Cohen-Macaulay Simplicial Complexes Research Articles

On virtually Cohen–Macaulay simplicial complexes

We examine virtual resolutions of Stanley–Reisner ideals for a product of projective spaces by providing sufficient conditions for a simplicial complex to be virtually Cohen–Macaulay (to have a virtual resolution with length equal to its codimension). In particular, we show that all balanced simplicial complexes are virtually Cohen–Macaulay.

Inequalities of invariants on Stanley–Reisner rings of Cohen–Macaulay simplicial complexes

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 ≤ reg ( Δ ) ⋅ type ( Δ ) holds for any ( d − 1 ) -dimensional Cohen–Macaulay simplicial complex Δ satisfying Δ = core ( Δ ) , where reg ( Δ ) (resp. type ( Δ ) ) denotes the Castelnuovo–Mumford regularity (resp. Cohen–Macaulay type) of the Stanley–Reisner ring k [ Δ ] . Moreover, for any given integers d , r , t satisfying r , t ≥ 2 and r ≤ d ≤ r t , we construct a Cohen–Macaulay simplicial complex Δ ( G ) as an independent complex of a graph G such that dim ⁡ ( Δ ( G ) ) = d − 1 , reg ( Δ ( G ) ) = r and type ( Δ ( G ) ) = t .

Combinatorial g-conjecture for interval subdivisions

We verify the g-conjecture for interval subdivisions of Cohen–Macaulay simplicial complexes, using purely combinatorial methods. More precisely, we show that the g-vector of the interval subdivision of a Cohen–Macaulay simplicial complex is an f-vector. Murai defined the -vector, starting from the -vector introduced by Novik for Buchsbaum simplicial complexes. We prove that the -vector of the interval subdivision of a Buchsbaum simplicial complex is an f-vector of some simplicial complex.

The stresses on centrally symmetric complexes and the lower bound theorems

In 1987, Stanley conjectured that if a centrally symmetric Cohen–Macaulay simplicial complex Δ of dimension d-1 satisfies h i (Δ)=d i for some i≥1, then h j (Δ)=d j for all j≥i. Much more recently, Klee, Nevo, Novik, and Zheng conjectured that if a centrally symmetric simplicial polytope P of dimension d satisfies g i (∂P)=d i-d i-1 for some d/2≥i≥1, then g j (∂P)=d j-d j-1 for all d/2≥j≥i. This note uses stress spaces to prove both of these conjectures.

English

for each squarefree monomial ideal I ⊂ S = k[x1, …, xn], we associate a simple finite graph GI by using the first linear syzygies of I. The nodes of GI are the generators of I, and two vertices ui and uj are adjacent if there exist variables x, y such that xui = yuj. In the cases, where GI is a cycle or a tree, we show that I has a linear resolution if and only if I has linear quotients and if and only if I is variable-decomposable. In addition, with the same assumption on GI, we characterize all squarefree monomial ideals with a linear resolution. Using our results, we characterize all Cohen-Macaulay codimension 2 monomial ideals with a linear resolution. As another application of our results, we also characterize all Cohen-Macaulay simplicial complexes in the case, where $${G_{\rm{\Delta }}} \cong {G_{{I_{\rm{\Delta }}} \vee }}$$ is a cycle or a tree.

Singularities and radical initial ideals

What kind of reduced monomial schemes can be obtained as a Gröbner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley–Reisner schemes associated to acyclic Cohen–Macaulay simplicial complexes. This would imply, in particular, that only curves of genus zero have such a degeneration. We prove this conjecture for degrevlex orders, for elliptic curves over totally real number fields, for boundaries of cross-polytopes, and for leafless graphs. We discuss consequences for rational and F-rational singularities of algebras with straightening laws.

A non-partitionable Cohen–Macaulay simplicial complex

A long-standing conjecture of Stanley states that every Cohen–Macaulay simplicial complex is partition- able. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.

