Stanley Reisner Ideals Research Articles

Gröbner bases, symmetric matrices, and type C Kazhdan–Lusztig varieties

AbstractWe study a class of combinatorially defined polynomial ideals that are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan–Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme‐theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan–Lusztig ideals that arise are exactly those where the opposite cell is 123‐avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley–Reisner ideals of subword complexes), and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams.

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.

Regularity of powers of Stanley–Reisner ideals of one‐dimensional simplicial complexes

AbstractLet Δ be a one‐dimensional simplicial complex. Let be the Stanley–Reisner ideal of Δ. We prove that for all and all intermediate ideals J generated by and some minimal generators of , we have

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.

Triangulations of polygons and stacked simplicial complexes: separating their Stanley–Reisner ideals

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.

Licci level Stanley-Reisner ideals with height three

Licci level Stanley-Reisner ideals with height three

Integral closure of powers of edge ideals and their regularity

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+2sif chark=24+2sif chark≠2, for all s≥1.

Gröbner geometry for skew-symmetric matrix Schubert varieties

Matrix Schubert varieties are the closures of the orbits of B×B acting on all n×n matrices, where B is the group of invertible lower triangular matrices. Extending work of Fulton, Knutson and Miller identified a Gröbner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes, and derived a related primary decomposition in terms of reduced pipe dreams. These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many other applications. We define skew-symmetric matrix Schubert varieties to be the nonempty intersections of matrix Schubert varieties with the subspace of skew-symmetric matrices. In analogy with Knutson and Miller's work, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gröbner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams. We show that these initial ideals are likewise the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials. Our methods differ from Knutson and Miller's and can be used to give new proofs of some of their results, as we explain at the end of this article.

Chordality, d-collapsibility, and componentwise linear ideals

Using the concept of d-collapsibility from combinatorial topology, we define chordal simplicial complexes and show that their Stanley-Reisner ideals are componentwise linear. Our construction is inspired by and an extension of “chordal clutters” which was defined by Bigdeli, Yazdan Pour and Zaare-Nahandi in 2017, and characterizes Betti tables of all ideals with a linear resolution in a polynomial ring.We show d-collapsible and d-representable complexes produce componentwise linear ideals for appropriate d. Along the way, we prove that there are generators that when added to the ideal, do not change Betti numbers in certain degrees.We then show that large classes of componentwise linear ideals, such as Gotzmann ideals and square-free stable ideals have chordal Stanley-Reisner complexes, that Alexander duals of vertex decomposable complexes are chordal, and conclude that the Betti table of every componentwise linear ideal is identical to that of the Stanley-Reisner ideal of a chordal complex.

A note on f-graphs

The notion of [Formula: see text]-ideal was introduced in [G. Q. Abbasi, S. Ahmad, I. Anwar and W. A. Baig, [Formula: see text]-Ideals of degree 2, Algebra Colloq. 19(1) (2012) 921–926] in 2012 and has been studied in many papers after that. In this paper, we have studied those graphs whose Stanley–Reisner ideals turn out to be [Formula: see text]-ideals. We give a characterization and construction of these graphs and show that, unlike [Formula: see text]-graphs, these graphs are always connected. We have also discussed when these graphs are complete bipartite graphs. Moreover, we classify those graphs for which both the facet ideal (the edge ideal) and the Stanley–Resiner ideal, are [Formula: see text]-ideals.

Level property of ordinary and symbolic powers of Stanley-Reisner ideals

In this paper, we prove that the t-th ordinary and/or symbolic power of a Stanley-Reisner ideal is level for some positive integer t≥3 if and only if IΔ is a complete intersection and equi-generated. For t=2, we give a characterization of level property of the second symbolic power IΔ(2) when Δ is a matroid complex of dimension one.

Random flag complexes and asymptotic syzygies

We use the probabilistic method to construct examples of conjectured phenomenon about asymptotic syzygies. In particular, we use the Stanley-Reisner ideals of random flag complexes to construct new examples of Ein and Lazarsfeld's nonvanishing for asymptotic syzygies and of Ein, Erman, and Lazarsfeld's conjectural on the asymptotic normal distribution of Betti numbers.

Matroid configurations and symbolic powers of their ideals

Star configurations are certain unions of linear subspaces of projective space that have been studied extensively. We develop a framework for studying a substantial generalization, which we call matroid configurations, whose ideals generalize Stanley-Reisner ideals of matroids. Such a matroid configuration is a union of complete intersections of a fixed codimension. Relating these to the Stanley-Reisner ideals of matroids and using methods of liaison theory allows us, in particular, to describe the Hilbert function and minimal generators of the ideal of, what we call, a hypersurface configuration. We also establish that the symbolic powers of the ideal of any matroid configuration are Cohen-Macaulay. As applications, we study ideals coming from certain complete hypergraphs and ideals derived from tetrahedral curves. We also consider Waldschmidt constants and resurgences. In particular, we determine the resurgence of any star configuration and many hypersurface configurations. Previously, the only non-trivial cases for which the resurgence was known were certain monomial ideals and ideals of finite sets of points. Finally, we point out a connection to secant varieties of varieties of reducible forms.

Stability of depths of symbolic powers of Stanley–Reisner ideals

We give a bound for n0(I) from which the depth function depthR/I(n) of the quotient ring by symbolic powers of a squarefree monomial ideal I stabilizes. In the unmixed codimension two case we show that depthR/I(n) is a non-increasing function of n and use this property to provide a sharp bound for n0(I).

Waldschmidt constants for Stanley–Reisner ideals of a class of simplicial complexes

We study the symbolic powers of the Stanley–Reisner ideal [Formula: see text] of a bipyramid [Formula: see text] over a [Formula: see text]-gon [Formula: see text]. Using a combinatorial approach, based on analysis of subtrees in [Formula: see text] we compute the Waldschmidt constant of [Formula: see text].

On the Stanley–Reisner ideal of an expanded simplicial complex

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.

A generalization of weight polynomials to matroids

Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid M. Our main result is that these polynomials are determined by Betti numbers associated with N0-graded minimal free resolutions of the Stanley–Reisner ideals of M and so-called elongations of M. Generalizing Greene’s theorem from coding theory, we show that the enumerator of a matroid is equivalent to its Tutte polynomial.

On the k-Buchsbaum property of powers of Stanley–Reisner ideals

AbstractLetS=K[x1,x2,…,xn] be a polynomial ring over a fieldK. Let Δ be a simplicial complex whose vertex set is contained in {1, 2,…,n}. For an integerk≥ 0, we investigate thek-Buchsbaum property of residue class ringsS/I(t); andS/Itfor the Stanley-Reisner idealI=IΔ. We characterize thek-Buchsbaumness of such rings in terms of the simplicial complex Δ and the powert. We also give a characterization in the case whereIis the edge ideal of a simple graph.

Dependence of Betti Numbers on Characteristic

We study the dependence of graded Betti numbers of monomial ideals on the characteristic of the base field. The examples we describe include bipartite ideals and Stanley–Reisner ideals of vertex-decomposable complexes. We give a description of bipartite graphs and, using discrete Morse theory, provide a way of looking at the homology of arbitrary simplicial complexes through bipartite ideals.

Broken circuit complexes and hyperplane arrangements

We study Stanley-Reisner ideals of broken circuits complexes and characterize those ones admitting a linear resolution or being complete intersections. These results will then be used to characterize arrangements whose Orlik-Terao ideal has the same properties. As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph. We also show that for an ordered matroid with disjoint minimal broken circuits, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.