Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals
Nguyen Cong Minh, Ngo Viet Trung
备注：
The published version of this paper contains a gap in the proofs of Theorem 2.5 and Theorem 3.5. This version corrects the proofs with almost the same arguments. Moreover, we have to modify the definition of tight complexes in Theorem 2.5. These changes don't affect other things in the published version. A corrigendum has been sent to the journal
We present criteria for the Cohen-Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen-Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen-Macaulayness of the second symbolic power or of all symbolic powers of a Stanley-Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen-Macaulay. In particular, all symbolic powers are Cohen-Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen-Macaulayness can pass from a symbolic power to another symbolic powers in different ways.
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.
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 $G$.
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.
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.
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 give an explicit presentation for each lower bound cluster algebra. Using this presentation, we show that each lower bound algebra Grobner degenerates to the Stanley-Reisner scheme of a vertex-decomposable ball or sphere, and is thus Cohen-Macaulay. Finally, we use Stanley-Reisner combinatorics and a result of Knutson-Lam-Speyer to show that all lower bound algebras are normal.
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.
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 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.
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…
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 $I_{B_n}$.
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…
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.
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…
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.
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 introduces two new notions of graded linear resolution and graded linear quotients, which generalize the concepts of linear resolution property and linear quotient for modules over the polynomial ring $A=k[x_1, \dots ,x_n]$. Besides, we compare graded linearity with componentwise linearity in general. For modules minimally generated by a regular sequence in a maximal ideal of $A$, we show that graded linear quotients imply graded linear resolution property for the colon ideals. On the other hand, we provide specific characterizations of graded linear resolution property for the Stan…
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 $\mathcal{H}$.
