We introduce the abstract notion of squarefree-power-like functions, which unify the sequences of squarefree ordinary and symbolic powers of squarefree monomial ideals. By employing the Tor-vanishing criteria for mixed sums of ideals, we establish sharp lower bounds for their Castelnuovo-Mumford regularity in terms of what we call the admissible set of the associated hypergraph. As an application, we derive the first general combinatorial lower bound for the regularity of squarefree symbolic powers of monomial ideals. In the setting of edge ideals, by exploiting the special combinatorial structures of block graphs and Cohen-Macaulay chordal graphs, we show that this bound turns into an exact formula for all squarefree symbolic powers of block graphs, as well as for the second squarefree symbolic powers of edge ideals of Cohen-Macaulay chordal graphs.
Assume that $G$ is a graph with edge ideal $I(G)$. For every integer $s\geq 1$, we denote the squarefree part of the $s$-th symbolic power of $I(G)$ by $I(G)^{\{s\}}$. We determine an upper bound for the regularity of $I(G)^{\{s\}}$ when $G$ is a chordal graph. If $G$ is a Cameron-Walker graphs, we compute ${\rm reg}(I(G)^{\{s\}}$ in terms of the induced matching number of $G$. Moreover, for any graph $G$, we provide sharp upper bounds for ${\rm reg}(I(G)^{\{2\}})$ and ${\rm reg}(I(G)^{\{3\}})$.
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, Montaño and Núñez-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 co…
The symbolic powers, in general, are not equal to the ordinary powers. Therefore, one interesting question here is for what classes of ideals ordinary and symbolic powers coincide? The answer to this question for squarefree monomial ideals may be packing property. In this paper, we classify all cubic path ideals for those the symbolic and ordinary powers coincide.
The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\geq 1$, the inequalities ${\rm sdepth} (S/I^{(ks)}) \leq {\rm sdepth} (S/I^{(s)})$ and ${\rm sdepth} (I^{(ks)}) \leq {\rm sdepth} (I^{(s)})$ hold. If moreover $I$ is unmixed of height $d$, then we show that for every integer $k\geq1$, ${\rm sdepth}(I^{(k+d)})\leq {\rm sdepth}(I^{(k)})$ and ${\rm sdepth}(S/I^{(k+d)})\leq {\rm sdepth}(S/I^{(k)})$. Finally, we consider the limit behavior of the Stanley depth o…
Comparing v-numbers of symbolic and ordinary powers of squarefree monomial ideals
Chennai Mathematical Institute（金奈数学研究所）；University of Manitoba（曼尼托巴大学）；Tulane University（杜兰大学）；Indian Institute of Technology Madras（印度马德拉斯理工学院）；Institute of Mathematics, VAST（越南科学技术研究院数学研究所）
We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuejols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.
Two-dimensional squarefree monomial ideals can be seen as the Stanley-Reisner ideals of graphs. The main results of this paper are combinatorial characterizations for the Cohen-Macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph.
We show that for any two proper monomial ideals I and J in the polynomial ring S = k[x_1, ..., x_n] the ring S/IJ is Golod. We also show that if I is squarefree then for large enough k the quotient S/I^{(k)} of S by the kth symbolic power of I is Golod. As an application we prove that the multiplication on the cohomology algebra of some classes of moment-angle complexes is trivial.
We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.
Symbolic powers are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups. As a result, perfect graphs play an important role in the theory, dual to the role played by perfect graphs in the theory of secants of monomial ideals. We use Gröbner degenerations as a tool to reduce questions about symbolic powers of arbitrary ideals to the monomial case. Among the applications are a new, unified approach to the Gröbner b…
We prove that the integral closures of the powers of a squarefree monomial ideal I equal the symbolic powers if and only if I is the edge ideal of a Fulkersonian hypergraph.
