Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Resolutions of Pinched Power Ideals

Hoài Đào, Jeff Mermin

TL;DR

This work develops symmetry-respecting, minimal polytopal resolutions for the dth power of the maximal ideal I = (x1,...,xn)^d and for its pinched variant I_hat obtained by deleting a central monomial m. By leveraging the complex-of-boxes, cyclic symmetry, and gluing lemmas, the authors construct explicit resolutions centered at m and then merge the m-containing facet to resolve I_hat, providing exact Betti-number data via a mapping-cone framework that couples with Koszul homology. The results yield concrete polytopal supports, including symmetric resolutions in central cases and explicit Betti formulas, while opening questions about multiple deletions, staircase diagrams, and box-plus resolutions in higher dimensions. Overall, the paper advances a canonical, symmetry-preserving approach to resolving a natural family of monomial ideals and lays groundwork for broader explorations of deletions and combinatorial structures in polynomial ideals.

Abstract

In this paper, we construct resolutions of ideals obtained by removing a small number of generators from the generators of $(x_1,\dots,x_n)^d$.

Resolutions of Pinched Power Ideals

TL;DR

This work develops symmetry-respecting, minimal polytopal resolutions for the dth power of the maximal ideal I = (x1,...,xn)^d and for its pinched variant I_hat obtained by deleting a central monomial m. By leveraging the complex-of-boxes, cyclic symmetry, and gluing lemmas, the authors construct explicit resolutions centered at m and then merge the m-containing facet to resolve I_hat, providing exact Betti-number data via a mapping-cone framework that couples with Koszul homology. The results yield concrete polytopal supports, including symmetric resolutions in central cases and explicit Betti formulas, while opening questions about multiple deletions, staircase diagrams, and box-plus resolutions in higher dimensions. Overall, the paper advances a canonical, symmetry-preserving approach to resolving a natural family of monomial ideals and lays groundwork for broader explorations of deletions and combinatorial structures in polynomial ideals.

Abstract

In this paper, we construct resolutions of ideals obtained by removing a small number of generators from the generators of .
Paper Structure (13 sections, 34 theorems, 50 equations, 24 figures)

This paper contains 13 sections, 34 theorems, 50 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Free resolutions
  4. Cellular resolutions and polyhedral complexes
  5. Borel ideals
  6. The complex-of-boxes resolution
  7. Notation for cyclic intervals
  8. Gluing of complexes
  9. Symmetric polytopal resolutions of $I=(x_{1},\dots, x_{n})^{n}$ and $I\smallsetminus x_{1}\cdots x_{n}$
  10. A cyclic resolution of $(x_1,\cdots,x_n)^d$
  11. A minimal resolution of the ideal obtained by removing $m= x_{1}^{d_{1}}\cdots x_{n}^{d_{n}}$ from the generators of $I= (x_1,\dots,x_n)^d$.
  12. Betti numbers
  13. Further questions

Key Result

Theorem 2.8

Let $X$ be a labelled polytopal complex. The free complex $\mathcal{F}_X$ supported on $X$ is a (polytopal) resolution if and only if $X_{\leq m}$ is acyclic for all $m$. In this case, it is a free resolution of the quotient module $\dfrac{S}{I}$, where $I=(\ell(v) : v\in X \text{ is a vertex})$ is

Figures (24)

  • Figure 1: The polytopal structures of ${\color{red}\Delta_{[a,c]}}\times {\color{blue}\Delta_{[c,d]}}$ and $\Delta_{[a,b]}\times \Delta_{[b,c]}\times \Delta_{[c,d]}$. The vertices are labelled with monomials rather than ordered tuples, using the labelling of Definition \ref{['d:boxlabels']}.
  • Figure 2: The polytopal structures of $\Delta_{[a,c]}\times \Delta_{[c]}\times \Delta_{[c,d]}$ and $\Delta_{[a,b]}\times \Delta_{[a,b]}$.
  • Figure 3: The Eliahou-Kervaire resolution (left) and the complex-of-boxes resolution (right) of $(a,b,c)^{3}$.
  • Figure 4: The complexes supporting the Eliahou-Kervaire resolution and the complex-of-boxes resolution of $(a,b,c,d)^{2}$. The Eliahou-Kervaire complex is not polytopal, since the cell containing $d^2$ contains $ac$ and $bc$ but none of the segment connecting them. The box complex consists of two tetrahedra and two triangular prisms.
  • Figure 5: The four boxes comprising the complex-of-boxes resolution of $(a,b,c,d)^{2}$.
  • ...and 19 more figures

Theorems & Definitions (105)

  • Definition 2.1
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • Remark 2.12
  • ...and 95 more