The Multiplicity Conjecture for Barycentric Subdivisions

For a simplicial complex Δ we study the effect of barycentric subdivision on ring theoretic invariants of its Stanley–Reisner ring. In particular, for Stanley–Reisner rings of barycentric subdivisions we verify a conjecture by Huneke and Herzog and Srinivasan, that relates the multiplicity of a standard graded k-algebra to the product of the maximal and minimal shifts in its minimal free resolution up to the height. On the way to proving the conjecture, we develop new and list well-known results on behavior of dimension, Hilbert series, multiplicity, local cohomology, depth, and regularity when passing from the Stanley–Reisner ring of Δ to the one of its barycentric subdivision.     

Some degenerations of Kazhdan–Lusztig ideals and multiplicities of Schubert varieties

We study Hilbert–Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Gröbner degenerations of the Kazhdan–Lusztig ideal. In the covexillary case, we give a manifestly positive combinatorial rule for multiplicity by establishing (with a Gröbner basis) a reduced limit whose Stanley–Reisner simplicial complex is homeomorphic to a shellable ball or sphere. We show that multiplicity counts the number of facets of this complex. We also obtain a formula for the Hilbert series of the local ring. In particular, our work gives a multiplicity rule for Grassmannian Schubert varieties, providing alternative statements and proofs to formulae of Lakshmibai and Weyman (1990) [26], Rosenthal and Zelevinsky (2001) [37], Krattenthaler (2001) [22], Kodiyalam and Raghavan (2003) [21], Kreiman and Lakshmibai (2004) [24], Ikeda and Naruse (2009) [13] and Woo and Yong (2009) [40]. We suggest extensions of our methodology to the general case.     

Buchsbaum Stanley–Reisner Rings and Cohen–Macaulay Covers

First, we give a new criterion for Buchsbaum Stanley–Reisner rings to have linear resolutions. Next, we prove that every (d ? 1)-dimensional complex Δ of initial degree d is contained in the same dimensional Cohen–Macaulay complex whose (d ? 1)th reduced homology is isomorphic to that of Δ. We call such a simplicial complex a Cohen–Macaulay cover of Δ. And we also show that all the intermediate complexes between Δ and its Cohen–Macaulay cover are Buchsbaum provided that Δ is Buchsbaum. As an application, we determine the h-vectors of the 3-dimensional Buchsbaum Stanley–Reisner rings with initial degree 3.     

New Classes of Set-theoretic Complete Intersection Monomial Ideals

Let Δ be a simplicial complex and χ be an s-coloring of Δ. Biermann and Van Tuyl have introduced the simplicial complex Δ_χ. As a corollary of Theorems 5 and 7 in their 2013 article, we obtain that the Stanley–Reisner ring of Δ_χ over a field is Cohen–Macaulay. In this note, we generalize this corollary by proving that the Stanley–Reisner ideal of Δ_χ over a field is set-theoretic complete intersection. This also generalizes a result of Macchia.     

Degenerations to unobstructed Fano Stanley–Reisner schemes

We construct degenerations of Mukai varieties and linear sections thereof to special unobstructed Fano Stanley–Reisner schemes corresponding to convex deltahedra. This can be used to find toric degenerations of rank one index one Fano threefolds. Furthermore, we show that the Stanley–Reisner ring of the boundary complex of the dual polytope of the associahedron has trivial \(T^2\) . This can be used to find new toric degenerations of linear sections of \(G(2,n)\) .     

Schubert Patches Degenerate to Subword Complexes

We study the intersections of general Schubert varieties X _w with permuted big cells, and give an inductive degeneration of each such “Schubert patch” to a Stanley–Reisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using the results of Knutson [Kn07] on automatically reduced degenerations, or through more standard cohomology-vanishing arguments. The underlying simplicial complex of the Stanley–Reisner scheme is a subword complex, as introduced for slightly di_erent purposes in Knutson–Miller [KnM05], and is homeomorphic to a ball. This gives a new proof of the Andersen–Jantzen–Soergel/Billey and Graham/Willems formulae for restrictions of equivariant Schubert classes to fixed points.     