We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo-Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this we show the before-now unknown convergence of Stanley depths of integral closure powers. …
When a monomial ideal has linear quotients with respect to an admissible order of increasing support-degree, we provide two proofs of different flavors to show that it is componentwise support-linear. We also introduce the variable decomposable monomial ideals. In squarefree case, these ideals correspond to the vertex decomposable simplicial complexes. We study the relationships of the variable decomposable ideals with weakly polymatroidal ideals, weakly stable ideals and ideals with linear quotients. We also investigate the componentwise properties of all these ideals.
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), w…
Let $I$ be a homogeneous ideal in a polynomial ring over a field. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. Motivated by results about ordinary powers of $I$, we study the asymptotic behavior of the regularity function $\text{reg}~ (I^{(n)})$ and the maximal generating degree function $ω(I^{(n)})$, when $I$ is a monomial ideal. It is known that both functions are eventually quasi-linear. We show that, in addition, the sequences $\{\text{reg}~ I^{(n)}/n\}_n$ and $\{ω(I^{(n)})/n\}_n$ converge to the same limit, which can be described combinatorially. We construct an example of an equidi…
Let $R$ be a commutative Noetherian ring and let ${\bf x} :=x_1,\ldots,x_d$ be a regular $R$-sequence contained in the Jacobson radical of $R$. An ideal $I$ of $R$ is said to be a monomial ideal with respect to ${\bf x}$ if it is generated by a set of monomials $x_1^{e_1}\ldots x_d^{e_d}$. It is shown that, if ${\bf x}R$ is a prime ideal of $R$, then each monomial ideal $I$ has a canonical and unique decomposition as an irredundant finite intersection of primary ideals of the form $x^{e_1}_{τ(1)}R+\dots+x^{e_s}_{τ(s)}R$, where $τ$ is a permutation of $\{1,\ldots,d\}$, $s\in\{1,\ldots,d\}$ and …
Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. In 1982, R. Stanley associated a combinatorial invariant to any finitely generated $\mathbb{Z}^n$-graded $S$-module which is now called Stanley depth. Stanley conjectured that this invariant is an upper bound for the depth of module. Stanley's conjecture has been disproved by Duval et al. \cite{abcj}, and the counterexample is a quotient of squarefree monomial ideals. On the other hand, there are evidences showing that Stanley's inequality can be true for high powers of mono…
This article concerns monomial ideals fixed by differential operators of affine semi-group rings over $\mathbb{C}$. We give a complete characterization of when this happens. Perhaps surprisingly, every monomial ideal is fixed by an infinite set of homogeneous differential operators and is in fact determined by them. This opens up a new tool for studying monomial ideals. We explore applications of this to (mixed) multiplier ideals and other variants as well as give examples of detecting ideal membership in integrally closed powers and symbolic powers of squarefree monomial ideals.
We study basic properties of monomial ideals with linear quotients. It is shown that if the monomial ideal $I$ has linear quotients, then the squarefree part of $I$ and each component of $I$ as well as $\mm I$ have linear quotients, where $\mm$ is the graded maximal ideal of the polynomial ring. As an analogy to the Rearrangement Lemma of Björner and Wachs we also show that for a monomial ideal with linear quotients the admissible order of the generators can be chosen degree increasingly. As a generalization of the facet ideal of a forest, we define monomial ideals of forest type and show that…
In this paper, we show for a monomial ideal $I$ of $K[x_1,x_2,\ldots,x_n]$ that the integral closure $\ol{I}$ is a monomial ideal of Borel type (Borel-fixed, strongly stable, lexsegment, or universal lexsegment respectively), if $I$ has the same property. We also show that the $k^{th}$ symbolic power $I^{(k)}$ of $I$ preserves the properties of Borel type, Borel-fixed and strongly stable, and $I^{(k)}$ is lexsegment if $I$ is stably lexsegment. For a monomial ideal $I$ and a monomial prime ideal $P$, a new ideal $J(I, P)$ is studied, which also gives a clear description of the primary decompos…
