s = 2,3, {\rm reg} I^{(s)} & \le {\rm reg} I + 2s - 2,\text{ for } s = 2,3,4. \end{align*}
... Usually, the Cohen-Macaulay property has a nice combinatorial interpretation. In the case that the defining ideal of the algebra is a monomial ideal, Hochster's formula [10] and its extension by Takayama [17] are powerful tools to investigate the homological properties of the algebra. In the case that the defining ideal is a binomial prime ideal, one may use the squarefree divisor complex [3] or one may use Gröbner basis theory to reduce the problem to the case of monomial ideals. ...
We recall numerical criteria for Cohen–Macaulayness related to system of parameters and introduce monomial ideals of König type which include the edge ideals of König graphs. We show that a monomial ideal is of König type if and only if its corresponding residue class ring admits a system of parameters whose elements are of the form xi-xj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i-x_j$$\end{document}. This provides an algebraic characterization of König graphs. We use this special parameter systems for the study of the edge ideal of König graphs and the study of the order complex of a certain family of posets. Finally, for any simplicial complex Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document} we introduce a system of parameters for K[Δ]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K[\Delta ]$$\end{document} with a universal construction principle, independent of the base field and only dependent on the faces of Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}. This system of parameters is an efficient tool to test Cohen–Macaulayness of the Stanley–Reisner ring of a simplicial complex.
... Usually the Cohen-Macaulay property has a nice combinatorial interpretation. In the case that the defining ideal of the algebra is a monomial ideal, Hochster's formula [10] and its extension by Takayama [17] are powerful tools to investigate the homological properties of the algebra. In the case that the defining ideal is a binomial prime ideal, one may use the squarefree divisor complex [3] or one may use Gröbner basis theory to reduce the problem to the case of monomial ideals. ...
We recall a numerical criteria for Cohen-Macaulayness related to system of parameters, and introduce monomial ideals of König type which include the edge ideals of König graphs. We show that a monomial ideal is of König type if and only if its corresponding residue class ring admits a system of parameters whose elements are of the form x i −x j. This provides an algebraic characterization of König graphs. We use this special parameter systems for the study of the edge ideal of König graphs and the study of the order complex of a certain family of posets. Finally, for any simplicial complex ∆ we introduce a system of parameters for K[∆] with a universal construction principle, independent of the base field and only dependent on the faces of ∆. This system of parameters is an efficient tool to test Cohen-Macaulayness of the Stanley-Reisner ring of a simplicial complex.
... Let (I ) denote the simplicial complex of all F ⊆ [r ] such that x F / ∈ √ I . The famous Takayama's formula [21] can be stated as follows. ...
Let I be a two-dimensional squarefree monomial ideal of a polynomial ring S. We evaluate the geometric regularity, ai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_i$$\end{document}-invariants of S/In\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S/I^n$$\end{document} for i≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\ge 2$$\end{document}. It turns out that they are all linear functions in n from n=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2$$\end{document}. Also, it is shown that g-reg(S/In)=reg(S/I(n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ g-reg }(S/I^n)={\text {reg}}(S/I^{(n)})$$\end{document} for all n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 1$$\end{document}.
... . The above definition of a degree complex is simpler than the original construction in [34]. Moreover, the original result contains additional conditions on a. ...
This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function depth R/I^(t) = dim R - pd I^(t) - 1, where I^(t) denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and pd denotes the projective dimension. It has been an open question whether the function depth R/I^(t) is non-increasing if I is a squarefree monomial ideal. We show that depth R/I^(t) is almost non-increasing in the sense that depth R/I^(s) \ge depth R/I^(t) for all s \ge 1 and t \in E(s), where E(s) = \cup_{i \ge 1} {t \in N| i(s-1)+1 \le t \le is} (which contains all integers t \ge (s-1)^2+1). The range E(s) is the best possible since we can find squarefree monomial ideals I such that depth R/I^(s) < depth R/I^(t) for t \not\in E(s), which gives a negative answer to the above question. Another open question asks whether the function depth R/I^(t) is always constant for t \gg 0. We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that I^(t) is integrally closed for t \gg 0 (e.g. if I is a squarefree monomial ideal), then depth R/I^(t) is constant for t \gg 0 with lim_{t \to \infty} depth R/I^(t) = dim R - dim \oplus_{t \ge 0} I^(t)/m I^(t). Our last result (which is the main contribution of this paper) shows that for any positive numerical function \phi(t) which is periodic for t \gg 0, there exist a polynomial ring R and a homogeneous ideal I such that depth R/I^(t) = \phi(t) for all t \ge 1. As a consequence, for any non-negative numerical function \psi(t) which is periodic for t \gg 0, there is a homogeneous ideal I and a number c such that pd I^(t) = \psi(t) + c for all t \ge 1.