The Stanley–Reisner Ideals of Polygons as Set-Theoretic Complete Intersections

We show that the Stanley–Reisner ideal of the one-dimensional simplicial complex whose diagram is an n-gon is always a set-theoretic complete intersection in any positive characteristic.     

Broken circuit complexes and hyperplane arrangements

We study Stanley–Reisner ideals of broken circuit complexes and characterize those ones admitting linear resolutions or being complete intersections. These results will then be used to characterize hyperplane 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 a matroid with a complete intersection broken circuit complex, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik–Solomon algebra.     

Skeletons of monomial ideals

In analogy to the skeletons of a simplicial complex and their Stanley–Reisner ideals we introduce the skeletons of an arbitrary monomial ideal I ? S = K [x₁, …, x_n ]. This allows us to compute the depth of S /I in terms of its skeleton ideals. We apply these techniques to show that Stanley's conjecture on Stanley decompositions of S /I holds provided it holds whenever S /I is Cohen–Macaulay. We also discuss a conjecture of Soleyman Jahan and show that it suffices to prove his conjecture for monomial ideals with linear resolution (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simplicial Girth and Pure Resolutions

Generalizing the notion of the girth of a graph, a sequence of simplicial girths is assigned to each simplicial complex. Given a simplicial girth, lower bounds on higher simplicial girths are proven. When a simplicial girth is given and the Stanley–Reisner ring has a pure resolution, upper bounds on the number of vertices are proven.     

Stanley–Reisner resolution of constant weight linear codes

Given a constant weight linear code, we investigate its weight hierarchy and the Stanley–Reisner resolution of its associated matroid regarded as a simplicial complex. We also exhibit conditions on the higher weights sufficient to conclude that the code is of constant weight.     

Lyubeznik Table of Sequentially Cohen–Macaulay Rings

We prove that sequentially Cohen–Macaulay rings in positive characteristic, as well as sequentially Cohen–Macaulay Stanley–Reisner rings in any characteristic, have trivial Lyubeznik table. Some other configurations of Lyubeznik tables are also provided depending on the deficiency modules of the ring.     

Constructible Ideals

Based on the study of simplicial complexes, one may naturally define the constructible monomial ideals. We connect the square-free constructible ideal with the Stanley–Reisner ideal of the Alexander dual associated to a constructible simplicial complex. We give some properties of constructible ideals, and we compute the Betti numbers. We prove that all monomial ideals with linear quotients are constructible ideals. We also show that all constructible ideals have a linear resolution.     

Codismantlability and projective dimension of the Stanley–Reisner ring of special hypergraphs

In this paper, 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.     

Betti numbers of Stanley–Reisner rings determine hierarchical Markov degrees

There are two seemingly unrelated ideals associated with a simplicial complex Δ: one is the Stanley–Reisner ideal I _Δ , the monomial ideal generated by minimal non-faces of Δ, well-known in combinatorial commutative algebra; the other is the toric ideal I _M(Δ) of the facet subring of Δ, whose generators give a Markov basis for the hierarchical model defined by Δ, playing a prominent role in algebraic statistics. In this note we show that the complexity of the generators of I _M(Δ) is determined by the Betti numbers of I _Δ . The unexpected connection between the syzygies of the Stanley–Reisner ideal and degrees of minimal generators of the toric ideal provide a framework for further exploration of the connection between the model and its many relatives in algebra and combinatorics.     

A Gorenstein simplicial complex for symmetric minors

We show that the ideal generated by the (n - 2) minors of a general symmetric n by n matrix has an initial ideal that is the Stanley–Reisner ideal of the boundary complex of a simplicial polytope and has the same graded Betti numbers.     

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.