Stanley-Reisner Ring Research Articles

Cohomology operations for moment-angle complexes and resolutions of Stanley–Reisner rings

A fundamental result in toric topology identifies the cohomology ring of the moment-angle complex Z K \mathcal {Z}_K associated to a simplicial complex K K with the Koszul homology of the Stanley–Reisner ring of K K . By studying cohomology operations induced by the standard torus action on the moment-angle complex, we extend this to a topological interpretation of the minimal free resolution of the Stanley–Reisner ring. The exterior algebra module structure in cohomology induced by the torus action recovers the linear part of the minimal free resolution, and we show that higher cohomology operations induced by the action (in the sense of Goresky–Kottwitz–MacPherson [Invent. Math. 131 (1998), pp. 25–83]) can be assembled into an explicit differential on the resolution. Describing these operations in terms of Hochster’s formula, we recover and extend a result due to Katthän [Mathematics 7 (2019), no. 7, p. 605]. We then apply all of this to study the equivariant formality of torus actions on moment-angle complexes. For these spaces, we obtain complete algebraic and combinatorial characterisations of which subtori of the naturally acting torus act equivariantly formally.

Cohomology of smooth toric varieties: Naturality

Building on the recent computation of the cohomology rings of smooth toric varieties and partial quotients of moment-angle complexes, we investigate the naturality properties of the resulting isomorphism between the cohomology of such a space and the torsion product involving the Stanley–Reisner ring. If 2 is invertible in the chosen coefficient ring, then the isomorphism is natural with respect to toric morphisms, which for partial quotients are defined in analogy with toric varieties. In general there are deformation terms that we describe explicitly.

Algebraic invariants of interval subdivided complexes

We study ring-theoretic properties of the Stanley–Reisner rings of simplicial complexes that undergo the interval subdivisions. In particular, we establish results concerning computational description of dimension, Hilbert series, multiplicity, local cohomology, depth and regularity of the Stanley–Reisner ring of interval subdivided simplicial complex. We also prove results regarding the non-vanishing of graded Betti numbers and bounds on Betti numbers for these Stanley–Reisner rings.

Levelness versus nearly Gorensteinness of homogeneous rings

Levelness and nearly Gorensteinness are well-studied properties of graded rings as a generalized notion of Gorensteinness. In this paper, we compare the strength of these properties. For any Cohen-Macaulay homogeneous affine semigroup ring R, we give a necessary condition for R to be non-Gorenstein and nearly Gorenstein in terms of the h-vector of R and we show that if R is nearly Gorenstein with Cohen-Macaulay type 2, then it is level. We also show that if Cohen-Macaulay type is more than 2, there are 2-dimensional counterexamples. Moreover, we characterize nearly Gorensteinness of Stanley-Reisner rings of low-dimensional simplicial complexes.

The first cotangent cohomology module for matroids

We find a combinatorial formula which computes the first cotangent cohomology module of Stanley–Reisner rings associated to matroids. For arbitrary simplicial complexes we provide upper bounds for the dimensions of the multigraded components of T^1 . For specific degrees we prove that these bounds are reached if and only if the simplicial complex is a matroid, obtaining thus a new characterization for matroids. Furthermore, the graded first cotangent cohomology turns out to be a complete invariant for nondiscrete matroids.

Tight Complexes Are Golod

Abstract Golodness of a simplicial complex is defined algebraically in terms of the Stanley–Reisner ring, and it has been a long-standing problem to find its combinatorial characterization. Tightness of a simplicial complex is a combinatorial analogue of a tight embedding of a manifold into Euclidean space, and has been studied in connection to minimal manifold triangulations. In this paper, by using an idea from toric topology, we prove that tight complexes are Golod, and as a corollary, we obtain that a triangulation of a closed connected orientable manifold is Golod if and only if it is tight, which is a combinatorial characterization of Golodness for manifold triangulations.

On the dimension of dual modules of local cohomology and the Serre's condition for the unmixed Stanley–Reisner ideals of small height

In this paper, we focus on the dimension of dual modules of local cohomology of Stanley–Reisner rings to obtain a new vector. This vector contains important information on the Serre's condition (Sr) and the CMt property as well as the depth of Stanley–Reisner rings. We prove some results in this regard including lower bounds for the depth of Stanley–Reisner rings. Further, we give a characterization of (d−1)-dimensional simplicial complexes with codimension two which are (Sd−3) but they are not Cohen–Macaulay. By using this characterization, we obtain a condition to equality of projective dimension of the Stanley–Reisner rings and the arithmetical rank of their Stanley–Reisner ideals. Moreover, our characterization allows us to compute the h-vectors and give a negative answer to a known question regarding these vectors.