... , F k , which means the set {F : F ⊆ F i for some 1 ≤ i ≤ k.} Let I be a monomial ideal of the polynomial ring S =: K[x 1 , . . . , x r ] and m the maximal homogeneous ideal of R. The famous Takayama lemma can be stated as follows [16]. Put G a = {i ∈ [r] : a i < 0} ...
Let $G$ be a simplicial complex of dimension one and $I_G$ its Stanley-Reiner ideals. We evaluate the geometric regularity, $a_i$-invariants for $i\geq 1$ of the power $I_G^n$. It turns out they are all linear functions from $n=2$. Moreover, $\mbox{g-reg}(R/I_G^n)=\reg(R/I_G^{(n)})$ holds for all $G$ and $n\geq 1$.
... .. , α n ) ∈ Z n. In order to compute the i−th local cohomology module of R/I, we use the formula of Takayama [22]. A simplicial complex on [n] = {1,. ...
We prove that the depth functions of cover ideals of balanced hyper- gragh have the non-increasing property. Furthermore, we also give a bound for the index of depth stability of these ideals.
... Our approach is based on a generalized Hochster's formula for computing local cohomology modules of arbitrary monomial ideals formulated by Takayama [24]. Using this formula we are able to investigate the depth of powers of monomial ideals via the integer solutions of certain systems of linear inequalities. ...
We prove that the cover ideals of all unimodular hypergraphs have the non-increasing depth function property. Furthermore, we show that the index of depth stability of these ideals is bounded by the number of variables.
... A main tool in the study of the set of associated primes and the depth of rings is using local cohomology modules. In the setting of monomial ideals, one often uses a generalized version of a Hochster's formula given by Takayama in [16]. Let us recall this formula here. ...
Let $I$ be a monomial ideal $I$ in a polynomial ring $R = k[x_1,...,x_r]$. In this paper we give an upper bound on $\overline{\dstab} (I)$ in terms of $r$ and the maximal generating degree $d(I)$ of $I$ such that $\depth R/\overline{I^n}$ is constant for all $n\geqslant \overline{\dstab}(I)$. As an application, we classify the class of monomial ideals $I$ such that $\overline{I^n}$ is Cohen-Macaulay for some integer $n\gg 0$.
Let I(G) be the edge ideal of a simple graph G. We prove that \(I(G)^2\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 3\). Similarly, we show that \(I(G)^3\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 4\). We deduce these characterizations by establishing a general formula for the regularity of powers of edge ideals of gap-free graphs \({{\,\textrm{reg}\,}}(I(G)^s) = \max ({{\,\textrm{reg}\,}}I(G) + s-1,2s)\), for \(s =2,3\).
Let I(G) be the edge ideal of a simple graph G over a field k. We prove thatreg(I(G)s‾)=reg(I(G)s), for all s≤4. For Stanley-Reisner ideals of one-dimensional simplicial complexes, we compute the regularity of integral closure of all powers. We further provide an example of a graph G such thatregI(G)s=regI(G)s‾=regI(G)(s)={5+2s if chark=24+2s if chark≠2, for all s≥1.
Let I=I(G) be the edge ideal of a simple graph G. We prove thatreg(I(s))=reg(Is) for s=2,3, where I(s) is the s-th symbolic power of I. As a consequence, we prove the following boundsregIs≤regI+2s−2, for s=2,3,regI(s)≤regI+2s−2, for s=2,3,4.