A balanced non-partitionable Cohen–Macaulay complex

In a recent article, Duval, Goeckner, Klivans and Martin disproved the longstanding conjecture by Stanley, that every Cohen–Macaulay simplicial complex is partitionable. We construct counterexamples to this conjecture that are even balanced, i.e. their underlying graph has a minimal coloring. This answers a question by Duval et al. in the negative.

Face numbers of sequentially Cohen–Macaulay complexes and Betti numbers of componentwise linear ideals

A numerical characterization is given of the so-called h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result characterizes the number of faces of various dimensions and codim ...

Improved bounds for the regularity of edge ideals of graphs

Let G be a graph with n vertices, let \(S={\mathbb {K}}[x_1,\dots ,x_n]\) be the polynomial ring in n variables over a field \({\mathbb {K}}\) and let I(G) denote the edge ideal of G. For every collection \({\mathcal {H}}\) of connected graphs with \(K_2\in {\mathcal {H}}\), we introduce the notions of \({{\mathrm{ind-match}}}_{{\mathcal {H}}}(G)\) and \({{\mathrm{min-match}}}_{{\mathcal {H}}}(G)\). It will be proved that the inequalities \({{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\le \mathrm{reg}(S/I(G))\le {{\mathrm{min-match}}}_{\{K_2, C_5\}}(G)\) are true. Moreover, we show that if G is a Cohen–Macaulay graph with girth at least five, then \(\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\). Furthermore, we prove that if G is a paw-free and doubly Cohen–Macaulay graph, then \(\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\) if and only if every connected component of G is either a complete graph or a 5-cycle graph. Among other results, we show that for every doubly Cohen–Macaulay simplicial complex, the equality \(\mathrm{reg}({\mathbb {K}}[\Delta ])=\mathrm{dim}({\mathbb {K}}[\Delta ])\) holds.

Hilbert functions of monomial ideals containing a regular sequence

Let M be an ideal in K[x1,...,x n] (K is a field) generated by products of linear forms and containing a homogeneous regular sequence of some length. We prove that ideals containing M satisfy the Eisenbud–Green–Harris conjecture and moreover prove that the Cohen–Macaulay property is preserved. We conclude that monomial ideals satisfy this conjecture. We obtain that the h-vector of Cohen–Macaulay simplicial complex Δ is the h-vector of Cohen–Macaulay (a 1 - 1,..., a t - 1)-balanced simplicial complex, where t is the height of the Stanley–Reisner ideal of Δ and (a 1,..., a t) is the type of some regular sequence contained in this ideal.

A non-partitionable Cohen–Macaulay simplicial complex

A long-standing conjecture of Stanley states that every Cohen–Macaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.

Recursive Betti numbers for Cohen–Macaulay d-partite clutters arising from posets

A natural extension of bipartite graphs are d-partite clutters, where d≥2 is an integer. For a poset P, Ene, Herzog and Mohammadi introduced the d-partite clutter CP,d of multichains of length d in P, showing that it is Cohen–Macaulay. We prove that the cover ideal of CP,d admits an xi-splitting, determining a recursive formula for its Betti numbers and generalizing a result of Francisco, Hà and Van Tuyl on the cover ideal of Cohen–Macaulay bipartite graphs. Moreover we prove a Betti splitting result for the Alexander dual of a Cohen–Macaulay simplicial complex. Interesting examples are given, in particular the first example of ideal that does not admit Betti splitting in any characteristic.

Uniformly Cohen–Macaulay simplicial complexes and almost Gorenstein* simplicial complexes

In this paper, we study simplicial complexes whose Stanley–Reisner rings are almost Gorenstein and have a-invariant zero. We call such a simplicial complex an almost Gorenstein* simplicial complex. To study the almost Gorenstein* property, we introduce a new class of simplicial complexes which we call uniformly Cohen–Macaulay simplicial complexes. A d-dimensional simplicial complex Δ is said to be uniformly Cohen–Macaulay if it is Cohen–Macaulay and, for any facet F of Δ, the simplicial complex Δ∖{F} is Cohen–Macaulay of dimension d. We investigate fundamental algebraic, combinatorial and topological properties of these simplicial complexes, and show that almost Gorenstein* simplicial complexes must be uniformly Cohen–Macaulay. By using this fact, we show that every almost Gorenstein* simplicial complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for simplicial complexes of dimension ≤2.