Product decompositions of moment-angle manifolds and B-rigidity

AbstractA simple polytope P is called B-rigid if its combinatorial type is determined by the cohomology ring of the moment-angle manifold $\mathcal {Z}_P$ over P. We show that any tensor product decomposition of this cohomology ring is geometrically realized by a product decomposition of the moment-angle manifold up to equivariant diffeomorphism. As an application, we find that B-rigid polytopes are closed under products, generalizing some recent results in the toric topology literature. Algebraically, our proof establishes that the Koszul homology of a Gorenstein Stanley–Reisner ring admits a nontrivial tensor product decomposition if and only if the underlying simplicial complex decomposes as a join of full subcomplexes.

The Cartier core map for Cartier algebras

Let R be a commutative Noetherian F-finite ring of prime characteristic and let D be a Cartier algebra. We define a self-map on the Frobenius split locus of the pair (R,D) by sending a point P to the splitting prime of (RP,DP). We prove this map is continuous, containment preserving, and fixes the D-compatible ideals. We show this map can be extended to arbitrary ideals J, where in the Frobenius split case it gives the largest D-compatible ideal contained in J. Finally, we apply Glassbrenner's criterion to prove that the prime uniformly F-compatible ideals of a Stanley-Reisner rings are the sums of its minimal primes.

Algebraic h-vectors of simplicial complexes through local cohomology, part 1

Given an infinite field k and a simplicial complex Δ, a common theme in studying the f- and h-vectors of Δ has been the consideration of the Hilbert series of the Stanley–Reisner ring k[Δ] modulo a generic linear system of parameters Θ. Historically, these computations have been restricted to special classes of complexes (most typically triangulations of spheres or manifolds). We provide a compact topological expression of hd−1a(Δ), the dimension over k in degree d−1 of k[Δ]/(Θ), for any complex Δ of dimension d−1. In the process, we provide tools and techniques for the possible extension to other coefficients in the Hilbert series.

Arnold’s Piecewise Linear Filtrations, Analogues of Stanley–Reisner Rings and Simplicial Newton Polyhedra

In 1974, the author proved that the codimension of the ideal (g1,g2,…,gd) generated in the group algebra K[Zd] over a field K of characteristic 0 by generic Laurent polynomials having the same Newton polytope Γ is equal to d!×Volume(Γ). Assuming that Newtons polytope is simplicial and super-convenient (that is, containing some neighborhood of the origin), the author strengthens the 1974 result by explicitly specifying the set Bsh of monomials of cardinality d!×Volume(Γ), whose equivalence classes form a basis of the quotient algebra K[Zd]/(g1,g2,…,gd). The set Bsh is constructed inductively from any shelling sh of the polytope Γ. Using the Bsh structure, we prove that the associated graded K -algebra grΓ(K[Zd]) constructed from the Arnold–Newton filtration of K -algebra K[Zd] has the Cohen–Macaulay property. This proof is a generalization of B. Kind and P. Kleinschmitt’s 1979 proof that Stanley–Reisner rings of simplicial complexes admitting shelling are Cohen–Macaulay. Finally, we prove that for generic Laurent polynomials (f1,f2,…,fd) with the same Newton polytope Γ, the set Bsh defines a monomial basis of the quotient algebra K[Zd]/(g1,g2,…,gd).

Solution to a conjecture on edge rings with 2-linear resolutions

ABSTRACT For a graph the edge ring is , where and I(G) is generated by . The conjecture we treat is the following. Conjecture 1. If has a 2-linear resolution, then the projective dimension of , pd , equals the maximal degree of a vertex in G. As far as we know, this conjecture is first mentioned in a paper by Gitler and Valencia [7, Conjecture 4.13], and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq [1]. That there are counterexamples was noted already by Moradi and Kiani [9]. By interpreting as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds.

Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem

A result of G. Walker and R. Wood states that the space of indecomposable elements in degree 2 n -1-n of the polynomial algebra 𝔽 2 [x 1 ,...,x n ], considered as a module over the mod 2 Steenrod algebra, is isomorphic to the Steinberg representation of GL n (𝔽 2 ). We generalize this result to all finite fields by studying a family of finite quotient rings R n,k , k∈ℕ * , of 𝔽 q [x 1 ,...,x n ], where each R n,k is defined as a quotient of the Stanley–Reisner ring of a matroid complex. By considering a variant of R n,k , we also show that the space of indecomposable elements of 𝔽 q [x 1 ,...,x n ] in degree q n-1 -n has dimension equal to that of a complex cuspidal representation of GL n (𝔽 q ), that is (q-1)(q 2 -1)⋯(q n-1 -1).

