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A level initial ideal of the 2-minors determinantal ideal

Francesco Bisio

TL;DR

The paper analyzes the determinantal ideal $I_2$ of $2$-minors of a generic $m\times n$ matrix and constructs a Gröbner deformation by selecting a degrevlex order so that the initial ideal $I=\mathrm{in}_<(I_2)$ is square-free. The quotient $S/I$ acquires a Stanley–Reisner interpretation via a complex $\Delta$, which is a quasimanifold; $\Delta$ is a sphere when $m=n$ and a ball when $m<n$, and the paper proves that $S/I$ is level by applying Gräbe's canonical-module description, showing $\omega_{S/I}$ is generated in degree $n-m$ and that $a(S/I)=-n$. The Betti tables of $I_2$ versus its initial ideals are compared, and it is shown that, for $m \ge 3$, $n \ge 4$, there is no initial ideal sharing the same Betti table as $I_2$ (in characteristic $0$, with extensions to other characteristics); this highlights limits of initial ideals for Betti geometry. Finally, the complex $\Delta$ is shown to be shellable in the rectangular case $m<n$, via a constructive two-way shelling order on facets, strengthening the link between determinantal algebras and combinatorial topology.

Abstract

For $\Bbbk$ a field, let $X$ a $m \times n$ matrix of variables and $S=\Bbbk[X].$ We consider the determinantal ideal $I_2 \subseteq S$ generated by the $2$-minors of $X.$ In this paper we find a suitable monomial order over $S$ such that $I$, the initial ideal of $I_2$ with respect to that order, is level, namely, it is Cohen-Macaulay and the socle of an Artinian reduction of the $\mathbb{N}$-graded algebra $S/I$ is concentrated in only one degree. Moreover, we compare the Betti tables of $I_2$ with the tables of its initial ideals. In the last section, we prove the shellability of the simplicial complex naturally associated to $S/I$ in the case $m<n.$

A level initial ideal of the 2-minors determinantal ideal

TL;DR

The paper analyzes the determinantal ideal of -minors of a generic matrix and constructs a Gröbner deformation by selecting a degrevlex order so that the initial ideal is square-free. The quotient acquires a Stanley–Reisner interpretation via a complex , which is a quasimanifold; is a sphere when and a ball when , and the paper proves that is level by applying Gräbe's canonical-module description, showing is generated in degree and that . The Betti tables of versus its initial ideals are compared, and it is shown that, for , , there is no initial ideal sharing the same Betti table as (in characteristic , with extensions to other characteristics); this highlights limits of initial ideals for Betti geometry. Finally, the complex is shown to be shellable in the rectangular case , via a constructive two-way shelling order on facets, strengthening the link between determinantal algebras and combinatorial topology.

Abstract

For a field, let a matrix of variables and We consider the determinantal ideal generated by the -minors of In this paper we find a suitable monomial order over such that , the initial ideal of with respect to that order, is level, namely, it is Cohen-Macaulay and the socle of an Artinian reduction of the -graded algebra is concentrated in only one degree. Moreover, we compare the Betti tables of with the tables of its initial ideals. In the last section, we prove the shellability of the simplicial complex naturally associated to in the case

Paper Structure

This paper contains 4 sections, 9 theorems, 26 equations, 20 figures.

Table of Contents

  1. Introduction
  2. A Gröbner deformation of the determinantal ring
  3. The canonical module and the levelness
  4. Shellability

Key Result

Theorem 2.9

Let $\Gamma$ be a simplicial complex over a field $\Bbbk.$$\Gamma$ is Cohen-Macaulay if and only if $\widetilde{H}_i(\textup{lk}_\Gamma(\sigma);\Bbbk)=0,$ for all $\sigma \in \Gamma$ and all $i \neq \textup{dim}(\textup{lk}_\Gamma(\sigma)).$

Figures (20)

  • Figure 1: A simplicial ball which is not level. As we will see, the minimal generators of the canonical module correspond to the minimal faces that are not contained in the boundary of the ball
  • Figure 2: Case $m=5,$$n=7;$$r^F=\{2,3,5\},$$c^F = \{1,4,6,7\}$ and $F=\{(2,1),(3,1),(3,4),(3,6),(5,6),(5,7)\}.$
  • Figure 3: $v$ is a right (yellow) or left (green) turning vertex of $F.$
  • Figure 4: $v$ is a vertical vertex of $F.$
  • Figure 5: Case $m=5,$$n=9;$$\sigma = \{(2,6),(2,7),(4,8),(5,9)\}$ is an example of an element of C.
  • ...and 15 more figures

Theorems & Definitions (33)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.2
  • Definition 2.3
  • Example 2.4: $m=3, n=4$
  • Remark 2.5
  • Definition 2.8
  • Theorem 2.9: Reisner's criterion
  • Proposition 2.10
  • proof
  • ...and 23 more