Determinantal facet ideals

Advances in Mathematics, 1992

Communications in Algebra, 2000

ABSTRACT We give a Gröbner basis for the ideal of 2-minors of a 2 × n utiatrix of linear forms. The minimal free resolution of such an ideal is obtained in [4] when the corresponding Kronecker-Weierstrass normal form has no iiilpotent blocks. For the general case, using this result, the Grobner basis and the Eliahou-Kervaire resolution for stable monomial ideals, we obtain a free resolution with the expected regularity. For a specialization of the defining ideal of ordinary pinch points, as a special case of these ideals, we provide a minimal free resolution explicitly in terms of certain Koszul complex.

Bulletin of the Iranian Mathematical Society, 2018

For a simplicial complex ∆, the affect of the expansion functor on combinatorial properties of ∆ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal I(∆) and its Alexander dual which we denote by J∆ to see how the expansion functor alter the algebraic properties of these ideals. It is shown that for any expansion ∆ α the ideals J∆ and J∆α have the same total Betti numbers and their Cohen-Macaulayness are equivalent, which implies that the regularities of the ideals I(∆) and I(∆ α) are equal. Moreover, the projective dimensions of I(∆) and I(∆ α) are compared. In the sequel for a graph G, some properties that are equivalent in G and its expansions are presented and for a Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable) graph G, we give some conditions for adding or removing a vertex from G, so that the remaining graph is still Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable).

2004

Let ∆ be an abstract finite simplicial complex with vertices X1, . . . , Xn. Let k be a field throughout this chapter. Let R denote the polynomial ring k[X1, X2, . . . , Xn], where, by abuse of notation, we regard the vertices X1, X2, . . . , Xn as indeterminates over k. Let I∆ be the ideal of R generated by the monomials Xi1 . . . Xir , i1 < i2 < . . . < ir such that {Xi1 , . . . , Xir} is not a face of ∆. The face ring of ∆ is the quotient ring k[∆] := R/I∆. Since I∆ is a homogeneous ideal, k[∆] is a graded ring. In this section we will prove Stanley’s formula for the Hilbert series of k[∆]. In some sense, this formula opened up the connection of Commutative Algebra with Combinatorics. We will exhibit the power of Hilbert series methods by giving an elementary proof of Dehn-Sommerville equations towards the end of this section. We begin by establishing the primary decomposition of I∆. (1.1) Definition. Let F be a face of a simplicial complex ∆. Let PF denote the prime ide...

Algebra, Geometry and Their Interactions, 2007

We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex ∆. This correspondence gives us a new method, distinct from the Stanley-Reisner correspondence, to associate to a square-free monomial ideal a simplicial complex. In this context, the monomial ideal is called the facet ideal of ∆. Of particular interest is the case that all the facets have dimension one. Here, the simplicial complex is a simple graph G, and the facet ideal is usually called the edge ideal of G. Many people have been interested in understanding how the combinatorial data or structure of ∆ appears in or affects the minimal graded free resolution of the associated facet ideal. In the first part of this paper, we describe the current state-of-the-art with respect to this program by collecting together many of the relevant results. We sketch the main details of many of the proofs and provide pointers to the relevant literature for the remainder. In the second part we introduce some open questions which will hopefully inspire future research on this topic.

Contemporary Mathematics, 2003

arXiv: Commutative Algebra, 2020

Using SAGBI basis techniques, we find Gr\"obner bases for the presentation ideals of the Rees algebra and special fiber ring of a closed determinantal facet ideal. In particular, we show that closed determinantal facet ideals are of fiber type and their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains, and have rational singularities.

MATHEMATICA SCANDINAVICA, 2019

We show that the ideal generated by maximal minors (i.e., $k+1$-minors) of a $(k+1) \times n$ Vandermonde matrix is radical and Cohen-Macaulay. Note that this ideal is generated by all Specht polynomials with shape $(n-k,1, …,1)$.

Communications in Algebra, 2002

ABSTRACT Let be a pluri-circulant matrix, a concatenation of circulant matrices with entries in a commutative ring. Let be the submatrix of the first rows of We provide some basic “determinantal identities” and prove that every -minor of is a -linear combination of the “basic” -minors. When is generic, we show that the set of basic -minors of and the set of “weakly ordered” maximal minors of both form minimal generating sets for the ideal of -minors of . This implies that the quotient ring by the ideal of -minors of a generic circulant matrix is Cohen-Macaulay. When has two blocks, we show that the weakly ordered maximal minors of and the weakly ordered maximal minors of form minimal Gröbner bases for the ideals of maximal and sub-maximal minors of , respectively. The motivation for studying this class of determinantal ideals comes from the study of generic multiple points.

arXiv (Cornell University), 2015

Let ∆ be a simplicial complex. We study the expansions of ∆ mainly to see how the algebraic and combinatorial properties of ∆ and its expansions are related to each other. It is shown that ∆ is Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum or k-decomposable, if and only if an arbitrary expansion of ∆ has the same property. Moreover, some homological invariants like the regularity and the projective dimension of the Stanley-Reisner ideals of ∆ and those of their expansions are compared.