Frobenius methods in combinatorics

We survey results produced from the interaction between methods in prime characteristic and combinatorial commutative algebra. We showcase results for edge ideals, toric varieties, Stanley-Reisner rings, and initial ideals that were proven via Frobenius. We also discuss results for monomial ideals obtained using Frobenius-like maps. Finally, we present results for F-pure rings that were inspired by work done for Stanley-Reisner rings.

Polyhedral products over finite posets

Polyhedral products were defined by Bahri, Bendersky, Cohen, and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper, we give a very general homotopy theoretic construction of polyhedral products over arbitrary pointed posets. We show that under certain restrictions on the poset P, which include all known cases, the cohomology of the resulting spaces can be computed as an inverse limit over P of the cohomology of the building blocks. This motivates the definition of an analogous algebraic construction—the polyhedral tensor product. We show that for a large family of posets, the cohomology of the polyhedral product is given by the polyhedral tensor product. We then restrict attention to polyhedral posets, a family of posets that includes face posets of simplicial complexes, and simplicial posets, as well as many others. We define the Stanley–Reisner ring of a polyhedral poset and show that, as in the classical cases, these rings occur as the cohomology of certain polyhedral products over the poset in question. For any pointed poset P, we construct a simplicial poset s(P), and show that if P is a polyhedral poset, then polyhedral products over P coincide up to homotopy with the corresponding polyhedral products over s(P).

On the topological structure of the (non-)finitely generated locus of Frobenius algebras emerging from Stanley–Reisner rings

Let U be the set of prime ideals P of the completion of a Stanley–Reisner ring S, such that the localization at P of the Frobenius algebra of the injective hull of the residue field of S is a finitely generated algebra. We give a partial answer to a conjecture made by M. Katzman about the openness of U. Specifically, we show that U has non-empty interior and we present some sufficient conditions for a principal open set D(f) to be contained in U, and for intersections of closed and principal opens to be contained in the complement of U.

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 .

Betti numbers of fat forests and their Alexander dual

Let k be a field and R=k[x_1,ldots ,x_n]/I=S/I a graded ring. Then R has a t-linear resolution if I is generated by homogeneous elements of degree t, and all higher syzygies are linear. Thus, R has a t-linear resolution if mathrm{Tor}^S_{i,j}(S/I,k)=0 if jne i+t-1. For a graph G on {1,ldots ,n}, the edge algebra is k[x_1,ldots ,x_n]/I, where I is generated by those x_ix_j for which { i,j} is an edge in G. We want to determine the Betti numbers of edge rings with 2-linear resolution. But we want to do that by looking at the edge ring as a Stanley–Reisner ring. For a simplicial complex Delta on [mathbf{n}]={1,ldots ,n} and a field k, the Stanley–Reisner ring k[Delta ] is k[x_1,ldots ,x_n]/I, where I is generated by the squarefree monomials x_{i_1}ldots x_{i_k} for which { i_1,ldots ,i_k} does not belong to Delta . Which Stanley–Reisner rings that are edge rings with 2-linear resolution are known. Their associated complexes has had different names in the literature. We call them fat forests here. We determine the Betti numbers of many fat forests and compare our result with what is known. We also consider Betti numbers of Alexander duals of fat forests.

Deriving some properties of Stanley-Reisner rings from their squarefree zero-divisor graphs

Let ΔΔ be a simplicial complex, IΔIΔ its Stanley-Reisner ideal and R=K[Δ]R=K[Δ] its Stanley-Reisner ring over a field KK. In 2018, the author introduced the squarefree zero-divisor graph of RR, denoted by Γsf(R)Γsf(R), and proved that if ΔΔ and Δ′Δ′ are two simplicial complexes, then the graphs Γsf(K[Δ])Γsf(K[Δ]) and Γsf(K[Δ′])Γsf(K[Δ′]) are isomorphic if and only if the rings K[Δ]K[Δ] and K[Δ′]K[Δ′] are isomorphic. Here we derive some algebraic properties of RR using combinatorial properties of Γsf(R)Γsf(R). In particular, we state combinatorial conditions on Γsf(R)Γsf(R) which are necessary or sufficient for RR to be Cohen-Macaulay. Moreover, we investigate when Γsf(R)Γsf(R) is in some well-known classes of graphs and show that in these cases, IΔIΔ has a linear resolution or is componentwise linear. Also we study the diameter and girth of Γsf(R)Γsf(R) and their algebraic interpretations.

Graded Linearity of Stanley–Reisner Ring of Broken Circuit Complexes

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[x1, …, xn]. 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 Stanley–Reisner ring of broken circuit complexes and generalize the results of Van Le and Römer, related to the decomposition of matroids into the direct sum of uniform matroids. Specifically, we show that the matroid can be stratified such that each stratum has a decomposition into uniform matroids. We also present analogs of our results for the Orlik–Terao ideal of hyperplane arrangements which are translations of the corresponding results on matroids.