Let G be a simple graph and I its edge ideal. We survey recent results on the problem of computing the regularity of symbolic powers of I and its relation to the regularity of ordinary powers. We prove a rigidity property of the regularity for intermediate ideals lying between ordinary powers and symbolic powers for small powers. We then propose some related problems and provide illustrating examples.KeywordsSymbolic powersOrdinary powersEdge idealsRegularity2010 Mathematics Subject Classification
13D0213D0513H99
Let $\Delta$ be an one-dimensional simplicial complex on $\{1,2,\ldots,s\}$ and $S$ the polynomial ring $K[x_1,\ldots,x_s]$ over a field $K$. The explicit formula for $a_0(S/I_{\Delta}^n)$ is presented when $\mathrm{girth}(\Delta)\geq 4$. If $\mathrm{girth}(\Delta)=3$ we characterize the simplicial complexes $\Delta$ for which $a_0(S/I_{\Delta}^n)=3n-1$ or $3n-2$.
We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I = I_\Delta$ is the Stanley-Reisner ideal of a simplicial complex $\Delta$, then $\reg(I^{(n)}) \leqslant \delta(n-1) +b$ for all $n\geqslant 1$, where $\delta = \lim\limits_{n\to\infty} \reg(I^{(n)})/n$, and $b = \max\{\reg(I_\Gamma) \mid \Gamma \text{ is a subcomplex of } \Delta \text{ with } \F(\Gamma) \subseteq \F(\Delta)\}$. This bound is sharp for any $n$.
The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integer coefficients depending linearly on n. An integer \(N_*\) is determined such that the optima of these integer programs are a quasi-linear function of n for all \(n\ge N_*\). Using results in the first part, one can bound in the second part the indices of stability of the Castelnuovo–Mumford regularities of integral closures of powers of a monomial ideal and that of symbolic powers of a square-free monomial ideal.
Castelnuovo-Mumford regularity (or simply regularity) is an important invariant in commutative algebra and algebraic geometry. Computing or finding bounds for the regularity is a difficult problem. In the next three chapters, we shall address the regularity of ordinary and symbolic powers of squarefree monomial ideals.
In this article, we consider ideals of the form I=∩1≤i<j≤nPi,jwi,j of a polynomial ring R=K[x1,…,xn] over a field, where Pi,j is an ideal generated by variables {x1,…,xn}∖{xi,xj} and wi,j is a non-negative integer for all i, j. We will give explicit formulas for computing the ai-invariants ai(R/I),i=1,2, and the Castelnuovo-Mumford regularity reg(I) in the case wi,j takes a value α or β, where α>β>0.
Let K \mathbb {K} be a field and let S = K [ x 1 , … , x n ] S=\mathbb {K}[x_1,\dots ,x_n] be the polynomial ring in n n variables over K \mathbb {K} . Assume that I ⊂ S I\subset S is a squarefree monomial ideal. For every integer k ≥ 1 k\geq 1 , we denote the k k -th symbolic power of I I by I ( k ) I^{(k)} . Recently, Montaño and Núñez-Betancourt (2018), and independently Nguyen and Trung (to appear), proved that for every pair of integers k , i ≥ 1 k, i\geq 1 , d e p t h ( S / I ( k ) ) ≤ d e p t h ( S / I ( ⌈ k i ⌉ ) ) . \begin{equation*} \mathrm {depth}(S/I^{(k)})\leq \mathrm {depth}(S/I^{(\lceil \frac {k}{i}\rceil )}). \end{equation*} We provide an alternative proof for this inequality. Moreover, we re-prove the known results that the sequence { d e p t h ( S / I ( k ) ) } k = 1 ∞ \{\mathrm {depth}(S/I^{(k)})\}_{k=1}^{\infty } is convergent and min k d e p t h ( S / I ( k ) ) = lim k → ∞ d e p t h ( S / I ( k ) ) = n − ℓ s ( I ) , \begin{equation*} \min _k\mathrm {depth}(S/I^{(k)})=\lim _{k\rightarrow \infty }\mathrm {depth}(S/I^{(k)})=n-\ell _s(I), \end{equation*} where ℓ s ( I ) \ell _s(I) denotes the symbolic analytic spread of I I . We also determine an upper bound for the index of depth stability of symbolic powers of I I . Next, we consider the Stanley depth of symbolic powers and prove that the sequences { s d e p t h ( S / I ( k ) ) } k = 1 ∞ \{\mathrm {sdepth}(S/I^{(k)})\}_{k=1}^{\infty } and { s d e p t h ( I ( k ) ) } k = 1 ∞ \{\mathrm {sdepth}(I^{(k)})\}_{k=1}^{\infty } are convergent and the limit of each sequence is equal to its minimum. Furthermore, we determine an upper bound for the indices of sdepth stability of symbolic powers.