Betti Posets and the Stanley Depth

Let S be a polynomial ring and let $$I \subseteq S$$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of I determines the Stanley projective dimension of S / I or I. Our main result is that this conjecture implies the Stanley conjecture for I, and it also implies that $${{\mathrm{sdepth}}}S/I \ge {{\mathrm{depth}}}S/I - 1.$$ Recently, Duval et al. (A non-partitionable Cohen–Macaulay simplicial complex, arXiv:1504.04279 , 2015) found a counterexample to the Stanley conjecture, and their counterexample satisfies $${{\mathrm{sdepth}}}S/I = {{\mathrm{depth}}}S/I - 1$$ . So if our conjecture is true, then the conclusion is best possible.

Partial coloring, vertex decomposability and sequentially Cohen-Macaulay simplicial complexes

In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding ``whiskers'' to graphs. In this paper, we study a similar construction for building a simplicial complex $\Delta_\chi$ from a coloring $\chi$ of a subset of the vertices of $\Delta$ and give necessary and sufficient conditions for this construction to produce vertex decomposable simplicial complexes. We apply this work to strengthen and give new proofs about sequentially Cohen-Macaulay edge ideals of graphs.

The behavior of Stanley depth under polarization

Let K be a field, R=K[X1,…,Xn] be the polynomial ring and J⊊I be two monomial ideals in R. In this paper we show thatsdepthI/J−depthI/J=sdepthIp/Jp−depthIp/Jp, where sdepthI/J denotes the Stanley depth and Ip denotes the polarization. This solves a conjecture by Herzog [9] and reduces the famous Stanley conjecture (for modules of the form I/J) to the squarefree case. As a consequence, the Stanley conjecture for algebras of the form R/I and the well-known combinatorial conjecture that every Cohen–Macaulay simplicial complex is partitionable are equivalent.

On a conjecture by Kalai

We show that monomial ideals generated in degree two satisfy a conjecture by Eisenbud, Green and Harris. In particular, we give a partial answer to a conjecture of Kalai by proving that h-vectors of flag Cohen-Macaulay simplicial complexes are h-vectors of Cohen-Macaulay balanced simplicial complexes.

Geometric two-dimensional duality groups

We consider a finite, aspherical, 2-dimensional Cohen–Macaulay simplicial complex $\Delta$ and we find additional conditions that imply the universal cover $\tilde\Delta$ has one end. In order to find these additional conditions we use a form of “Zeeman Duality”. The context is an attempt to better understand duality groups.

A GENERALIZATION OF THE SWARTZ EQUALITY

AbstractFor a given (d−1)-dimensional simplicial complex Γ, we denote its h-vector by h(Γ)=(h0(Γ),h1(Γ),. . .,hd(Γ)) and set h−1(Γ)=0. The known Swartz equality implies that if Δ is a (d−1)-dimensional Buchsbaum simplicial complex over a field, then for every 0 ≤ i ≤ d, the inequality ihi(Δ)+(d−i+1)hi−1(Δ) ≥ 0 holds true. In this paper, by using these inequalities, we give a simple proof for a result of Terai (N. Terai, On h-vectors of Buchsbaum Stanley–Reisner rings, Hokkaido Math. J. 25(1) (1996), 137–148) on the h-vectors of Buchsbaum simplicial complexes. We then generalize the Swartz equality (E. Swartz, Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes, SIAM J. Discrete Math. 18(3) (2004/05), 647–661), which in turn leads to a generalization of the above-mentioned inequalities for Cohen–Macaulay simplicial complexes in co-dimension t.