Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gorenstein Special Fiber Rings of Ladder Determinantal Modules

Louiza Fouli, Kuei-Nuan Lin, Haydee Lindo, Maral Mostafazadehfard

TL;DR

This work addresses when the special fiber ring $\mathcal{F}(M)$ of a ladder determinantal module is Gorenstein, expressing precise combinatorial conditions in terms of the ladder matrix data. The authors leverage Sagbi degeneration to reduce to a Hibi ring, so Gorensteinness corresponds to the purity of the join-irreducible poset associated to the distributive lattice $\mathcal{L}\times[r]$. They provide explicit necessary and sufficient conditions on the ladder data (via $u_i,v_i,\epsilon_i,\theta_i$) and describe the join-irreducible posets $P_{\mathcal{L}}$ and $P_{\mathcal{L}\times[r]}$, enabling computation of invariants such as regularity, $a$-invariant, and the reduction number. The paper also proves $F$-regularity in the Gorenstein case and offers examples where $\mathcal{F}(L)$ is $F$-regular without being Gorenstein, thereby connecting determinantal, combinatorial, and singularity-theoretic aspects of these algebras.

Abstract

A ladder determinantal module is an arbitrary direct sum of ideals of maximal minors of a generic ladder matrix. In this article, we give necessary and sufficient conditions for the special fiber ring of such modules to be Gorenstein. These conditions are expressed in terms of data obtained from the underlying matrix.

Gorenstein Special Fiber Rings of Ladder Determinantal Modules

TL;DR

This work addresses when the special fiber ring of a ladder determinantal module is Gorenstein, expressing precise combinatorial conditions in terms of the ladder matrix data. The authors leverage Sagbi degeneration to reduce to a Hibi ring, so Gorensteinness corresponds to the purity of the join-irreducible poset associated to the distributive lattice . They provide explicit necessary and sufficient conditions on the ladder data (via ) and describe the join-irreducible posets and , enabling computation of invariants such as regularity, -invariant, and the reduction number. The paper also proves -regularity in the Gorenstein case and offers examples where is -regular without being Gorenstein, thereby connecting determinantal, combinatorial, and singularity-theoretic aspects of these algebras.

Abstract

A ladder determinantal module is an arbitrary direct sum of ideals of maximal minors of a generic ladder matrix. In this article, we give necessary and sufficient conditions for the special fiber ring of such modules to be Gorenstein. These conditions are expressed in terms of data obtained from the underlying matrix.
Paper Structure (4 sections, 17 theorems, 42 equations, 1 figure)

This paper contains 4 sections, 17 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. The join-irreducible poset
  4. The Gorenstein property

Key Result

Theorem 2.2

Let $M$ and $N$ be as in def:Ladder.

Figures (1)

  • Figure 1: The poset $P_\mathcal{L}$ derived directly from matrices $\mathbf{X}_{\mathbb{S}}$ and $\mathbf{X}_{P_\mathcal{L}}$.

Theorems & Definitions (47)

  • Definition 2.1
  • Theorem 2.2: LinShenLadder
  • Remark 2.3
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Corollary 3.6
  • ...and 37 more