This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function depthR/I(t)=dimR-pdI(t)-1, where I(t) denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and pd denotes the projective dimension. It has been an open question whether the function depthR/I(t) is non-increasing if I is a squarefree monomial ideal. We show that depthR/I(t) is almost non-increasing in the sense that depthR/I(s)≥depthR/I(t) for all s≥1 and t∈E(s), where E(s)=⋃i≥1{t∈N|i(s-1)+1≤t≤is}(which contains all integers t≥(s-1)2+1). The range E(s) is the best possible since we can find squarefree monomial ideals I such that depthR/I(s)
Let R be a standard graded algebra over a field k, with irrelevant maximal ideal \(\mathfrak {m}\), and I a homogeneous R-ideal. We study the asymptotic vanishing behavior of the graded components of the local cohomology modules \(\{{\text {H}}^{i}_{\mathfrak {m}}(R/I^n)\}_{n\in {\mathbb {N}}}\) for \(i<\dim R/I\). We show that, when \({{\,\mathrm{{{char}}}\,}}k= 0\), R / I is Cohen–Macaulay, and I is a complete intersection locally on \({{\,\mathrm{{Spec}}\,}}R {\setminus }\{\mathfrak {m}\}\), the lowest degrees of the modules \(\{{\text {H}}^{i}_{\mathfrak {m}}(R/I^n)\}_{n\in {\mathbb {N}}}\) are bounded by a linear function whose slope is controlled by the generating degrees of the dual of \(I/I^2\). Our result is a direct consequence of a related bound for symmetric powers of locally free modules. If no assumptions are made on the ideal or the field k, we show that the complexity of the sequence of lowest degrees is at most polynomial, provided they are finite. Our methods also provide a result on stabilization of maps between local cohomology of consecutive powers of ideals.
We survey recent studies on the Castelnuovo–Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of \(\text {reg}I(G)\) and the asymptotic linear function \(\text {reg}I(G)^q\), for \(q \ge 1\), in terms of combinatorial data of the given graph G.
Let K be a field and S=K[x1,…,xn] be the polynomial ring in n variables over K. For any monomial ideal I, we denote its integral closure by I¯. Assume that G is a graph with edge ideal I(G). We prove that the modules S/I(G)k¯ and I(G)k¯/I(G)k+1¯ satisfy Stanley’s inequality for every integer k≫0. If G is a non-bipartite graph, we show that the ideals I(G)k¯ satisfy Stanley’s inequality for all k≫0. For every connected bipartite graph G (with at least one edge), we prove that sdepth(I(G)k)≥2, for any positive integer k≤girth(G)/2+1. This result partially answers a question asked in Seyed Fakhari (J Algebra 489:463–474, 2017). For any proper monomial ideal I of S, it is shown that the sequence {depth(Ik¯/Ik+1¯)}k=0∞ is convergent and limk→∞depth(Ik¯/Ik+1¯)=n-ℓ(I), where ℓ(I) denotes the analytic spread of I. Furthermore, it is proved that for any monomial ideal I, there exists an integer s such that depth(S/Ism)≤depth(S/I¯),for every integer m≥1. We also determine a value s for which the above inequality holds. If I is an integrally closed ideal, we show that depth(S/Im)≤depth(S/I), for every integer m≥1. As a consequence, we obtain that for any integrally closed monomial ideal I and any integer m≥1, we have Ass(S/I)⊆Ass(S/Im).
This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial complexes, integral points in polytopes and graphs.
Let $R$ be a standard graded algebra over a field $k$, with irrelevant maximal ideal $\mathfrak{m}$, and $I$ a homogeneous $R$-ideal. We study the asymptotic vanishing behavior of the graded components of the local cohomology modules $\{\text{H}^i_{\mathfrak{m}}(R/I^n)\}_{n\in \mathbb{N}}$. If these modules are Noetherian for $n\gg 0$, we show that, when $\text{char}\, k= 0$, $R/I$ is Cohen-Macaulay, and $I$ is locally a complete intersection, their lowest degrees are bounded by a linear function whose slope is controlled by the largest generating degree of the dual of the conormal module of $I$. Our result is a direct consequence of a related bound for symmetric powers of locally free modules. If no assumptions are made on the ideal, we show that the complexity of the sequence of lowest degrees is at most polynomial.
We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The first type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain applying a family of functors to a single module. For the second type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limit. We also have a version for contravariant functors. In all the introduced spectral sequences we provide sufficient conditions to ensure their degeneration at their second page. As a consequence we obtain some decomposition theorems that greatly generalize the well-known decomposition formula for local cohomology modules given by Hochster.
Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $J(G)$ is the cover ideal of $G$ and $J(G)^{(k)}$ is its $k$-th symbolic power. We prove that the sequences $\{{\rm sdepth}(S/J(G)^{(k)})\}_{k=1}^\infty$ and $\{{\rm sdepth}(J(G)^{(k)})\}_{k=1}^\infty$ are non-increasing and hence convergent. Suppose that $\nu_{o}(G)$ denotes the ordered matching number of $G$. We show that for every integer $k\geq 2\nu_{o}(G)-1$, the modules $J(G)^{(k)}$ and $S/J(G)^{(k)}$ satisfy the Stanley's inequality. We also provide an alternative proof for \cite[Theorem 3.4]{hktt} which states that ${\rm depth}(S/J(G)^{(k)})=n-\nu_{o}(G)-1$, for every integer $k\geq 2\nu_{o}(G)-1$.
Algebraic and combinatorial properties of a monomial ideal and its radical are compared.
Given non-negative integers β 0 ,β 1 ,···,β d we construct a d-dimensional Buchsbaum complex Δ over ℤ such that H ˜ i (Δ;ℤ)≅ℤ β i for all 0≤i≤d. This demonstrates (via work of P. Schenzel [Math. Z. 178, 125-142 (1981; Zbl 0472.13012)]) the existence of Stanley-Reisner rings with arbitrarily prescribed Betti numbers for local cohomology.
This text offers an overview of two of the main topics in the connections between commutative algebra and combinatorics. The first concerns the solutions of linear equations in non-negative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the upper bound conjecture for spheres. An introductory chapter giving background information in algebra, combinatorics and toplogy aims to broaden access to this material for non-specialists. This edition contains a chapter surveying more recent work related to face rings, focusing on applications to f-vectors. Also included is information on subcomplexes and subdivisions of simplicial complexes, and an application to spline theory.
Let S = k[x1,…,xn] be a polynomial ring, and let ωS be its canonical module. First, we will define squarefreeness for n-graded S-modules. A Stanley–Reisner ring k[Δ] = S/IΔ, its syzygy module Syzi(k[Δ]), and ExtiS(k[Δ], ωS) are always squarefree. This notion will simplify some standard arguments in the Stanley–Reisner ring theory. Next, we will prove that the i-linear strand of the minimal free resolution of a Stanley–Reisner ideal IΔ ⊂ S has the “same information” as the module structure of ExtiS(k[Δ∨ ], ωS), where Δ∨ is the Alexander dual of Δ. In particular, if k[Δ] has a linear resolution, we can describe its minimal free resolution using the module structure of the canonical module of k[Δ∨ ], which is Cohen–Macaulay in this case. We can also give a new interpretation of a result of Herzog and co-workers, which states that k[Δ] is sequentially Cohen–Macaulay if and only if IΔ∨ is componentwise linear.
Alexander duality for Stanely-Reisner rings and squarefree N n -graded modules-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan E-mail address: takayama@se
- K Yanagawa
K. Yanagawa, Alexander duality for Stanely-Reisner rings and squarefree N n -graded modules, J. Algebra 225, No. 2, (2000), 630-645. Yukihide Takayama, Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan E-mail address: takayama@se.ritsumei.ac.jp 15